{"id":154,"date":"2019-08-20T17:03:25","date_gmt":"2019-08-20T21:03:25","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/polar-coordinates\/"},"modified":"2022-06-01T10:39:34","modified_gmt":"2022-06-01T14:39:34","slug":"polar-coordinates","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/polar-coordinates\/","title":{"raw":"Polar Coordinates","rendered":"Polar Coordinates"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Plot points using polar coordinates.<\/li>\n \t<li>Convert from polar coordinates to rectangular coordinates.<\/li>\n \t<li>Convert from rectangular coordinates to polar coordinates.<\/li>\n \t<li>Transform equations between polar and rectangular forms.<\/li>\n \t<li>Identify and graph polar equations by converting to rectangular equations.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137411387\">Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind (see <a class=\"autogenerated-content\" href=\"#Figure_08_03_001\">(Figure)<\/a>). How can the sailor indicate his location to the Coast Guard? In this section, we will investigate a method of representing location that is different from a standard coordinate grid.<\/p>\n\n<div id=\"Figure_08_03_001\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150014\/CNX_Precalc_Figure_08_03_001.jpg\" alt=\"An illustration of a boat on the polar grid.\" width=\"487\" height=\"402\"> <strong>Figure 1.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165135551681\" class=\"bc-section section\">\n<h3>Plotting Points Using Polar Coordinates<\/h3>\n<p id=\"fs-id1165135666727\">When we think about plotting points in the plane, we usually think of <span class=\"no-emphasis\">rectangular coordinates<\/span>[latex]\\,\\left(x,y\\right)\\,[\/latex]in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates, which are points labeled[latex]\\,\\left(r,\\theta \\right)\\,[\/latex]and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.<\/p>\n<p id=\"fs-id1165134166599\">The <span class=\"no-emphasis\">polar grid<\/span> is scaled as the unit circle with the positive <em>x-<\/em>axis now viewed as the polar axis and the origin as the pole. The first coordinate[latex]\\,r\\,[\/latex]is the radius or length of the directed line segment from the pole. The angle[latex]\\,\\theta ,[\/latex] measured in radians, indicates the direction of[latex]\\,r.\\,[\/latex]We move counterclockwise from the polar axis by an angle of[latex]\\,\\theta ,[\/latex]and measure a directed line segment the length of[latex]\\,r\\,[\/latex]in the direction of[latex]\\,\\theta .\\,[\/latex]Even though we measure[latex]\\,\\theta \\,[\/latex]first and then[latex]\\,r,[\/latex] the polar point is written with the <em>r<\/em>-coordinate first. For example, to plot the point[latex]\\,\\left(2,\\frac{\\pi }{4}\\right),[\/latex]we would move[latex]\\,\\frac{\\pi }{4}\\,[\/latex]units in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in <a class=\"autogenerated-content\" href=\"#Figure_08_03_002\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_03_002\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150040\/CNX_Precalc_Figure_08_03_002.jpg\" alt=\"Polar grid with point (2, pi\/4) plotted.\" width=\"487\" height=\"398\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"Example_08_03_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137735842\">\n<div id=\"fs-id1165133141305\">\n<h3>Plotting a Point on the Polar Grid<\/h3>\n<p id=\"fs-id1165135169219\">Plot the point[latex]\\,\\left(3,\\frac{\\pi }{2}\\right)\\,[\/latex]on the polar grid.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137929939\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137929939\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137929939\"]\n<p id=\"fs-id1165137655995\">The angle[latex]\\,\\frac{\\pi }{2}\\,[\/latex]is found by sweeping in a counterclockwise direction 90\u00b0 from the polar axis. The point is located at a length of 3 units from the pole in the[latex]\\,\\frac{\\pi }{2}\\,[\/latex]direction, as shown in <a class=\"autogenerated-content\" href=\"#Figure_08_03_003\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_03_003\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150042\/CNX_Precalc_Figure_08_03_003.jpg\" alt=\"Polar grid with point (3, pi\/2) plotted.\" width=\"487\" height=\"369\"> <strong>Figure 3.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137704675\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_03_01\">\n<div id=\"fs-id1165134225866\">\n\nPlot the point[latex]\\,\\left(2,\\,\\frac{\\pi }{3}\\right)\\,[\/latex]in the <span class=\"no-emphasis\">polar grid<\/span>.\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"52548\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"52548\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150052\/CNX_Precalc_Figure_08_03_004.jpg\" alt=\"Polar grid with point (2, pi\/3) plotted.\">[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137812135\">\n<div>\n<h3>Plotting a Point in the Polar Coordinate System with a Negative Component<\/h3>\n<p id=\"fs-id1165135663316\">Plot the point[latex]\\,\\left(-2,\\,\\frac{\\pi }{6}\\right)\\,[\/latex]on the polar grid.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137634114\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137634114\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137634114\"]\n<p id=\"fs-id1165135386384\">We know that[latex]\\,\\frac{\\pi }{6}\\,[\/latex]is located in the first quadrant. However,[latex]\\,r=-2.\\,[\/latex]We can approach plotting a point with a negative[latex]\\,r\\,[\/latex]in two ways:<\/p>\n\n<ol id=\"eip-id3158355\" type=\"1\">\n \t<li>Plot the point[latex]\\,\\left(2,\\frac{\\pi }{6}\\right)\\,[\/latex]by moving[latex]\\,\\frac{\\pi }{6}\\,[\/latex]in the counterclockwise direction and extending a directed line segment 2 units into the first quadrant. Then retrace the directed line segment back through the pole, and continue 2 units into the third quadrant;<\/li>\n \t<li>Move[latex]\\,\\frac{\\pi }{6}\\,[\/latex]in the counterclockwise direction, and draw the directed line segment from the pole 2 units in the negative direction, into the third quadrant.<\/li>\n<\/ol>\n<p id=\"eip-id3772311\">See <a class=\"autogenerated-content\" href=\"#Figure_08_03_005\">(Figure)<\/a>(a). Compare this to the graph of the polar coordinate[latex]\\,\\left(2,\\frac{\\pi }{6}\\right)\\,[\/latex]shown in <a class=\"autogenerated-content\" href=\"#Figure_08_03_005\">(Figure)<\/a>(b).<\/p>\n\n<div id=\"Figure_08_03_005\" class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150055\/CNX_Precalc_Figure_08_03_005.jpg\" alt=\"Two polar grids. Points (2, pi\/6) and (-2, pi\/6) are plotted. They are reflections across the origin in Q1 and Q3.\" width=\"731\" height=\"403\"> <strong>Figure 4.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137723371\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_03_02\">\n<div id=\"fs-id1165137544173\">\n<p id=\"fs-id1165137544174\">Plot the points[latex]\\,\\left(3,-\\frac{\\pi }{6}\\right)[\/latex]and[latex]\\,\\left(2,\\frac{9\\pi }{4}\\right)\\,[\/latex]on the same polar grid.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137444539\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137444539\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137444539\"]<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150108\/CNX_Precalc_Figure_08_03_006.jpg\" alt=\"Points (2, 9pi\/4) and (3, -pi\/6) are plotted in the polar grid.\">[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134149808\" class=\"bc-section section\">\n<h3>Converting from Polar Coordinates to Rectangular Coordinates<\/h3>\n<p id=\"fs-id1165135313604\">When given a set of <span class=\"no-emphasis\">polar coordinates<\/span>, we may need to convert them to <span class=\"no-emphasis\">rectangular coordinates<\/span>. To do so, we can recall the relationships that exist among the variables[latex]\\,x,\\,y,\\,r,\\,[\/latex]and[latex]\\,\\theta .[\/latex]<\/p>\n\n<div id=\"fs-id1165135315618\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ \\mathrm{cos}\\,\\theta =\\frac{x}{r}\\to x=r\\mathrm{cos}\\,\\theta \\end{array}\\hfill \\\\ \\mathrm{sin}\\,\\theta =\\frac{y}{r}\\to y=r\\mathrm{sin}\\,\\theta \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137410071\">Dropping a perpendicular from the point in the plane to the <em>x-<\/em>axis forms a right triangle, as illustrated in <a class=\"autogenerated-content\" href=\"#Figure_08_03_007\">(Figure)<\/a>. An easy way to remember the equations above is to think of[latex]\\,\\mathrm{cos}\\,\\theta \\,[\/latex]as the adjacent side over the hypotenuse and[latex]\\,\\mathrm{sin}\\,\\theta \\,[\/latex]as the opposite side over the hypotenuse.<\/p>\n\n<div id=\"Figure_08_03_007\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150111\/CNX_Precalc_Figure_08_03_007.jpg\" alt=\"Comparison between polar coordinates and rectangular coordinates. There is a right triangle plotted on the x,y axis. The sides are a horizontal line on the x-axis of length x, a vertical line extending from thex-axis to some point in quadrant 1, and a hypotenuse r extending from the origin to that same point in quadrant 1. The vertices are at the origin (0,0), some point along the x-axis at (x,0), and that point in quadrant 1. This last point is (x,y) or (r, theta), depending which system of coordinates you use.\" width=\"487\" height=\"290\"> <strong>Figure 5.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165134105991\" class=\"textbox key-takeaways\">\n<h3>Converting from Polar Coordinates to Rectangular Coordinates<\/h3>\n<p id=\"fs-id1165134230340\">To convert polar coordinates[latex]\\,\\left(r,\\,\\theta \\right)\\,[\/latex]to rectangular coordinates[latex]\\,\\left(x,\\,y\\right),[\/latex] let<\/p>\n\n<div id=\"fs-id1165135696992\" class=\"unnumbered aligncenter\">[latex]\\mathrm{cos}\\,\\theta =\\frac{x}{r}\\to x=r\\mathrm{cos}\\,\\theta [\/latex]<\/div>\n<div id=\"fs-id1165137461742\" class=\"unnumbered aligncenter\">[latex]\\mathrm{sin}\\,\\theta =\\frac{y}{r}\\to y=r\\mathrm{sin}\\,\\theta [\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165131880216\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165133371754\"><strong>Given polar coordinates, convert to rectangular coordinates.\n<\/strong><\/p>\n\n<ol id=\"fs-id1165133184235\" type=\"1\">\n \t<li>Given the polar coordinate[latex]\\,\\left(r,\\theta \\right),[\/latex] write[latex]\\,x=r\\mathrm{cos}\\,\\theta \\,[\/latex]and[latex]\\,y=r\\mathrm{sin}\\,\\theta .[\/latex]<\/li>\n \t<li>Evaluate[latex]\\,\\mathrm{cos}\\,\\theta \\,[\/latex]and[latex]\\,\\mathrm{sin}\\,\\theta .[\/latex]<\/li>\n \t<li>Multiply[latex]\\,\\mathrm{cos}\\,\\theta \\,[\/latex]by[latex]\\,r\\,[\/latex]to find the <em>x-<\/em>coordinate of the rectangular form.<\/li>\n \t<li>Multiply[latex]\\,\\mathrm{sin}\\,\\theta \\,[\/latex]by[latex]\\,r\\,[\/latex]to find the <em>y-<\/em>coordinate of the rectangular form.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_08_03_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137585396\">\n<div id=\"fs-id1165137585398\">\n<h3>Writing Polar Coordinates as Rectangular Coordinates<\/h3>\n<p id=\"fs-id1165135546106\">Write the polar coordinates[latex]\\,\\left(3,\\frac{\\pi }{2}\\right)\\,[\/latex]as rectangular coordinates.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134393743\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134393743\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134393743\"]Use the equivalent relationships.\n<div id=\"fs-id1165137894480\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ x=r\\mathrm{cos}\\,\\theta \\end{array}\\hfill \\\\ x=3\\mathrm{cos}\\,\\frac{\\pi }{2}=0\\hfill \\\\ y=r\\mathrm{sin}\\,\\theta \\hfill \\\\ y=3\\mathrm{sin}\\,\\frac{\\pi }{2}=3\\hfill \\end{array}[\/latex]<\/div>\nThe rectangular coordinates are[latex]\\,\\left(0,3\\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_08_03_008\">(Figure)<\/a>.\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150122\/CNX_Precalc_Figure_08_03_008.jpg\" alt=\"Illustration of (3, pi\/2) in polar coordinates and (0,3) in rectangular coordinates - they are the same point!\" width=\"975\" height=\"404\"> <strong>Figure 6.<\/strong>[\/caption]\n<p id=\"fs-id1165132921461\"><span id=\"fs-id1165135536558\"><\/span>[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135536607\">\n<div id=\"fs-id1165135536609\">\n<h3>Writing Polar Coordinates as Rectangular Coordinates<\/h3>\n<p id=\"fs-id1165137832785\">Write the polar coordinates[latex]\\,\\left(-2,0\\right)\\,[\/latex]as rectangular coordinates.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137599665\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137599665\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137599665\"]\n<p id=\"fs-id1165137460411\">See <a class=\"autogenerated-content\" href=\"#Figure_08_03_009\">(Figure)<\/a>. Writing the polar coordinates as rectangular, we have<\/p>\n\n<div id=\"fs-id1165135151267\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x=r\\mathrm{cos}\\,\\theta \\hfill \\\\ x=-2\\mathrm{cos}\\left(0\\right)=-2\\hfill \\\\ \\hfill \\\\ y=r\\mathrm{sin}\\,\\theta \\hfill \\\\ y=-2\\mathrm{sin}\\left(0\\right)=0\\hfill \\end{array}[\/latex]<\/div>\nThe rectangular coordinates are also[latex]\\,\\left(-2,0\\right).[\/latex]\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150129\/CNX_Precalc_Figure_08_03_009.jpg\" alt=\"Illustration of (-2, 0) in polar coordinates and (-2,0) in rectangular coordinates - they are the same point!\" width=\"731\" height=\"375\"> <strong>Figure 7.<\/strong>[\/caption]\n<p id=\"fs-id1165134279361\">[\/hidden-answer]<\/p>\n\n<div id=\"Figure_08_03_009\" class=\"medium\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134489724\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_03_03\">\n<div id=\"fs-id1165135648760\">\n<p id=\"fs-id1165135648761\">Write the polar coordinates[latex]\\,\\left(-1,\\frac{2\\pi }{3}\\right)\\,[\/latex]as rectangular coordinates.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135470020\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135470020\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135470020\"]\n<p id=\"fs-id1165135470021\">[latex]\\left(x,y\\right)=\\left(\\frac{1}{2},-\\frac{\\sqrt{3}}{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134319712\" class=\"bc-section section\">\n<h3>Converting from Rectangular Coordinates to Polar Coordinates<\/h3>\n<p id=\"fs-id1165133359265\">To convert <span class=\"no-emphasis\">rectangular coordinates<\/span> to <span class=\"no-emphasis\">polar coordinates<\/span>, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.<\/p>\n\n<div id=\"fs-id1165135551669\" class=\"textbox key-takeaways\">\n<h3>Converting from Rectangular Coordinates to Polar Coordinates<\/h3>\n<p id=\"fs-id1165135259493\">Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated in <a class=\"autogenerated-content\" href=\"#Figure_08_03_010\">(Figure)<\/a>.<\/p>\n\n<div id=\"fs-id1165134230382\">[latex]\\begin{array}{l}\\mathrm{cos}\\,\\theta =\\frac{x}{r}\\,\\,\\text{ or}\\,\\,\\,x=r\\mathrm{cos}\\,\\theta \\hfill \\\\ \\mathrm{sin}\\,\\theta =\\frac{y}{r}\\,\\,\\text{ or}\\,\\,\\,y=r\\mathrm{sin}\\,\\theta \\hfill \\\\ \\,\\,\\,\\,\\,\\,{r}^{2}={x}^{2}+{y}^{2}\\hfill \\\\ \\mathrm{tan}\\,\\theta =\\frac{y}{x}\\,\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Figure_08_03_010\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150144\/CNX_Precalc_Figure_08_03_010new.jpg\" alt=\"\" width=\"487\" height=\"298\"> <strong>Figure 8.<\/strong>[\/caption]\n\n<\/div>\n<\/div>\n<div id=\"Example_08_03_05\" class=\"textbox examples\">\n<div id=\"fs-id1165134303364\">\n<div id=\"fs-id1165137589072\">\n<h3>Writing Rectangular Coordinates as Polar Coordinates<\/h3>\n<p id=\"fs-id1165135634043\">Convert the rectangular coordinates[latex]\\,\\left(3,3\\right)\\,[\/latex]to polar coordinates.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137733557\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137733557\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137733557\"]\n<p id=\"fs-id1165137733559\">We see that the original point[latex]\\,\\left(3,3\\right)\\,[\/latex]is in the first quadrant. To find[latex]\\,\\theta ,\\,[\/latex]use the formula[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{y}{x}.\\,[\/latex]This gives<\/p>\n\n<div id=\"fs-id1165137460645\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\mathrm{tan}\\,\\theta =\\frac{3}{3}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\mathrm{tan}\\,\\theta =1\\hfill \\\\ \\,\\,\\,{\\mathrm{tan}}^{-1}\\left(1\\right)=\\frac{\\pi }{4}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137639677\">To find[latex]\\,r,\\,[\/latex]we substitute the values for[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]into the formula[latex]\\,r=\\sqrt{{x}^{2}+{y}^{2}}.\\,[\/latex]We know that[latex]\\,r\\,[\/latex]must be positive, as[latex]\\,\\frac{\\pi }{4}\\,[\/latex]is in the first quadrant. Thus<\/p>\n\n<div id=\"fs-id1165135689471\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ r=\\sqrt{{3}^{2}+{3}^{2}}\\end{array}\\hfill \\\\ r=\\sqrt{9+9}\\hfill \\\\ r=\\sqrt{18}=3\\sqrt{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135332735\">So,[latex]\\,r=3\\sqrt{2}\\,\\,[\/latex]and[latex]\\,\\theta \\text{=}\\frac{\\pi }{4},\\,[\/latex]giving us the polar point[latex]\\,\\left(3\\sqrt{2},\\frac{\\pi }{4}\\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_08_03_011\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_03_011\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150158\/CNX_Precalc_Figure_08_03_011.jpg\" alt=\"Illustration of (3rad2, pi\/4) in polar coordinates and (3,3) in rectangular coordinates - they are the same point!\" width=\"975\" height=\"375\"> <strong>Figure 9.<\/strong>[\/caption]\n\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133078085\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137574155\">There are other sets of polar coordinates that will be the same as our first solution. For example, the points[latex]\\,\\left(-3\\sqrt{2},\\,\\frac{5\\pi }{4}\\right)\\,[\/latex]and[latex]\\,\\left(3\\sqrt{2},-\\frac{7\\pi }{4}\\right)\\,[\/latex]will coincide with the original solution of[latex]\\,\\left(3\\sqrt{2},\\,\\frac{\\pi }{4}\\right).\\,[\/latex]The point[latex]\\,\\left(-3\\sqrt{2},\\,\\frac{5\\pi }{4}\\right)\\,[\/latex]indicates a move further counterclockwise by[latex]\\,\\pi ,\\,[\/latex]which is directly opposite[latex]\\,\\frac{\\pi }{4}.\\,[\/latex]The radius is expressed as[latex]\\,-3\\sqrt{2}.\\,[\/latex]However, the angle[latex]\\,\\frac{5\\pi }{4}\\,[\/latex]is located in the third quadrant and, as[latex]\\,r\\,[\/latex]is negative, we extend the directed line segment in the opposite direction, into the first quadrant. This is the same point as[latex]\\,\\left(3\\sqrt{2},\\,\\,\\frac{\\pi }{4}\\right).\\,[\/latex]The point[latex]\\,\\left(3\\sqrt{2},\\,-\\frac{7\\pi }{4}\\right)\\,[\/latex]is a move further clockwise by[latex]\\,-\\frac{7\\pi }{4},\\,[\/latex]from[latex]\\,\\frac{\\pi }{4}.\\,[\/latex]The radius,[latex]\\,3\\sqrt{2},\\,[\/latex]is the same.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132957131\" class=\"bc-section section\">\n<h3>Transforming Equations between Polar and Rectangular Forms<\/h3>\n<p id=\"fs-id1165137726540\">We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation.<\/p>\n\n<div id=\"fs-id1165133356035\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165134422164\"><strong>Given an equation in polar form, graph it using a graphing calculator.\n<\/strong><\/p>\n\n<ol id=\"fs-id1165137810313\" type=\"1\">\n \t<li>Change the <strong>MODE<\/strong> to <strong>POL<\/strong>, representing polar form.<\/li>\n \t<li>Press the <strong>Y= <\/strong>button to bring up a screen allowing the input of six equations:[latex]\\,{r}_{1},\\,\\,{r}_{2},\\,\\,.\\,\\,.\\,\\,.\\,\\,,\\,\\,{r}_{6}.[\/latex]<\/li>\n \t<li>Enter the polar equation, set equal to[latex]\\,r.[\/latex]<\/li>\n \t<li>Press <strong>GRAPH<\/strong>.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_08_03_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135485797\">\n<div id=\"fs-id1165135503665\">\n<h3>Writing a Cartesian Equation in Polar Form<\/h3>\n<p id=\"fs-id1165135503671\">Write the Cartesian equation[latex]\\,{x}^{2}+{y}^{2}=9\\,[\/latex]in polar form.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134418996\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134418996\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134418996\"]\n<p id=\"fs-id1165134418998\">The goal is to eliminate[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]from the equation and introduce[latex]\\,r\\,[\/latex]and [latex]\\,\\theta .\\,[\/latex]Ideally, we would write the equation[latex]\\,r\\,[\/latex]as a function of[latex]\\,\\theta .\\,[\/latex]To obtain the polar form, we will use the relationships between[latex]\\,\\left(x,y\\right)\\,[\/latex]and[latex]\\,\\left(r,\\theta \\right).\\,[\/latex]Since[latex]\\,x=r\\mathrm{cos}\\,\\theta \\,[\/latex] and[latex]\\,y=r\\mathrm{sin}\\,\\theta ,\\,[\/latex]we can substitute and solve for[latex]\\,r.[\/latex]<\/p>\n\n<div id=\"fs-id1165134193511\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\text{ }{\\left(r\\mathrm{cos}\\,\\theta \\right)}^{2}+{\\left(r\\mathrm{sin}\\,\\theta \\right)}^{2}=9\\hfill &amp; \\hfill \\\\ \\text{ }{r}^{2}{\\mathrm{cos}}^{2}\\theta +{r}^{2}{\\mathrm{sin}}^{2}\\theta =9\\hfill &amp; \\hfill \\\\ \\text{ }{r}^{2}\\left({\\mathrm{cos}}^{2}\\theta +{\\mathrm{sin}}^{2}\\theta \\right)=9\\hfill &amp; \\hfill \\\\ \\,\\text{ }{r}^{2}\\left(1\\right)=9 \\hfill &amp; {\\text{Substitute cos}}^{2}\\theta +{\\mathrm{sin}}^{2}\\theta =1.\\hfill \\\\ \\text{ }r=\u00b13\\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\hfill &amp; \\text{Use the square root property}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137757670\">Thus,[latex]\\,{x}^{2}+{y}^{2}=9,r=3,[\/latex]and[latex]\\,r=-3\\,[\/latex]should generate the same graph. See <a class=\"autogenerated-content\" href=\"#Figure_08_03_016\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_03_016\" class=\"medium\">[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150200\/CNX_Precalc_Figure_08_03_016.jpg\" alt=\"Plotting a circle of radius 3 with center at the origin in polar and rectangular coordinates. It is the same in both systems.\" width=\"731\" height=\"360\"> <strong>Figure 10. <\/strong>(a) Cartesian form[latex]\\,{x}^{2}+{y}^{2}=9\\,[\/latex](b) Polar form[latex]\\,r=3[\/latex][\/caption]<\/div>\n<p id=\"fs-id1165133192137\">To graph a circle in rectangular form, we must first solve for[latex]\\,y.[\/latex]<\/p>\n\n<div id=\"fs-id1165137835799\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {x}^{2}+{y}^{2}=9\\end{array}\\hfill \\\\ \\text{ }{y}^{2}=9-{x}^{2}\\hfill \\\\ \\text{ }y=\u00b1\\sqrt{9-{x}^{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135620839\">Note that this is two separate functions, since a circle fails the vertical line test. Therefore, we need to enter the positive and negative square roots into the calculator separately, as two equations in the form[latex]\\,{Y}_{1}=\\sqrt{9-{x}^{2}}\\,[\/latex]and[latex]\\,{Y}_{2}=-\\sqrt{9-{x}^{2}}.\\,[\/latex]Press <strong>GRAPH.<\/strong>[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137473589\">\n<div id=\"fs-id1165137473591\">\n<h3>Rewriting a Cartesian Equation as a Polar Equation<\/h3>\n<p id=\"fs-id1165137928681\">Rewrite the <span class=\"no-emphasis\">Cartesian equation<\/span>[latex]\\,{x}^{2}+{y}^{2}=6y\\,[\/latex]as a polar equation.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135547690\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135547690\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135547690\"]\n<p id=\"fs-id1165135547693\">This equation appears similar to the previous example, but it requires different steps to convert the equation.<\/p>\n<p id=\"fs-id1165137698161\">We can still follow the same procedures we have already learned and make the following substitutions:<\/p>\n\n<div id=\"fs-id1165134168283\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{r}^{2}=6y\\hfill &amp; \\text{Use }{x}^{2}+{y}^{2}={r}^{2}.\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{r}^{2}=6r\\mathrm{sin}\\,\\theta \\hfill &amp; \\text{Substitute}\\,y=r\\mathrm{sin}\\,\\theta .\\hfill \\\\ \\text{ }{r}^{2}-6r\\mathrm{sin}\\,\\theta =0\\hfill &amp; \\text{Set equal to 0}.\\hfill \\\\ \\text{ }r\\left(r-6\\mathrm{sin}\\,\\theta \\right)=0\\hfill &amp; \\text{Factor and solve}.\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,r=0\\hfill &amp; \\text{We reject }r=0,\\,\\text{as it only represents one point, }\\left(0,0\\right).\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,r=6\\mathrm{sin}\\,\\theta \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\hfill &amp; \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135360250\">Therefore, the equations[latex]\\,{x}^{2}+{y}^{2}=6y\\,[\/latex]and[latex]\\,r=6\\mathrm{sin}\\,\\theta \\,[\/latex] should give us the same graph. See <a class=\"autogenerated-content\" href=\"#Figure_08_03_012\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_03_012\" class=\"wp-caption aligncenter\">[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150203\/CNX_Precalc_Figure_08_03_012.jpg\" alt=\"Plots of the equations stated above - the plots are the same in both rectangular and polar coordinates. They are circles.\" width=\"975\" height=\"328\"> <strong>Figure 11. <\/strong>(a) Cartesian form[latex]\\,{x}^{2}+{y}^{2}=6y[\/latex](b) polar form[latex]\\,r=6\\mathrm{sin}\\,\\theta [\/latex][\/caption]<\/div>\n<p id=\"fs-id1165135702699\">The Cartesian or <span class=\"no-emphasis\">rectangular equation<\/span> is plotted on the rectangular grid, and the <span class=\"no-emphasis\">polar equation<\/span> is plotted on the polar grid. Clearly, the graphs are identical.[\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_08\" class=\"textbox examples\">\n<div id=\"fs-id1165135330933\">\n<div id=\"fs-id1165135330936\">\n<h3>Rewriting a Cartesian Equation in Polar Form<\/h3>\n<p id=\"fs-id1165135547678\">Rewrite the Cartesian equation[latex]\\,y=3x+2\\,[\/latex]as a polar equation.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135201018\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135201018\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135201018\"]\n<p id=\"fs-id1165135394016\">We will use the relationships[latex]\\,x=r\\mathrm{cos}\\,\\theta \\,[\/latex] and [latex]\\,y=r\\mathrm{sin}\\,\\theta .[\/latex]<\/p>\n\n<div id=\"fs-id1165135514618\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\text{ }y=3x+2\\hfill &amp; \\hfill \\\\ \\text{ }r\\mathrm{sin}\\,\\theta =3r\\mathrm{cos}\\,\\theta +2\\hfill &amp; \\hfill \\\\ \\,r\\mathrm{sin}\\,\\theta -3r\\mathrm{cos}\\,\\theta =2\\hfill &amp; \\hfill \\\\ r\\left(\\mathrm{sin}\\,\\theta -3\\mathrm{cos}\\,\\theta \\right)=2\\hfill &amp; \\text{Isolate }r.\\hfill \\\\ \\text{ }\\,\\,\\,r=\\frac{2}{\\mathrm{sin}\\,\\theta -3\\mathrm{cos}\\,\\theta }\\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\hfill &amp; \\text{Solve for }r.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134496398\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_03_04\">\n<div id=\"fs-id1165135413673\">\n<p id=\"fs-id1165135413674\">Rewrite the Cartesian equation[latex]\\,{y}^{2}=3-{x}^{2}\\,[\/latex]in polar form.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135564036\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135564036\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135564036\"]\n<p id=\"fs-id1165135564037\">[latex]r=\\sqrt{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137422423\" class=\"bc-section section\">\n<h3>Identify and Graph Polar Equations by Converting to Rectangular Equations<\/h3>\n<p id=\"fs-id1165137735564\">We have learned how to convert rectangular coordinates to polar coordinates, and we have seen that the points are indeed the same. We have also transformed polar equations to rectangular equations and vice versa. Now we will demonstrate that their graphs, while drawn on different grids, are identical.<\/p>\n\n<div id=\"Example_08_03_09\" class=\"textbox examples\">\n<div id=\"fs-id1165134433326\">\n<div id=\"fs-id1165135512717\">\n<h3>Graphing a Polar Equation by Converting to a Rectangular Equation<\/h3>\n<p id=\"fs-id1165135512722\">Covert the polar equation[latex]\\,r=2\\mathrm{sec}\\,\\theta \\,[\/latex] to a rectangular equation, and draw its corresponding graph.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135580397\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135580397\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135580397\"]\n<p id=\"fs-id1165135580399\">The conversion is<\/p>\n\n<div id=\"fs-id1165137697050\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }r=2\\mathrm{sec}\\,\\theta \\hfill \\\\ \\,\\,\\,\\,\\text{ }r=\\frac{2}{\\mathrm{cos}\\,\\theta }\\hfill \\\\ r\\mathrm{cos}\\,\\theta =2\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x=2\\hfill \\end{array}[\/latex]<\/div>\nNotice that the equation[latex]\\,r=2\\mathrm{sec}\\,\\theta \\,[\/latex]drawn on the polar grid is clearly the same as the vertical line[latex]\\,x=2\\,[\/latex]drawn on the rectangular grid (see <a class=\"autogenerated-content\" href=\"#Figure_08_03_013\">(Figure)<\/a>). Just as[latex]\\,x=c\\,[\/latex]is the standard form for a vertical line in rectangular form,[latex]\\,r=c\\mathrm{sec}\\,\\theta \\,[\/latex]is the standard form for a vertical line in polar form.\n<div id=\"Figure_08_03_013\" class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150210\/CNX_Precalc_Figure_08_03_013.jpg\" alt=\"Plots of the equations stated above - the plots are the same in both rectangular and polar coordinates. They are lines.\" width=\"731\" height=\"408\"> <strong>Figure 12. <\/strong>(a) Polar grid (b) Rectangular coordinate system[\/caption]\n\n<\/div>\n<p id=\"fs-id1165135689446\">A similar discussion would demonstrate that the graph of the function[latex]\\,r=2\\mathrm{csc}\\,\\theta \\,[\/latex] will be the horizontal line[latex]\\,y=2.\\,[\/latex]In fact,[latex]\\,r=c\\mathrm{csc}\\,\\theta \\,[\/latex] is the standard form for a horizontal line in polar form, corresponding to the rectangular form[latex]\\,y=c.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_10\" class=\"textbox examples\">\n<div id=\"fs-id1165137532816\">\n<div id=\"fs-id1165137442988\">\n<h3>Rewriting a Polar Equation in Cartesian Form<\/h3>\n<p id=\"fs-id1165137442993\">Rewrite the polar equation[latex]\\,r=\\frac{3}{1-2\\mathrm{cos}\\,\\theta }\\,[\/latex]as a Cartesian equation.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137435019\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137435019\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137435019\"]\n<p id=\"fs-id1165137435021\">The goal is to eliminate[latex]\\,\\theta \\,[\/latex]and[latex]\\,r,[\/latex]and introduce[latex]\\,x\\,[\/latex] and [latex]\\,y.\\,[\/latex]We clear the fraction, and then use substitution. In order to replace[latex]\\,r\\,[\/latex] with [latex]\\,x\\,[\/latex]and[latex]\\,y,[\/latex] we must use the expression[latex]\\,{x}^{2}+{y}^{2}={r}^{2}.[\/latex]<\/p>\n\n<div id=\"fs-id1165132946700\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{llll}\\text{ }r=\\frac{3}{1-2\\mathrm{cos}\\,\\theta }\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ r\\left(1-2\\mathrm{cos}\\,\\theta \\right)=3\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ \\,\\,\\,\\,\\,\\,r\\left(1-2\\left(\\frac{x}{r}\\right)\\right)=3\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Use }\\mathrm{cos}\\,\\theta =\\frac{x}{r}\\text{ to eliminate }\\theta .\\hfill \\\\ \\text{ }\\,\\,\\,r-2x=3\\hfill &amp; \\hfill &amp; \\hfill &amp; \\hfill \\\\ \\text{ }r=3+2x\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Isolate }r.\\hfill \\\\ \\text{ }{r}^{2}={\\left(3+2x\\right)}^{2}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Square both sides}.\\hfill \\\\ \\text{ }\\,{x}^{2}+{y}^{2}={\\left(3+2x\\right)}^{2}\\hfill &amp; \\hfill &amp; \\hfill &amp; \\text{Use }{x}^{2}+{y}^{2}={r}^{2}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134368154\">The Cartesian equation is[latex]\\,{x}^{2}+{y}^{2}={\\left(3+2x\\right)}^{2}.\\,[\/latex]However, to graph it, especially using a graphing calculator or computer program, we want to isolate[latex]\\,y.[\/latex]<\/p>\n\n<div id=\"fs-id1165134101989\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}={\\left(3+2x\\right)}^{2}\\hfill \\\\ \\text{ }{y}^{2}={\\left(3+2x\\right)}^{2}-{x}^{2}\\hfill \\\\ \\text{ }y=\u00b1\\sqrt{{\\left(3+2x\\right)}^{2}-{x}^{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134107035\">When our entire equation has been changed from[latex]\\,r\\,[\/latex]and[latex]\\,\\theta \\,[\/latex]to [latex]\\,x\\,[\/latex] and [latex]\\,y,\\,[\/latex]we can stop, unless asked to solve for[latex]\\,y\\,[\/latex]or simplify. See <a class=\"autogenerated-content\" href=\"#Figure_08_03_015\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_03_015\" class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150212\/CNX_Precalc_Figure_08_03_015.jpg\" alt=\"Plots of the equations stated above - the plots are the same in both rectangular and polar coordinates. They are hyperbolas.\" width=\"975\" height=\"481\"> <strong>Figure 13.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id1165135355014\">The \u201chour-glass\u201d shape of the graph is called a <em>hyperbola<\/em>. Hyperbolas have many interesting geometric features and applications, which we will investigate further in <a class=\"target-chapter\" href=\"\/contents\/3315e118-f58b-48f1-a06e-f03612d38b40\">Analytic Geometry<\/a>.[\/hidden-answer]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134116776\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134116781\">In this example, the right side of the equation can be expanded and the equation simplified further, as shown above. However, the equation cannot be written as a single function in Cartesian form. We may wish to write the rectangular equation in the hyperbola\u2019s standard form. To do this, we can start with the initial equation.<\/p>\n\n<div id=\"fs-id1165135453309\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\,\\text{ }{x}^{2}+{y}^{2}={\\left(3+2x\\right)}^{2}\\hfill &amp; \\hfill \\\\ \\,\\,\\text{ }{x}^{2}+{y}^{2}-{\\left(3+2x\\right)}^{2}=0\\hfill &amp; \\hfill \\\\ {x}^{2}+{y}^{2}-\\left(9+12x+4{x}^{2}\\right)=0\\hfill &amp; \\hfill \\\\ \\,\\text{ }{x}^{2}+{y}^{2}-9-12x-4{x}^{2}=0\\hfill &amp; \\hfill \\\\ \\,\\text{ }-3{x}^{2}-12x+{y}^{2}=9\\hfill &amp; \\text{Multiply through by }-1.\\hfill \\\\ \\,\\,\\text{ }\\,3{x}^{2}+12x-{y}^{2}=-9\\hfill &amp; \\hfill \\\\ \\,\\,\\text{ }\\,3\\left({x}^{2}+4x+\\,\\,\\,\\,\\,\\,\\,\\,\\,\\right)-{y}^{2}=-9\\hfill &amp; \\text{Organize terms to complete the square for}\\,x.\\hfill \\\\ \\,\\,\\text{ }\\,3\\left({x}^{2}+4x+4\\right)-{y}^{2}=-9+12\\hfill &amp; \\hfill \\\\ \\text{ }3{\\left(x+2\\right)}^{2}-{y}^{2}=3\\hfill &amp; \\hfill \\\\ \\,\\text{ }{\\left(x+2\\right)}^{2}-\\frac{{y}^{2}}{3}=1\\hfill &amp; \\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133074971\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_03_05\">\n<div id=\"fs-id1165134495101\">\n<p id=\"fs-id1165134495102\">Rewrite the polar equation[latex]\\,r=2\\mathrm{sin}\\,\\theta \\,[\/latex] in Cartesian form.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135378641\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135378641\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135378641\"]\n<p id=\"fs-id1165135378642\">[latex]{x}^{2}+{y}^{2}=2y\\,[\/latex]or, in the standard form for a circle,[latex]\\,{x}^{2}+{\\left(y-1\\right)}^{2}=1[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_11\" class=\"textbox examples\">\n<div id=\"fs-id1165135456744\">\n<div id=\"fs-id1165135456746\">\n<h3>Rewriting a Polar Equation in Cartesian Form<\/h3>\n<p id=\"fs-id1165135456752\">Rewrite the polar equation[latex]\\,r=\\mathrm{sin}\\left(2\\theta \\right)\\,[\/latex]in Cartesian form.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134437217\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134437217\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134437217\"]\n<div id=\"fs-id1165134437221\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\text{ }r=\\mathrm{sin}\\left(2\\theta \\right)\\hfill &amp; \\text{Use the double angle identity for sine}.\\hfill \\\\ \\text{ }r=2\\mathrm{sin}\\,\\theta \\mathrm{cos}\\,\\theta \\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\hfill &amp; \\text{Use }\\mathrm{cos}\\,\\theta =\\frac{x}{r}\\text{ and }\\mathrm{sin}\\,\\theta =\\frac{y}{r}.\\hfill \\\\ \\text{ }r=2\\left(\\frac{x}{r}\\right)\\left(\\frac{y}{r}\\right)\\hfill &amp; \\text{Simplify}.\\hfill \\\\ \\text{ }r=\\frac{2xy}{{r}^{2}}\\hfill &amp; \\text{ Multiply both sides by }{r}^{2}.\\hfill \\\\ \\text{ }{r}^{3}=2xy\\hfill &amp; \\hfill \\\\ {\\left(\\sqrt{{x}^{2}+{y}^{2}}\\right)}^{3}=2xy\\hfill &amp; \\text{As}\\,{x}^{2}+{y}^{2}={r}^{2},r=\\sqrt{{x}^{2}+{y}^{2}}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137697887\">This equation can also be written as<\/p>\n\n<div id=\"fs-id1165137697890\" class=\"unnumbered aligncenter\">[latex]{\\left({x}^{2}+{y}^{2}\\right)}^{\\frac{3}{2}}=2xy\\,\\text{or}\\,{x}^{2}+{y}^{2}={\\left(2xy\\right)}^{\\frac{2}{3}}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135664937\" class=\"precalculus media\">\n<p id=\"fs-id1165133361920\">Access these online resources for additional instruction and practice with polar coordinates.<\/p>\n\n<ul id=\"fs-id1165133361923\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/intropolar\">Introduction to Polar Coordinates<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/polarrect\">Comparing Polar and Rectangular Coordinates<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135407353\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id3984733\" summary=\"..\">\n<tbody>\n<tr>\n<td>Conversion formulas<\/td>\n<td>[latex]\\begin{array}{ll}\\hfill &amp; \\mathrm{cos}\\,\\theta =\\frac{x}{r}\\to x=r\\mathrm{cos}\\,\\theta \\hfill \\\\ \\hfill &amp; \\mathrm{sin}\\,\\theta =\\frac{y}{r}\\to y=r\\mathrm{sin}\\,\\theta \\hfill \\\\ \\hfill &amp; \\,\\,\\,\\,\\,\\,\\,{r}^{2}={x}^{2}+{y}^{2}\\hfill \\\\ \\hfill &amp; \\mathrm{tan}\\,\\theta =\\frac{y}{x}\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137707236\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137707243\">\n \t<li>The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.<\/li>\n \t<li>To plot a point in the form[latex]\\,\\left(r,\\theta \\right),\\,\\theta &gt;0,\\,[\/latex]move in a counterclockwise direction from the polar axis by an angle of [latex]\\,\\theta ,\\,[\/latex]and then extend a directed line segment from the pole the length of [latex]\\,r\\,[\/latex] in the direction of [latex]\\,\\theta .\\,[\/latex]If[latex]\\,\\theta \\,[\/latex]is negative, move in a clockwise direction, and extend a directed line segment the length of [latex]\\,r\\,[\/latex] in the direction of[latex]\\,\\theta .[\/latex] See <a class=\"autogenerated-content\" href=\"#Example_08_03_01\">(Figure)<\/a>.<\/li>\n \t<li>If[latex]\\,r\\,[\/latex]is negative, extend the directed line segment in the opposite direction of[latex]\\,\\theta .\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_03_02\">(Figure)<\/a>.<\/li>\n \t<li>To convert from polar coordinates to rectangular coordinates, use the formulas[latex]\\,x=r\\mathrm{cos}\\,\\theta \\,[\/latex]and[latex]\\,y=r\\mathrm{sin}\\,\\theta .\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_03_03\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_03_04\">(Figure)<\/a>.<\/li>\n \t<li>To convert from rectangular coordinates to polar coordinates, use one or more of the formulas:[latex]\\,\\mathrm{cos}\\,\\theta =\\frac{x}{r},\\mathrm{sin}\\,\\theta =\\frac{y}{r},\\mathrm{tan}\\,\\theta =\\frac{y}{x},\\,[\/latex]and[latex]\\,r=\\sqrt{{x}^{2}+{y}^{2}}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_03_05\">(Figure)<\/a>.<\/li>\n \t<li>Transforming equations between polar and rectangular forms means making the appropriate substitutions based on the available formulas, together with algebraic manipulations. See <a class=\"autogenerated-content\" href=\"#Example_08_03_06\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_08_03_07\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_08_03_08\">(Figure)<\/a>.<\/li>\n \t<li>Using the appropriate substitutions makes it possible to rewrite a polar equation as a rectangular equation, and then graph it in the rectangular plane. See <a class=\"autogenerated-content\" href=\"#Example_08_03_09\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_08_03_10\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_08_03_11\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165134113788\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165134113791\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165134113796\">\n<div id=\"fs-id1165135481176\">\n<p id=\"fs-id1165135481179\">How are polar coordinates different from rectangular coordinates?<\/p>\n\n<\/div>\n<div id=\"fs-id1165135481183\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135481183\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135481183\"]\n<p id=\"fs-id1165135481185\">For polar coordinates, the point in the plane depends on the angle from the positive <em>x-<\/em>axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137634226\">\n<div id=\"fs-id1165137634229\">\n<p id=\"fs-id1165137634231\">How are the polar axes different from the <em>x<\/em>- and <em>y<\/em>-axes of the Cartesian plane?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133075606\">\n<div id=\"fs-id1165133075608\">\n<p id=\"fs-id1165134193488\">Explain how polar coordinates are graphed.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134193492\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134193492\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134193492\"]\n<p id=\"fs-id1165134193494\">Determine[latex]\\,\\theta \\,[\/latex]for the point, then move[latex]\\,r\\,[\/latex]units from the pole to plot the point. If[latex]\\,r\\,[\/latex]is negative, move[latex]\\,r\\,[\/latex]units from the pole in the opposite direction but along the same angle. The point is a distance of[latex]\\,r\\,[\/latex]away from the origin at an angle of[latex]\\,\\theta \\,[\/latex]from the polar axis.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135316122\">\n<div id=\"fs-id1165135316124\">\n<p id=\"fs-id1165135316126\">How are the points[latex]\\,\\left(3,\\frac{\\pi }{2}\\right)\\,[\/latex]and[latex]\\,\\left(-3,\\frac{\\pi }{2}\\right)\\,[\/latex]related?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135403191\">\n<div id=\"fs-id1165135403193\">\n<p id=\"fs-id1165135403195\">Explain why the points[latex]\\,\\left(-3,\\frac{\\pi }{2}\\right)\\,[\/latex]and[latex]\\,\\left(3,-\\frac{\\pi }{2}\\right)\\,[\/latex]are the same.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135319445\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135319445\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135319445\"]\n<p id=\"fs-id1165135319448\">The point[latex]\\,\\left(-3,\\frac{\\pi }{2}\\right)\\,[\/latex]has a positive angle but a negative radius and is plotted by moving to an angle of[latex]\\,\\frac{\\pi }{2}\\,[\/latex]and then moving 3 units in the negative direction. This places the point 3 units down the negative <em>y<\/em>-axis. The point[latex]\\,\\left(3,-\\frac{\\pi }{2}\\right)\\,[\/latex]has a negative angle and a positive radius and is plotted by first moving to an angle of[latex]\\,-\\frac{\\pi }{2}\\,[\/latex]and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative <em>y<\/em>-axis.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135329737\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165133143104\">For the following exercises, convert the given polar coordinates to Cartesian coordinates with[latex]\\,r&gt;0\\,[\/latex]and[latex]\\,0\\le \\theta \\le 2\\pi .\\,[\/latex]Remember to consider the quadrant in which the given point is located when determining[latex]\\,\\theta \\,[\/latex]for the point.<\/p>\n\n<div id=\"fs-id1165131986178\">\n<div id=\"fs-id1165131986180\">\n<p id=\"fs-id1165131986182\">[latex]\\left(7,\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134312184\">\n<div id=\"fs-id1165134312186\">\n<p id=\"fs-id1165134312188\">[latex]\\left(5,\\pi \\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135616318\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135616318\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135616318\"]\n<p id=\"fs-id1165135616321\">[latex]\\left(-5,0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134384438\">\n<div id=\"fs-id1165134384440\">[latex]\\left(6,-\\frac{\\pi }{4}\\right)[\/latex]<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134196150\">\n<p id=\"fs-id1165134196152\">[latex]\\left(-3,\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134383794\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134383794\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134383794\"]\n<p id=\"fs-id1165134383796\">[latex]\\left(-\\frac{3\\sqrt{3}}{2},-\\frac{3}{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134177068\">\n<div id=\"fs-id1165134177070\">\n<p id=\"fs-id1165134177072\">[latex]\\left(4,\\frac{7\\pi }{4}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165131961716\">For the following exercises, convert the given Cartesian coordinates to polar coordinates with[latex]\\,r&gt;0,\\,\\,0\\le \\theta &lt;2\\pi .\\,[\/latex]Remember to consider the quadrant in which the given point is located.<\/p>\n\n<div id=\"fs-id1165134312061\">\n<div id=\"fs-id1165134312063\">\n<p id=\"fs-id1165134312065\">[latex]\\left(4,2\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134279740\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134279740\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134279740\"]\n<p id=\"fs-id1165134279743\">[latex]\\left(2\\sqrt{5}, 0.464\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137696692\">\n<div id=\"fs-id1165137696694\">[latex]\\left(-4,6\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134113855\">\n<div id=\"fs-id1165134113857\">\n<p id=\"fs-id1165134113859\">[latex]\\left(3,-5\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135407446\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135407446\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135407446\"]\n<p id=\"fs-id1165135407448\">[latex]\\left(\\sqrt{34},5.253\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135434706\">\n<div id=\"fs-id1165135434708\">\n<p id=\"fs-id1165134380127\">[latex]\\left(-10,-13\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135390847\">\n<div id=\"fs-id1165135390849\">\n<p id=\"fs-id1165135254614\">[latex]\\left(8,8\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137890658\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137890658\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137890658\"]\n<p id=\"fs-id1165137890660\">[latex]\\left(8\\sqrt{2},\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165134153314\">For the following exercises, convert the given Cartesian equation to a polar equation.<\/p>\n\n<div id=\"fs-id1165135390660\">\n<div id=\"fs-id1165135390663\">\n<p id=\"fs-id1165135390665\">[latex]x=3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134237043\">\n<div id=\"fs-id1165134237046\">\n<p id=\"fs-id1165134237048\">[latex]y=4[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134174890\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134174890\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134174890\"]\n<p id=\"fs-id1165134174892\">[latex]r=4\\mathrm{csc}\\theta [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137896831\">\n<div id=\"fs-id1165137896834\">\n<p id=\"fs-id1165133277552\">[latex]y=4{x}^{2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135339520\">\n<div id=\"fs-id1165135339523\">\n<p id=\"fs-id1165135339525\">[latex]y=2{x}^{4}[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135404270\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135404270\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135404270\"]\n<p id=\"fs-id1165135698548\">[latex]r=\\sqrt[3]{\\frac{sin\\theta }{2co{s}^{4}\\theta }}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135538875\">\n<div id=\"fs-id1165135538877\">\n<p id=\"fs-id1165135538879\">[latex]{x}^{2}+{y}^{2}=4y[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134129902\">[latex]{x}^{2}+{y}^{2}=3x[\/latex]<\/div>\n<div id=\"fs-id1165132912574\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165132912574\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165132912574\"]\n<p id=\"fs-id1165132912576\">[latex]r=3\\mathrm{cos}\\theta [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134279393\">\n<div id=\"fs-id1165134279395\">\n<p id=\"fs-id1165134279397\">[latex]{x}^{2}-{y}^{2}=x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135531341\">\n<div id=\"fs-id1165134047499\">\n<p id=\"fs-id1165134047501\">[latex]{x}^{2}-{y}^{2}=3y[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135512559\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135512559\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135512559\"]\n<p id=\"fs-id1165135512561\">[latex]r=\\frac{3\\mathrm{sin}\\theta }{\\mathrm{cos}\\left(2\\theta \\right)}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135397902\">\n<div id=\"fs-id1165135397905\">\n<p id=\"fs-id1165135674058\">[latex]{x}^{2}+{y}^{2}=9[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135680173\">\n<div id=\"fs-id1165135680176\">\n<p id=\"fs-id1165135680178\">[latex]{x}^{2}=9y[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134325175\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134325175\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134325175\"]\n<p id=\"fs-id1165134325177\">[latex]r=\\frac{9\\mathrm{sin}\\theta }{{\\mathrm{cos}}^{2}\\theta }[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135570196\">\n<div id=\"fs-id1165135570198\">\n<p id=\"fs-id1165135570200\">[latex]{y}^{2}=9x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134039275\">\n<div id=\"fs-id1165134039278\">\n<p id=\"fs-id1165131895989\">[latex]9xy=1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134116891\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134116891\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134116891\"]\n<p id=\"fs-id1165134116893\">[latex]r=\\sqrt{\\frac{1}{9\\mathrm{cos}\\theta \\mathrm{sin}\\theta }}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165137619903\">For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.<\/p>\n\n<div id=\"fs-id1165135639210\">\n<div id=\"fs-id1165135639212\">\n<p id=\"fs-id1165135639214\">[latex]r=3\\mathrm{sin}\\,\\theta [\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137590001\">\n<div id=\"fs-id1165137590003\">\n<p id=\"fs-id1165137590005\">[latex]r=4\\mathrm{cos}\\,\\theta [\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137810361\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137810361\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137810361\"]\n<p id=\"fs-id1165137810363\">[latex]{x}^{2}+{y}^{2}=4x\\,[\/latex]or[latex]\\,\\frac{{\\left(x-2\\right)}^{2}}{4}+\\frac{{y}^{2}}{4}=1;[\/latex]circle<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134237180\">\n<p id=\"fs-id1165134081878\">[latex]r=\\frac{4}{\\mathrm{sin}\\,\\theta +7\\mathrm{cos}\\,\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165131893786\">\n<div id=\"fs-id1165131893788\">\n<p id=\"fs-id1165131893790\">[latex]r=\\frac{6}{\\mathrm{cos}\\,\\theta +3\\mathrm{sin}\\,\\theta }[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134393823\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134393823\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134393823\"]\n<p id=\"fs-id1165134393825\">[latex]3y+x=6;\\,[\/latex]line<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135351455\">\n<div id=\"fs-id1165135351457\">\n<p id=\"fs-id1165137731259\">[latex]r=2\\mathrm{sec}\\,\\theta [\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133035956\">\n<div id=\"fs-id1165133035958\">\n<p id=\"fs-id1165133035961\">[latex]r=3\\mathrm{csc}\\,\\theta [\/latex]<\/p>\n\n<div class=\"textbox shaded\">\n[reveal-answer q=\"343335\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"343335\"]\n<div id=\"fs-id1165133035956\">\n<div>\n<p id=\"fs-id1165137628658\">[latex]y=3;\\,[\/latex]line<\/p>\n\n<\/div>\n<\/div>\n[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135665421\">\n<div id=\"fs-id1165135665423\">\n<p id=\"fs-id1165135665425\">[latex]r=\\sqrt{r\\mathrm{cos}\\,\\theta +2}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134418774\">\n<div id=\"fs-id1165134418776\">\n<p id=\"fs-id1165134418778\">[latex]{r}^{2}=4\\mathrm{sec}\\,\\theta \\,\\mathrm{csc}\\,\\theta [\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135555427\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135555427\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135555427\"]\n<p id=\"fs-id1165137642992\">[latex]xy=4;\\,[\/latex]hyperbola<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134356832\">\n<div id=\"fs-id1165135546060\">[latex]r=4[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134358530\">\n<div id=\"fs-id1165134358532\">\n<p id=\"fs-id1165134358534\">[latex]{r}^{2}=4[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135702647\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135702647\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135702647\"]\n<p id=\"fs-id1165135702649\">[latex]{x}^{2}+{y}^{2}=4;\\,[\/latex]circle<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137838193\">\n<div id=\"fs-id1165137838195\">\n<p id=\"fs-id1165137838198\">[latex]r=\\frac{1}{4\\mathrm{cos}\\,\\theta -3\\mathrm{sin}\\,\\theta }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135381198\">\n<div id=\"fs-id1165135381200\">\n<p id=\"fs-id1165135381203\">[latex]r=\\frac{3}{\\mathrm{cos}\\,\\theta -5\\mathrm{sin}\\,\\theta }[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133318753\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133318753\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133318753\"]\n<p id=\"fs-id1165133318755\">[latex]x-5y=3;\\,[\/latex]line<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135202388\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165135202394\">For the following exercises, find the polar coordinates of the point.<\/p>\n\n<div id=\"fs-id1165135202397\">\n<div id=\"fs-id1165135202399\"><span id=\"fs-id1165133267806\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150215\/CNX_Precalc_Figure_08_03_201n.jpg\" alt=\"Polar coordinate system with a point located on the third concentric circle and pi\/2.\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134108566\">\n<div id=\"fs-id1165134108568\"><span id=\"fs-id1165134108574\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150230\/CNX_Precalc_Figure_08_03_202n.jpg\" alt=\"Polar coordinate system with a point located on the third concentric circle and midway between pi\/2 and pi in the second quadrant.\"><\/span><\/div>\n<div id=\"fs-id1165135203456\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135203456\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135203456\"]\n<p id=\"fs-id1165135203458\">[latex]\\left(3,\\frac{3\\pi }{4}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137900018\">\n<div id=\"fs-id1165137900019\"><span id=\"fs-id1165135445695\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150232\/CNX_Precalc_Figure_08_03_203n.jpg\" alt=\"Polar coordinate system with a point located midway between the first and second concentric circles and a third of the way between pi and 3pi\/2 (closer to pi).\"><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135397089\">\n<div id=\"fs-id1165135397091\"><span id=\"fs-id1165135397098\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150243\/CNX_Precalc_Figure_08_03_204n.jpg\" alt=\"Polar coordinate system with a point located on the fifth concentric circle and pi.\"><\/span><\/div>\n<div id=\"fs-id1165137445164\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137445164\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137445164\"]\n<p id=\"fs-id1165137445166\">[latex]\\left(5,\\pi \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137652655\">\n<div id=\"fs-id1165135468162\"><span id=\"fs-id1165135468169\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150257\/CNX_Precalc_Figure_08_03_205n.jpg\" alt=\"Polar coordinate system with a point located on the fourth concentric circle and a third of the way between 3pi\/2 and 2pi (closer to 3pi\/2).\"><\/span><\/div>\n<\/div>\n<p id=\"fs-id1165137554410\">For the following exercises, plot the points.<\/p>\n\n<div id=\"fs-id1165137554413\">\n<div id=\"fs-id1165137554415\">\n<p id=\"fs-id1165137554418\">[latex]\\left(-2,\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137653604\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137653604\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137653604\"]<span id=\"fs-id1165133318711\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150301\/CNX_Precalc_Figure_08_03_206.jpg\" alt=\"Polar coordinate system with a point located on the second concentric circle and two-thirds of the way between pi and 3pi\/2 (closer to 3pi\/2).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135494475\">\n<p id=\"fs-id1165135494477\">[latex]\\left(-1,-\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133178305\">\n<div id=\"fs-id1165133178307\">[latex]\\left(3.5,\\frac{7\\pi }{4}\\right)[\/latex]<\/div>\n<div id=\"fs-id1165137812544\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137812544\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137812544\"]<span id=\"fs-id1165135528493\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150303\/CNX_Precalc_Figure_08_03_208.jpg\" alt=\"Polar coordinate system with a point located midway between the third and fourth concentric circles and midway between 3pi\/2 and 2pi.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137398634\">\n<div id=\"fs-id1165137398637\">\n<p id=\"fs-id1165137398639\">[latex]\\left(-4,\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137862022\">\n<div id=\"fs-id1165137862024\">\n<p id=\"fs-id1165137862026\">[latex]\\left(5,\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134081844\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134081844\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134081844\"]<span id=\"fs-id1165134081850\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150306\/CNX_Precalc_Figure_08_03_210.jpg\" alt=\"Polar coordinate system with a point located on the fifth concentric circle and pi\/2.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165134167348\">\n<div id=\"fs-id1165134167350\">\n<p id=\"fs-id1165134167352\">[latex]\\left(4,\\frac{-5\\pi }{4}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134394439\">\n<div id=\"fs-id1165135648722\">\n<p id=\"fs-id1165135430921\">[latex]\\left(3,\\frac{5\\pi }{6}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137897886\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137897886\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137897886\"]<span id=\"fs-id1165137897892\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150308\/CNX_Precalc_Figure_08_03_212.jpg\" alt=\"Polar coordinate system with a point located on the third concentric circle and 2\/3 of the way between pi\/2 and pi (closer to pi).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137911004\">\n<p id=\"fs-id1165137911006\">[latex]\\left(-1.5,\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133104562\">\n<div id=\"fs-id1165133104564\">\n<p id=\"fs-id1165133104566\">[latex]\\left(-2,\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135414186\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135414186\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135414186\"]<span id=\"fs-id1165135414193\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150313\/CNX_Precalc_Figure_08_03_214.jpg\" alt=\"Polar coordinate system with a point located on the second concentric circle and midway between pi and 3pi\/2.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165134033194\">\n<div id=\"fs-id1165134033196\">\n<p id=\"fs-id1165134033198\">[latex]\\left(1,\\frac{3\\pi }{2}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165134081104\">For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.<\/p>\n\n<div>\n<div>[latex]5x-y=6[\/latex]<\/div>\n<div id=\"fs-id1165133306819\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133306819\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133306819\"]\n<p id=\"fs-id1165133306821\">[latex]r=\\frac{6}{5\\mathrm{cos}\\theta -\\mathrm{sin}\\theta }[\/latex]<\/p>\n<span id=\"fs-id1165134070906\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150317\/CNX_Precalc_Figure_08_03_222.jpg\" alt=\"Plot of given line in the polar coordinate grid\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135559505\">\n<div>\n<p id=\"fs-id1165135559509\">[latex]2x+7y=-3[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135349442\">\n<p id=\"fs-id1165135349444\">[latex]{x}^{2}+{\\left(y-1\\right)}^{2}=1[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137771343\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137771343\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137771343\"]\n<p id=\"fs-id1165137771345\">[latex]r=2\\mathrm{sin}\\theta [\/latex]<\/p>\n<span id=\"fs-id1165135407253\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150319\/CNX_Precalc_Figure_08_03_224.jpg\" alt=\"Plot of given circle in the polar coordinate grid\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134068940\">\n<div id=\"fs-id1165134068943\">\n<p id=\"fs-id1165134068945\">[latex]{\\left(x+2\\right)}^{2}+{\\left(y+3\\right)}^{2}=13[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135299829\">\n<div id=\"fs-id1165135299831\">\n<p id=\"fs-id1165135694348\">[latex]x=2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134279766\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134279766\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134279766\"]\n<p id=\"fs-id1165134279768\">[latex]r=\\frac{2}{\\mathrm{cos}\\theta }[\/latex]<\/p>\n<span id=\"fs-id1165131991391\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150322\/CNX_Precalc_Figure_08_03_226.jpg\" alt=\"Plot of given circle in the polar coordinate grid\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133077985\">\n<div id=\"fs-id1165133077987\">\n<p id=\"fs-id1165134224066\">[latex]{x}^{2}+{y}^{2}=5y[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133388197\">\n<div id=\"fs-id1165133388199\">\n<p id=\"fs-id1165133388201\">[latex]{x}^{2}+{y}^{2}=3x[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134185761\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134185761\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134185761\"]\n<p id=\"fs-id1165134185763\">[latex]r=3\\mathrm{cos}\\theta [\/latex]<\/p>\n<span id=\"fs-id1165137844424\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150328\/CNX_Precalc_Figure_08_03_228.jpg\" alt=\"Plot of given circle in the polar coordinate grid.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135547602\">For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.<\/p>\n\n<div id=\"fs-id1165135547607\">\n<div id=\"fs-id1165135547609\">\n<p id=\"fs-id1165135547611\">[latex]r=6[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135634353\">\n<div id=\"fs-id1165135634355\">\n<p id=\"fs-id1165135634357\">[latex]r=-4[\/latex]<\/p>\n\n<\/div>\n<div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1165134212018\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134212018\"]\n<p id=\"fs-id1165134212018\">[latex]{x}^{2}+{y}^{2}=16[\/latex]<\/p>\n<span id=\"fs-id1165137888093\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150344\/CNX_Precalc_Figure_08_03_230.jpg\" alt=\"Plot of circle with radius 4 centered at the origin in the rectangular coordinates grid.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137888104\">\n<div>[latex]\\theta =-\\frac{2\\pi }{3}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135518106\">\n<div id=\"fs-id1165135518108\">[latex]\\theta =\\frac{\\pi }{4}[\/latex]<\/div>\n<div id=\"fs-id1165133092668\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133092668\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133092668\"]\n<p id=\"fs-id1165133092670\">[latex]y=x[\/latex]<\/p>\n<span id=\"fs-id1165134037559\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150355\/CNX_Precalc_Figure_08_03_232.jpg\" alt=\"Plot of line y=x in the rectangular coordinates grid.\"><\/span>[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135519143\">\n<p id=\"fs-id1165135519145\">[latex]r=\\mathrm{sec}\\,\\theta [\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134192812\">\n<div id=\"fs-id1165134192815\">\n<p id=\"fs-id1165134192817\">[latex]r=-10\\mathrm{sin}\\,\\theta [\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134383713\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134383713\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134383713\"]\n[latex]{x}^{2}+{\\left(y+5\\right)}^{2}=25[\/latex]<span id=\"fs-id1165135255439\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150408\/CNX_Precalc_Figure_08_03_234.jpg\" alt=\"Plot of circle with radius 5 centered at (0,-5).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135452296\">\n<div id=\"fs-id1165135452299\">\n<p id=\"fs-id1165135452301\">[latex]r=3\\mathrm{cos}\\,\\theta [\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4>Technology<\/h4>\n<div id=\"fs-id1165134032293\">\n<div id=\"fs-id1165134032295\">\n\nUse a graphing calculator to find the rectangular coordinates of[latex]\\,\\left(2,-\\frac{\\pi }{5}\\right).\\,[\/latex]Round to the nearest thousandth.\n\n<\/div>\n<div id=\"fs-id1165135597693\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135597693\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135597693\"]\n<p id=\"fs-id1165135597695\">[latex]\\left(1.618,-1.176\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135192524\">\n<div id=\"fs-id1165135192526\">\n\nUse a graphing calculator to find the rectangular coordinates of[latex]\\,\\left(-3,\\frac{3\\pi }{7}\\right).\\,[\/latex]Round to the nearest thousandth.\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135354958\">\n<div id=\"fs-id1165135354960\">\n<p id=\"fs-id1165135354962\">Use a graphing calculator to find the polar coordinates of[latex]\\,\\left(-7,8\\right)\\,[\/latex]in degrees. Round to the nearest thousandth.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135538845\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135538845\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135538845\"]\n<p id=\"fs-id1165135538847\">[latex]\\left(10.630,131.186\u00b0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134391107\">\n<div id=\"fs-id1165137938461\">\n<p id=\"fs-id1165137938463\">Use a graphing calculator to find the polar coordinates of[latex]\\,\\left(3,-4\\right)\\,[\/latex]in degrees. Round to the nearest hundredth.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132973458\">\n<div id=\"fs-id1165132973460\">\n<p id=\"fs-id1165132973463\">Use a graphing calculator to find the polar coordinates of[latex]\\,\\left(-2,0\\right)\\,[\/latex]in radians. Round to the nearest hundredth.<\/p>\n\n<\/div>\n<div id=\"fs-id1165133389118\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133389118\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133389118\"]\n<p id=\"fs-id1165133389121\">[latex]\\,\\left(2,3.14\\right)or\\left(2,\\pi \\right)\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135424580\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165135424585\">\n<div id=\"fs-id1165135424587\">\n<p id=\"fs-id1165135424589\">Describe the graph of[latex]\\,r=a\\mathrm{sec}\\,\\theta ;a&gt;0.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135622466\">\n<div id=\"fs-id1165135622468\">\n<p id=\"fs-id1165135622470\">Describe the graph of[latex]\\,r=a\\mathrm{sec}\\,\\theta ;a&lt;0.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165132912586\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165132912586\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165132912586\"]\n<p id=\"fs-id1165132912588\">A vertical line with[latex]\\,a\\,[\/latex]units left of the <em>y<\/em>-axis.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135618284\">\n<div id=\"fs-id1165135618286\">\n<p id=\"fs-id1165131968582\">Describe the graph of[latex]\\,r=a\\mathrm{csc}\\,\\theta ;a&gt;0.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135538984\">\n<div id=\"fs-id1165135538986\">\n\nDescribe the graph of[latex]\\,r=a\\mathrm{csc}\\,\\theta ;a&lt;0.[\/latex]\n\n<\/div>\n<div id=\"fs-id1165132011254\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165132011254\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165132011254\"]\n<p id=\"fs-id1165132011256\">A horizontal line with[latex]\\,a\\,[\/latex]units below the <em>x<\/em>-axis.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135208493\">\n<div id=\"fs-id1165135208495\">\n<p id=\"fs-id1165131963828\">What polar equations will give an oblique line?<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135208500\">For the following exercise, graph the polar inequality.<\/p>\n\n<div>\n<div id=\"fs-id1165135186214\">\n<p id=\"fs-id1165135186216\">[latex]r&lt;4[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134177579\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134177579\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134177579\"]<span id=\"fs-id1165134177587\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150413\/CNX_Precalc_Figure_08_03_216.jpg\" alt=\"Graph of shaded circle of radius 4 with the edge not included (dotted line) - polar coordinate grid.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137734330\">\n<div id=\"fs-id1165137734332\">\n<p id=\"fs-id1165133310440\">[latex]0\\le \\theta \\le \\frac{\\pi }{4}[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134086016\">\n<div id=\"fs-id1165134086018\">\n<p id=\"fs-id1165134086020\">[latex]\\theta =\\frac{\\pi }{4},\\,r\\,\\ge \\,2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135154583\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135154583\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135154583\"]<span id=\"fs-id1165135154592\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150415\/CNX_Precalc_Figure_08_03_218.jpg\" alt=\"Graph of ray starting at (2, pi\/4) and extending in a positive direction along pi\/4 - polar coordinate grid.\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134378620\">[latex]\\theta =\\frac{\\pi }{4},\\,r\\,\\ge -3[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165131880477\">\n<div id=\"fs-id1165131880479\">\n<p id=\"fs-id1165131880481\">[latex]0\\le \\theta \\le \\frac{\\pi }{3},\\,r\\,&lt;\\,2[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133199370\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133199370\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133199370\"]<span id=\"fs-id1165135641642\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150420\/CNX_Precalc_Figure_08_03_220.jpg\" alt=\"Graph of the shaded region 0 to pi\/3 from r=0 to 2 with the edge not included (dotted line) - polar coordinate grid\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165137643355\">\n<div id=\"fs-id1165137643357\">\n<p id=\"fs-id1165137643360\">[latex]\\frac{-\\pi }{6}&lt;\\theta \\le \\frac{\\pi }{3},-3&lt;r\\,&lt;\\,2[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165135439812\">\n \t<dt>polar axis<\/dt>\n \t<dd id=\"fs-id1165134391590\">on the polar grid, the equivalent of the positive <em>x-<\/em>axis on the rectangular grid<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134391599\">\n \t<dt>polar coordinates<\/dt>\n \t<dd id=\"fs-id1165134041274\">on the polar grid, the coordinates of a point labeled[latex]\\,\\left(r,\\theta \\right),\\,[\/latex]where[latex]\\,\\theta \\,[\/latex]indicates the angle of rotation from the polar axis and[latex]\\,r\\,[\/latex]represents the radius, or the distance of the point from the pole in the direction of[latex]\\,\\theta [\/latex]<\/dd>\n<\/dl>\n<dl>\n \t<dt>pole<\/dt>\n \t<dd id=\"fs-id1165133307606\">the origin of the polar grid<\/dd>\n<\/dl>\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<p>In this section, you will:<\/p>\n<ul>\n<li>Plot points using polar coordinates.<\/li>\n<li>Convert from polar coordinates to rectangular coordinates.<\/li>\n<li>Convert from rectangular coordinates to polar coordinates.<\/li>\n<li>Transform equations between polar and rectangular forms.<\/li>\n<li>Identify and graph polar equations by converting to rectangular equations.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137411387\">Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind (see <a class=\"autogenerated-content\" href=\"#Figure_08_03_001\">(Figure)<\/a>). How can the sailor indicate his location to the Coast Guard? In this section, we will investigate a method of representing location that is different from a standard coordinate grid.<\/p>\n<div id=\"Figure_08_03_001\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150014\/CNX_Precalc_Figure_08_03_001.jpg\" alt=\"An illustration of a boat on the polar grid.\" width=\"487\" height=\"402\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165135551681\" class=\"bc-section section\">\n<h3>Plotting Points Using Polar Coordinates<\/h3>\n<p id=\"fs-id1165135666727\">When we think about plotting points in the plane, we usually think of <span class=\"no-emphasis\">rectangular coordinates<\/span>[latex]\\,\\left(x,y\\right)\\,[\/latex]in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce to polar coordinates, which are points labeled[latex]\\,\\left(r,\\theta \\right)\\,[\/latex]and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.<\/p>\n<p id=\"fs-id1165134166599\">The <span class=\"no-emphasis\">polar grid<\/span> is scaled as the unit circle with the positive <em>x-<\/em>axis now viewed as the polar axis and the origin as the pole. The first coordinate[latex]\\,r\\,[\/latex]is the radius or length of the directed line segment from the pole. The angle[latex]\\,\\theta ,[\/latex] measured in radians, indicates the direction of[latex]\\,r.\\,[\/latex]We move counterclockwise from the polar axis by an angle of[latex]\\,\\theta ,[\/latex]and measure a directed line segment the length of[latex]\\,r\\,[\/latex]in the direction of[latex]\\,\\theta .\\,[\/latex]Even though we measure[latex]\\,\\theta \\,[\/latex]first and then[latex]\\,r,[\/latex] the polar point is written with the <em>r<\/em>-coordinate first. For example, to plot the point[latex]\\,\\left(2,\\frac{\\pi }{4}\\right),[\/latex]we would move[latex]\\,\\frac{\\pi }{4}\\,[\/latex]units in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in <a class=\"autogenerated-content\" href=\"#Figure_08_03_002\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_03_002\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150040\/CNX_Precalc_Figure_08_03_002.jpg\" alt=\"Polar grid with point (2, pi\/4) plotted.\" width=\"487\" height=\"398\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"Example_08_03_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137735842\">\n<div id=\"fs-id1165133141305\">\n<h3>Plotting a Point on the Polar Grid<\/h3>\n<p id=\"fs-id1165135169219\">Plot the point[latex]\\,\\left(3,\\frac{\\pi }{2}\\right)\\,[\/latex]on the polar grid.<\/p>\n<\/div>\n<div id=\"fs-id1165137929939\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137655995\">The angle[latex]\\,\\frac{\\pi }{2}\\,[\/latex]is found by sweeping in a counterclockwise direction 90\u00b0 from the polar axis. The point is located at a length of 3 units from the pole in the[latex]\\,\\frac{\\pi }{2}\\,[\/latex]direction, as shown in <a class=\"autogenerated-content\" href=\"#Figure_08_03_003\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_03_003\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150042\/CNX_Precalc_Figure_08_03_003.jpg\" alt=\"Polar grid with point (3, pi\/2) plotted.\" width=\"487\" height=\"369\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137704675\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_03_01\">\n<div id=\"fs-id1165134225866\">\n<p>Plot the point[latex]\\,\\left(2,\\,\\frac{\\pi }{3}\\right)\\,[\/latex]in the <span class=\"no-emphasis\">polar grid<\/span>.<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150052\/CNX_Precalc_Figure_08_03_004.jpg\" alt=\"Polar grid with point (2, pi\/3) plotted.\" \/><\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137812135\">\n<div>\n<h3>Plotting a Point in the Polar Coordinate System with a Negative Component<\/h3>\n<p id=\"fs-id1165135663316\">Plot the point[latex]\\,\\left(-2,\\,\\frac{\\pi }{6}\\right)\\,[\/latex]on the polar grid.<\/p>\n<\/div>\n<div id=\"fs-id1165137634114\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135386384\">We know that[latex]\\,\\frac{\\pi }{6}\\,[\/latex]is located in the first quadrant. However,[latex]\\,r=-2.\\,[\/latex]We can approach plotting a point with a negative[latex]\\,r\\,[\/latex]in two ways:<\/p>\n<ol id=\"eip-id3158355\" type=\"1\">\n<li>Plot the point[latex]\\,\\left(2,\\frac{\\pi }{6}\\right)\\,[\/latex]by moving[latex]\\,\\frac{\\pi }{6}\\,[\/latex]in the counterclockwise direction and extending a directed line segment 2 units into the first quadrant. Then retrace the directed line segment back through the pole, and continue 2 units into the third quadrant;<\/li>\n<li>Move[latex]\\,\\frac{\\pi }{6}\\,[\/latex]in the counterclockwise direction, and draw the directed line segment from the pole 2 units in the negative direction, into the third quadrant.<\/li>\n<\/ol>\n<p id=\"eip-id3772311\">See <a class=\"autogenerated-content\" href=\"#Figure_08_03_005\">(Figure)<\/a>(a). Compare this to the graph of the polar coordinate[latex]\\,\\left(2,\\frac{\\pi }{6}\\right)\\,[\/latex]shown in <a class=\"autogenerated-content\" href=\"#Figure_08_03_005\">(Figure)<\/a>(b).<\/p>\n<div id=\"Figure_08_03_005\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150055\/CNX_Precalc_Figure_08_03_005.jpg\" alt=\"Two polar grids. Points (2, pi\/6) and (-2, pi\/6) are plotted. They are reflections across the origin in Q1 and Q3.\" width=\"731\" height=\"403\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137723371\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_03_02\">\n<div id=\"fs-id1165137544173\">\n<p id=\"fs-id1165137544174\">Plot the points[latex]\\,\\left(3,-\\frac{\\pi }{6}\\right)[\/latex]and[latex]\\,\\left(2,\\frac{9\\pi }{4}\\right)\\,[\/latex]on the same polar grid.<\/p>\n<\/div>\n<div id=\"fs-id1165137444539\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><img decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150108\/CNX_Precalc_Figure_08_03_006.jpg\" alt=\"Points (2, 9pi\/4) and (3, -pi\/6) are plotted in the polar grid.\" \/><\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134149808\" class=\"bc-section section\">\n<h3>Converting from Polar Coordinates to Rectangular Coordinates<\/h3>\n<p id=\"fs-id1165135313604\">When given a set of <span class=\"no-emphasis\">polar coordinates<\/span>, we may need to convert them to <span class=\"no-emphasis\">rectangular coordinates<\/span>. To do so, we can recall the relationships that exist among the variables[latex]\\,x,\\,y,\\,r,\\,[\/latex]and[latex]\\,\\theta .[\/latex]<\/p>\n<div id=\"fs-id1165135315618\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ \\mathrm{cos}\\,\\theta =\\frac{x}{r}\\to x=r\\mathrm{cos}\\,\\theta \\end{array}\\hfill \\\\ \\mathrm{sin}\\,\\theta =\\frac{y}{r}\\to y=r\\mathrm{sin}\\,\\theta \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137410071\">Dropping a perpendicular from the point in the plane to the <em>x-<\/em>axis forms a right triangle, as illustrated in <a class=\"autogenerated-content\" href=\"#Figure_08_03_007\">(Figure)<\/a>. An easy way to remember the equations above is to think of[latex]\\,\\mathrm{cos}\\,\\theta \\,[\/latex]as the adjacent side over the hypotenuse and[latex]\\,\\mathrm{sin}\\,\\theta \\,[\/latex]as the opposite side over the hypotenuse.<\/p>\n<div id=\"Figure_08_03_007\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150111\/CNX_Precalc_Figure_08_03_007.jpg\" alt=\"Comparison between polar coordinates and rectangular coordinates. There is a right triangle plotted on the x,y axis. The sides are a horizontal line on the x-axis of length x, a vertical line extending from thex-axis to some point in quadrant 1, and a hypotenuse r extending from the origin to that same point in quadrant 1. The vertices are at the origin (0,0), some point along the x-axis at (x,0), and that point in quadrant 1. This last point is (x,y) or (r, theta), depending which system of coordinates you use.\" width=\"487\" height=\"290\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165134105991\" class=\"textbox key-takeaways\">\n<h3>Converting from Polar Coordinates to Rectangular Coordinates<\/h3>\n<p id=\"fs-id1165134230340\">To convert polar coordinates[latex]\\,\\left(r,\\,\\theta \\right)\\,[\/latex]to rectangular coordinates[latex]\\,\\left(x,\\,y\\right),[\/latex] let<\/p>\n<div id=\"fs-id1165135696992\" class=\"unnumbered aligncenter\">[latex]\\mathrm{cos}\\,\\theta =\\frac{x}{r}\\to x=r\\mathrm{cos}\\,\\theta[\/latex]<\/div>\n<div id=\"fs-id1165137461742\" class=\"unnumbered aligncenter\">[latex]\\mathrm{sin}\\,\\theta =\\frac{y}{r}\\to y=r\\mathrm{sin}\\,\\theta[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165131880216\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165133371754\"><strong>Given polar coordinates, convert to rectangular coordinates.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1165133184235\" type=\"1\">\n<li>Given the polar coordinate[latex]\\,\\left(r,\\theta \\right),[\/latex] write[latex]\\,x=r\\mathrm{cos}\\,\\theta \\,[\/latex]and[latex]\\,y=r\\mathrm{sin}\\,\\theta .[\/latex]<\/li>\n<li>Evaluate[latex]\\,\\mathrm{cos}\\,\\theta \\,[\/latex]and[latex]\\,\\mathrm{sin}\\,\\theta .[\/latex]<\/li>\n<li>Multiply[latex]\\,\\mathrm{cos}\\,\\theta \\,[\/latex]by[latex]\\,r\\,[\/latex]to find the <em>x-<\/em>coordinate of the rectangular form.<\/li>\n<li>Multiply[latex]\\,\\mathrm{sin}\\,\\theta \\,[\/latex]by[latex]\\,r\\,[\/latex]to find the <em>y-<\/em>coordinate of the rectangular form.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_08_03_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137585396\">\n<div id=\"fs-id1165137585398\">\n<h3>Writing Polar Coordinates as Rectangular Coordinates<\/h3>\n<p id=\"fs-id1165135546106\">Write the polar coordinates[latex]\\,\\left(3,\\frac{\\pi }{2}\\right)\\,[\/latex]as rectangular coordinates.<\/p>\n<\/div>\n<div id=\"fs-id1165134393743\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>Use the equivalent relationships.<\/p>\n<div id=\"fs-id1165137894480\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ x=r\\mathrm{cos}\\,\\theta \\end{array}\\hfill \\\\ x=3\\mathrm{cos}\\,\\frac{\\pi }{2}=0\\hfill \\\\ y=r\\mathrm{sin}\\,\\theta \\hfill \\\\ y=3\\mathrm{sin}\\,\\frac{\\pi }{2}=3\\hfill \\end{array}[\/latex]<\/div>\n<p>The rectangular coordinates are[latex]\\,\\left(0,3\\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_08_03_008\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150122\/CNX_Precalc_Figure_08_03_008.jpg\" alt=\"Illustration of (3, pi\/2) in polar coordinates and (0,3) in rectangular coordinates - they are the same point!\" width=\"975\" height=\"404\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165132921461\"><span id=\"fs-id1165135536558\"><\/span><\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_04\" class=\"textbox examples\">\n<div id=\"fs-id1165135536607\">\n<div id=\"fs-id1165135536609\">\n<h3>Writing Polar Coordinates as Rectangular Coordinates<\/h3>\n<p id=\"fs-id1165137832785\">Write the polar coordinates[latex]\\,\\left(-2,0\\right)\\,[\/latex]as rectangular coordinates.<\/p>\n<\/div>\n<div id=\"fs-id1165137599665\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137460411\">See <a class=\"autogenerated-content\" href=\"#Figure_08_03_009\">(Figure)<\/a>. Writing the polar coordinates as rectangular, we have<\/p>\n<div id=\"fs-id1165135151267\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}x=r\\mathrm{cos}\\,\\theta \\hfill \\\\ x=-2\\mathrm{cos}\\left(0\\right)=-2\\hfill \\\\ \\hfill \\\\ y=r\\mathrm{sin}\\,\\theta \\hfill \\\\ y=-2\\mathrm{sin}\\left(0\\right)=0\\hfill \\end{array}[\/latex]<\/div>\n<p>The rectangular coordinates are also[latex]\\,\\left(-2,0\\right).[\/latex]<\/p>\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150129\/CNX_Precalc_Figure_08_03_009.jpg\" alt=\"Illustration of (-2, 0) in polar coordinates and (-2,0) in rectangular coordinates - they are the same point!\" width=\"731\" height=\"375\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 7.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165134279361\"><\/details>\n<\/p>\n<div id=\"Figure_08_03_009\" class=\"medium\"><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134489724\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_03_03\">\n<div id=\"fs-id1165135648760\">\n<p id=\"fs-id1165135648761\">Write the polar coordinates[latex]\\,\\left(-1,\\frac{2\\pi }{3}\\right)\\,[\/latex]as rectangular coordinates.<\/p>\n<\/div>\n<div id=\"fs-id1165135470020\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135470021\">[latex]\\left(x,y\\right)=\\left(\\frac{1}{2},-\\frac{\\sqrt{3}}{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134319712\" class=\"bc-section section\">\n<h3>Converting from Rectangular Coordinates to Polar Coordinates<\/h3>\n<p id=\"fs-id1165133359265\">To convert <span class=\"no-emphasis\">rectangular coordinates<\/span> to <span class=\"no-emphasis\">polar coordinates<\/span>, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.<\/p>\n<div id=\"fs-id1165135551669\" class=\"textbox key-takeaways\">\n<h3>Converting from Rectangular Coordinates to Polar Coordinates<\/h3>\n<p id=\"fs-id1165135259493\">Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated in <a class=\"autogenerated-content\" href=\"#Figure_08_03_010\">(Figure)<\/a>.<\/p>\n<div id=\"fs-id1165134230382\">[latex]\\begin{array}{l}\\mathrm{cos}\\,\\theta =\\frac{x}{r}\\,\\,\\text{ or}\\,\\,\\,x=r\\mathrm{cos}\\,\\theta \\hfill \\\\ \\mathrm{sin}\\,\\theta =\\frac{y}{r}\\,\\,\\text{ or}\\,\\,\\,y=r\\mathrm{sin}\\,\\theta \\hfill \\\\ \\,\\,\\,\\,\\,\\,{r}^{2}={x}^{2}+{y}^{2}\\hfill \\\\ \\mathrm{tan}\\,\\theta =\\frac{y}{x}\\,\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"Figure_08_03_010\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter small\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150144\/CNX_Precalc_Figure_08_03_010new.jpg\" alt=\"\" width=\"487\" height=\"298\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 8.<\/strong><\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_05\" class=\"textbox examples\">\n<div id=\"fs-id1165134303364\">\n<div id=\"fs-id1165137589072\">\n<h3>Writing Rectangular Coordinates as Polar Coordinates<\/h3>\n<p id=\"fs-id1165135634043\">Convert the rectangular coordinates[latex]\\,\\left(3,3\\right)\\,[\/latex]to polar coordinates.<\/p>\n<\/div>\n<div id=\"fs-id1165137733557\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137733559\">We see that the original point[latex]\\,\\left(3,3\\right)\\,[\/latex]is in the first quadrant. To find[latex]\\,\\theta ,\\,[\/latex]use the formula[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{y}{x}.\\,[\/latex]This gives<\/p>\n<div id=\"fs-id1165137460645\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\mathrm{tan}\\,\\theta =\\frac{3}{3}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\mathrm{tan}\\,\\theta =1\\hfill \\\\ \\,\\,\\,{\\mathrm{tan}}^{-1}\\left(1\\right)=\\frac{\\pi }{4}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137639677\">To find[latex]\\,r,\\,[\/latex]we substitute the values for[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]into the formula[latex]\\,r=\\sqrt{{x}^{2}+{y}^{2}}.\\,[\/latex]We know that[latex]\\,r\\,[\/latex]must be positive, as[latex]\\,\\frac{\\pi }{4}\\,[\/latex]is in the first quadrant. Thus<\/p>\n<div id=\"fs-id1165135689471\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ r=\\sqrt{{3}^{2}+{3}^{2}}\\end{array}\\hfill \\\\ r=\\sqrt{9+9}\\hfill \\\\ r=\\sqrt{18}=3\\sqrt{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135332735\">So,[latex]\\,r=3\\sqrt{2}\\,\\,[\/latex]and[latex]\\,\\theta \\text{=}\\frac{\\pi }{4},\\,[\/latex]giving us the polar point[latex]\\,\\left(3\\sqrt{2},\\frac{\\pi }{4}\\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_08_03_011\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_03_011\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150158\/CNX_Precalc_Figure_08_03_011.jpg\" alt=\"Illustration of (3rad2, pi\/4) in polar coordinates and (3,3) in rectangular coordinates - they are the same point!\" width=\"975\" height=\"375\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 9.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133078085\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165137574155\">There are other sets of polar coordinates that will be the same as our first solution. For example, the points[latex]\\,\\left(-3\\sqrt{2},\\,\\frac{5\\pi }{4}\\right)\\,[\/latex]and[latex]\\,\\left(3\\sqrt{2},-\\frac{7\\pi }{4}\\right)\\,[\/latex]will coincide with the original solution of[latex]\\,\\left(3\\sqrt{2},\\,\\frac{\\pi }{4}\\right).\\,[\/latex]The point[latex]\\,\\left(-3\\sqrt{2},\\,\\frac{5\\pi }{4}\\right)\\,[\/latex]indicates a move further counterclockwise by[latex]\\,\\pi ,\\,[\/latex]which is directly opposite[latex]\\,\\frac{\\pi }{4}.\\,[\/latex]The radius is expressed as[latex]\\,-3\\sqrt{2}.\\,[\/latex]However, the angle[latex]\\,\\frac{5\\pi }{4}\\,[\/latex]is located in the third quadrant and, as[latex]\\,r\\,[\/latex]is negative, we extend the directed line segment in the opposite direction, into the first quadrant. This is the same point as[latex]\\,\\left(3\\sqrt{2},\\,\\,\\frac{\\pi }{4}\\right).\\,[\/latex]The point[latex]\\,\\left(3\\sqrt{2},\\,-\\frac{7\\pi }{4}\\right)\\,[\/latex]is a move further clockwise by[latex]\\,-\\frac{7\\pi }{4},\\,[\/latex]from[latex]\\,\\frac{\\pi }{4}.\\,[\/latex]The radius,[latex]\\,3\\sqrt{2},\\,[\/latex]is the same.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132957131\" class=\"bc-section section\">\n<h3>Transforming Equations between Polar and Rectangular Forms<\/h3>\n<p id=\"fs-id1165137726540\">We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation.<\/p>\n<div id=\"fs-id1165133356035\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165134422164\"><strong>Given an equation in polar form, graph it using a graphing calculator.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1165137810313\" type=\"1\">\n<li>Change the <strong>MODE<\/strong> to <strong>POL<\/strong>, representing polar form.<\/li>\n<li>Press the <strong>Y= <\/strong>button to bring up a screen allowing the input of six equations:[latex]\\,{r}_{1},\\,\\,{r}_{2},\\,\\,.\\,\\,.\\,\\,.\\,\\,,\\,\\,{r}_{6}.[\/latex]<\/li>\n<li>Enter the polar equation, set equal to[latex]\\,r.[\/latex]<\/li>\n<li>Press <strong>GRAPH<\/strong>.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_08_03_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135485797\">\n<div id=\"fs-id1165135503665\">\n<h3>Writing a Cartesian Equation in Polar Form<\/h3>\n<p id=\"fs-id1165135503671\">Write the Cartesian equation[latex]\\,{x}^{2}+{y}^{2}=9\\,[\/latex]in polar form.<\/p>\n<\/div>\n<div id=\"fs-id1165134418996\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134418998\">The goal is to eliminate[latex]\\,x\\,[\/latex]and[latex]\\,y\\,[\/latex]from the equation and introduce[latex]\\,r\\,[\/latex]and [latex]\\,\\theta .\\,[\/latex]Ideally, we would write the equation[latex]\\,r\\,[\/latex]as a function of[latex]\\,\\theta .\\,[\/latex]To obtain the polar form, we will use the relationships between[latex]\\,\\left(x,y\\right)\\,[\/latex]and[latex]\\,\\left(r,\\theta \\right).\\,[\/latex]Since[latex]\\,x=r\\mathrm{cos}\\,\\theta \\,[\/latex] and[latex]\\,y=r\\mathrm{sin}\\,\\theta ,\\,[\/latex]we can substitute and solve for[latex]\\,r.[\/latex]<\/p>\n<div id=\"fs-id1165134193511\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\text{ }{\\left(r\\mathrm{cos}\\,\\theta \\right)}^{2}+{\\left(r\\mathrm{sin}\\,\\theta \\right)}^{2}=9\\hfill & \\hfill \\\\ \\text{ }{r}^{2}{\\mathrm{cos}}^{2}\\theta +{r}^{2}{\\mathrm{sin}}^{2}\\theta =9\\hfill & \\hfill \\\\ \\text{ }{r}^{2}\\left({\\mathrm{cos}}^{2}\\theta +{\\mathrm{sin}}^{2}\\theta \\right)=9\\hfill & \\hfill \\\\ \\,\\text{ }{r}^{2}\\left(1\\right)=9 \\hfill & {\\text{Substitute cos}}^{2}\\theta +{\\mathrm{sin}}^{2}\\theta =1.\\hfill \\\\ \\text{ }r=\u00b13\\begin{array}{cccc}& & & \\end{array}\\hfill & \\text{Use the square root property}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137757670\">Thus,[latex]\\,{x}^{2}+{y}^{2}=9,r=3,[\/latex]and[latex]\\,r=-3\\,[\/latex]should generate the same graph. See <a class=\"autogenerated-content\" href=\"#Figure_08_03_016\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_03_016\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150200\/CNX_Precalc_Figure_08_03_016.jpg\" alt=\"Plotting a circle of radius 3 with center at the origin in polar and rectangular coordinates. It is the same in both systems.\" width=\"731\" height=\"360\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 10. <\/strong>(a) Cartesian form[latex]\\,{x}^{2}+{y}^{2}=9\\,[\/latex](b) Polar form[latex]\\,r=3[\/latex]<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165133192137\">To graph a circle in rectangular form, we must first solve for[latex]\\,y.[\/latex]<\/p>\n<div id=\"fs-id1165137835799\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\begin{array}{l}\\\\ {x}^{2}+{y}^{2}=9\\end{array}\\hfill \\\\ \\text{ }{y}^{2}=9-{x}^{2}\\hfill \\\\ \\text{ }y=\u00b1\\sqrt{9-{x}^{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135620839\">Note that this is two separate functions, since a circle fails the vertical line test. Therefore, we need to enter the positive and negative square roots into the calculator separately, as two equations in the form[latex]\\,{Y}_{1}=\\sqrt{9-{x}^{2}}\\,[\/latex]and[latex]\\,{Y}_{2}=-\\sqrt{9-{x}^{2}}.\\,[\/latex]Press <strong>GRAPH.<\/strong><\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_07\" class=\"textbox examples\">\n<div id=\"fs-id1165137473589\">\n<div id=\"fs-id1165137473591\">\n<h3>Rewriting a Cartesian Equation as a Polar Equation<\/h3>\n<p id=\"fs-id1165137928681\">Rewrite the <span class=\"no-emphasis\">Cartesian equation<\/span>[latex]\\,{x}^{2}+{y}^{2}=6y\\,[\/latex]as a polar equation.<\/p>\n<\/div>\n<div id=\"fs-id1165135547690\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135547693\">This equation appears similar to the previous example, but it requires different steps to convert the equation.<\/p>\n<p id=\"fs-id1165137698161\">We can still follow the same procedures we have already learned and make the following substitutions:<\/p>\n<div id=\"fs-id1165134168283\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{r}^{2}=6y\\hfill & \\text{Use }{x}^{2}+{y}^{2}={r}^{2}.\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,{r}^{2}=6r\\mathrm{sin}\\,\\theta \\hfill & \\text{Substitute}\\,y=r\\mathrm{sin}\\,\\theta .\\hfill \\\\ \\text{ }{r}^{2}-6r\\mathrm{sin}\\,\\theta =0\\hfill & \\text{Set equal to 0}.\\hfill \\\\ \\text{ }r\\left(r-6\\mathrm{sin}\\,\\theta \\right)=0\\hfill & \\text{Factor and solve}.\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,r=0\\hfill & \\text{We reject }r=0,\\,\\text{as it only represents one point, }\\left(0,0\\right).\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\text{or}\\,r=6\\mathrm{sin}\\,\\theta \\begin{array}{cccc}& & & \\end{array}\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135360250\">Therefore, the equations[latex]\\,{x}^{2}+{y}^{2}=6y\\,[\/latex]and[latex]\\,r=6\\mathrm{sin}\\,\\theta \\,[\/latex] should give us the same graph. See <a class=\"autogenerated-content\" href=\"#Figure_08_03_012\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_03_012\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150203\/CNX_Precalc_Figure_08_03_012.jpg\" alt=\"Plots of the equations stated above - the plots are the same in both rectangular and polar coordinates. They are circles.\" width=\"975\" height=\"328\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 11. <\/strong>(a) Cartesian form[latex]\\,{x}^{2}+{y}^{2}=6y[\/latex](b) polar form[latex]\\,r=6\\mathrm{sin}\\,\\theta [\/latex]<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165135702699\">The Cartesian or <span class=\"no-emphasis\">rectangular equation<\/span> is plotted on the rectangular grid, and the <span class=\"no-emphasis\">polar equation<\/span> is plotted on the polar grid. Clearly, the graphs are identical.<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_08\" class=\"textbox examples\">\n<div id=\"fs-id1165135330933\">\n<div id=\"fs-id1165135330936\">\n<h3>Rewriting a Cartesian Equation in Polar Form<\/h3>\n<p id=\"fs-id1165135547678\">Rewrite the Cartesian equation[latex]\\,y=3x+2\\,[\/latex]as a polar equation.<\/p>\n<\/div>\n<div id=\"fs-id1165135201018\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135394016\">We will use the relationships[latex]\\,x=r\\mathrm{cos}\\,\\theta \\,[\/latex] and [latex]\\,y=r\\mathrm{sin}\\,\\theta .[\/latex]<\/p>\n<div id=\"fs-id1165135514618\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\text{ }y=3x+2\\hfill & \\hfill \\\\ \\text{ }r\\mathrm{sin}\\,\\theta =3r\\mathrm{cos}\\,\\theta +2\\hfill & \\hfill \\\\ \\,r\\mathrm{sin}\\,\\theta -3r\\mathrm{cos}\\,\\theta =2\\hfill & \\hfill \\\\ r\\left(\\mathrm{sin}\\,\\theta -3\\mathrm{cos}\\,\\theta \\right)=2\\hfill & \\text{Isolate }r.\\hfill \\\\ \\text{ }\\,\\,\\,r=\\frac{2}{\\mathrm{sin}\\,\\theta -3\\mathrm{cos}\\,\\theta }\\begin{array}{cccc}& & & \\end{array}\\hfill & \\text{Solve for }r.\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134496398\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_03_04\">\n<div id=\"fs-id1165135413673\">\n<p id=\"fs-id1165135413674\">Rewrite the Cartesian equation[latex]\\,{y}^{2}=3-{x}^{2}\\,[\/latex]in polar form.<\/p>\n<\/div>\n<div id=\"fs-id1165135564036\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135564037\">[latex]r=\\sqrt{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137422423\" class=\"bc-section section\">\n<h3>Identify and Graph Polar Equations by Converting to Rectangular Equations<\/h3>\n<p id=\"fs-id1165137735564\">We have learned how to convert rectangular coordinates to polar coordinates, and we have seen that the points are indeed the same. We have also transformed polar equations to rectangular equations and vice versa. Now we will demonstrate that their graphs, while drawn on different grids, are identical.<\/p>\n<div id=\"Example_08_03_09\" class=\"textbox examples\">\n<div id=\"fs-id1165134433326\">\n<div id=\"fs-id1165135512717\">\n<h3>Graphing a Polar Equation by Converting to a Rectangular Equation<\/h3>\n<p id=\"fs-id1165135512722\">Covert the polar equation[latex]\\,r=2\\mathrm{sec}\\,\\theta \\,[\/latex] to a rectangular equation, and draw its corresponding graph.<\/p>\n<\/div>\n<div id=\"fs-id1165135580397\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135580399\">The conversion is<\/p>\n<div id=\"fs-id1165137697050\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\text{ }r=2\\mathrm{sec}\\,\\theta \\hfill \\\\ \\,\\,\\,\\,\\text{ }r=\\frac{2}{\\mathrm{cos}\\,\\theta }\\hfill \\\\ r\\mathrm{cos}\\,\\theta =2\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x=2\\hfill \\end{array}[\/latex]<\/div>\n<p>Notice that the equation[latex]\\,r=2\\mathrm{sec}\\,\\theta \\,[\/latex]drawn on the polar grid is clearly the same as the vertical line[latex]\\,x=2\\,[\/latex]drawn on the rectangular grid (see <a class=\"autogenerated-content\" href=\"#Figure_08_03_013\">(Figure)<\/a>). Just as[latex]\\,x=c\\,[\/latex]is the standard form for a vertical line in rectangular form,[latex]\\,r=c\\mathrm{sec}\\,\\theta \\,[\/latex]is the standard form for a vertical line in polar form.<\/p>\n<div id=\"Figure_08_03_013\" class=\"medium\">\n<figure style=\"width: 731px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150210\/CNX_Precalc_Figure_08_03_013.jpg\" alt=\"Plots of the equations stated above - the plots are the same in both rectangular and polar coordinates. They are lines.\" width=\"731\" height=\"408\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 12. <\/strong>(a) Polar grid (b) Rectangular coordinate system<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165135689446\">A similar discussion would demonstrate that the graph of the function[latex]\\,r=2\\mathrm{csc}\\,\\theta \\,[\/latex] will be the horizontal line[latex]\\,y=2.\\,[\/latex]In fact,[latex]\\,r=c\\mathrm{csc}\\,\\theta \\,[\/latex] is the standard form for a horizontal line in polar form, corresponding to the rectangular form[latex]\\,y=c.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_10\" class=\"textbox examples\">\n<div id=\"fs-id1165137532816\">\n<div id=\"fs-id1165137442988\">\n<h3>Rewriting a Polar Equation in Cartesian Form<\/h3>\n<p id=\"fs-id1165137442993\">Rewrite the polar equation[latex]\\,r=\\frac{3}{1-2\\mathrm{cos}\\,\\theta }\\,[\/latex]as a Cartesian equation.<\/p>\n<\/div>\n<div id=\"fs-id1165137435019\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137435021\">The goal is to eliminate[latex]\\,\\theta \\,[\/latex]and[latex]\\,r,[\/latex]and introduce[latex]\\,x\\,[\/latex] and [latex]\\,y.\\,[\/latex]We clear the fraction, and then use substitution. In order to replace[latex]\\,r\\,[\/latex] with [latex]\\,x\\,[\/latex]and[latex]\\,y,[\/latex] we must use the expression[latex]\\,{x}^{2}+{y}^{2}={r}^{2}.[\/latex]<\/p>\n<div id=\"fs-id1165132946700\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{llll}\\text{ }r=\\frac{3}{1-2\\mathrm{cos}\\,\\theta }\\hfill & \\hfill & \\hfill & \\hfill \\\\ r\\left(1-2\\mathrm{cos}\\,\\theta \\right)=3\\hfill & \\hfill & \\hfill & \\hfill \\\\ \\,\\,\\,\\,\\,\\,r\\left(1-2\\left(\\frac{x}{r}\\right)\\right)=3\\hfill & \\hfill & \\hfill & \\text{Use }\\mathrm{cos}\\,\\theta =\\frac{x}{r}\\text{ to eliminate }\\theta .\\hfill \\\\ \\text{ }\\,\\,\\,r-2x=3\\hfill & \\hfill & \\hfill & \\hfill \\\\ \\text{ }r=3+2x\\hfill & \\hfill & \\hfill & \\text{Isolate }r.\\hfill \\\\ \\text{ }{r}^{2}={\\left(3+2x\\right)}^{2}\\hfill & \\hfill & \\hfill & \\text{Square both sides}.\\hfill \\\\ \\text{ }\\,{x}^{2}+{y}^{2}={\\left(3+2x\\right)}^{2}\\hfill & \\hfill & \\hfill & \\text{Use }{x}^{2}+{y}^{2}={r}^{2}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134368154\">The Cartesian equation is[latex]\\,{x}^{2}+{y}^{2}={\\left(3+2x\\right)}^{2}.\\,[\/latex]However, to graph it, especially using a graphing calculator or computer program, we want to isolate[latex]\\,y.[\/latex]<\/p>\n<div id=\"fs-id1165134101989\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{x}^{2}+{y}^{2}={\\left(3+2x\\right)}^{2}\\hfill \\\\ \\text{ }{y}^{2}={\\left(3+2x\\right)}^{2}-{x}^{2}\\hfill \\\\ \\text{ }y=\u00b1\\sqrt{{\\left(3+2x\\right)}^{2}-{x}^{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134107035\">When our entire equation has been changed from[latex]\\,r\\,[\/latex]and[latex]\\,\\theta \\,[\/latex]to [latex]\\,x\\,[\/latex] and [latex]\\,y,\\,[\/latex]we can stop, unless asked to solve for[latex]\\,y\\,[\/latex]or simplify. See <a class=\"autogenerated-content\" href=\"#Figure_08_03_015\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_03_015\" class=\"wp-caption aligncenter\">\n<figure style=\"width: 975px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150212\/CNX_Precalc_Figure_08_03_015.jpg\" alt=\"Plots of the equations stated above - the plots are the same in both rectangular and polar coordinates. They are hyperbolas.\" width=\"975\" height=\"481\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 13.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165135355014\">The \u201chour-glass\u201d shape of the graph is called a <em>hyperbola<\/em>. Hyperbolas have many interesting geometric features and applications, which we will investigate further in <a class=\"target-chapter\" href=\"\/contents\/3315e118-f58b-48f1-a06e-f03612d38b40\">Analytic Geometry<\/a>.<\/details>\n<\/p>\n<\/div>\n<div id=\"fs-id1165134116776\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1165134116781\">In this example, the right side of the equation can be expanded and the equation simplified further, as shown above. However, the equation cannot be written as a single function in Cartesian form. We may wish to write the rectangular equation in the hyperbola\u2019s standard form. To do this, we can start with the initial equation.<\/p>\n<div id=\"fs-id1165135453309\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\,\\text{ }{x}^{2}+{y}^{2}={\\left(3+2x\\right)}^{2}\\hfill & \\hfill \\\\ \\,\\,\\text{ }{x}^{2}+{y}^{2}-{\\left(3+2x\\right)}^{2}=0\\hfill & \\hfill \\\\ {x}^{2}+{y}^{2}-\\left(9+12x+4{x}^{2}\\right)=0\\hfill & \\hfill \\\\ \\,\\text{ }{x}^{2}+{y}^{2}-9-12x-4{x}^{2}=0\\hfill & \\hfill \\\\ \\,\\text{ }-3{x}^{2}-12x+{y}^{2}=9\\hfill & \\text{Multiply through by }-1.\\hfill \\\\ \\,\\,\\text{ }\\,3{x}^{2}+12x-{y}^{2}=-9\\hfill & \\hfill \\\\ \\,\\,\\text{ }\\,3\\left({x}^{2}+4x+\\,\\,\\,\\,\\,\\,\\,\\,\\,\\right)-{y}^{2}=-9\\hfill & \\text{Organize terms to complete the square for}\\,x.\\hfill \\\\ \\,\\,\\text{ }\\,3\\left({x}^{2}+4x+4\\right)-{y}^{2}=-9+12\\hfill & \\hfill \\\\ \\text{ }3{\\left(x+2\\right)}^{2}-{y}^{2}=3\\hfill & \\hfill \\\\ \\,\\text{ }{\\left(x+2\\right)}^{2}-\\frac{{y}^{2}}{3}=1\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133074971\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_03_05\">\n<div id=\"fs-id1165134495101\">\n<p id=\"fs-id1165134495102\">Rewrite the polar equation[latex]\\,r=2\\mathrm{sin}\\,\\theta \\,[\/latex] in Cartesian form.<\/p>\n<\/div>\n<div id=\"fs-id1165135378641\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135378642\">[latex]{x}^{2}+{y}^{2}=2y\\,[\/latex]or, in the standard form for a circle,[latex]\\,{x}^{2}+{\\left(y-1\\right)}^{2}=1[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_03_11\" class=\"textbox examples\">\n<div id=\"fs-id1165135456744\">\n<div id=\"fs-id1165135456746\">\n<h3>Rewriting a Polar Equation in Cartesian Form<\/h3>\n<p id=\"fs-id1165135456752\">Rewrite the polar equation[latex]\\,r=\\mathrm{sin}\\left(2\\theta \\right)\\,[\/latex]in Cartesian form.<\/p>\n<\/div>\n<div id=\"fs-id1165134437217\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1165134437221\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}\\text{ }r=\\mathrm{sin}\\left(2\\theta \\right)\\hfill & \\text{Use the double angle identity for sine}.\\hfill \\\\ \\text{ }r=2\\mathrm{sin}\\,\\theta \\mathrm{cos}\\,\\theta \\begin{array}{cccc}& & & \\end{array}\\hfill & \\text{Use }\\mathrm{cos}\\,\\theta =\\frac{x}{r}\\text{ and }\\mathrm{sin}\\,\\theta =\\frac{y}{r}.\\hfill \\\\ \\text{ }r=2\\left(\\frac{x}{r}\\right)\\left(\\frac{y}{r}\\right)\\hfill & \\text{Simplify}.\\hfill \\\\ \\text{ }r=\\frac{2xy}{{r}^{2}}\\hfill & \\text{ Multiply both sides by }{r}^{2}.\\hfill \\\\ \\text{ }{r}^{3}=2xy\\hfill & \\hfill \\\\ {\\left(\\sqrt{{x}^{2}+{y}^{2}}\\right)}^{3}=2xy\\hfill & \\text{As}\\,{x}^{2}+{y}^{2}={r}^{2},r=\\sqrt{{x}^{2}+{y}^{2}}.\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137697887\">This equation can also be written as<\/p>\n<div id=\"fs-id1165137697890\" class=\"unnumbered aligncenter\">[latex]{\\left({x}^{2}+{y}^{2}\\right)}^{\\frac{3}{2}}=2xy\\,\\text{or}\\,{x}^{2}+{y}^{2}={\\left(2xy\\right)}^{\\frac{2}{3}}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135664937\" class=\"precalculus media\">\n<p id=\"fs-id1165133361920\">Access these online resources for additional instruction and practice with polar coordinates.<\/p>\n<ul id=\"fs-id1165133361923\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/intropolar\">Introduction to Polar Coordinates<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/polarrect\">Comparing Polar and Rectangular Coordinates<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135407353\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"eip-id3984733\" summary=\"..\">\n<tbody>\n<tr>\n<td>Conversion formulas<\/td>\n<td>[latex]\\begin{array}{ll}\\hfill & \\mathrm{cos}\\,\\theta =\\frac{x}{r}\\to x=r\\mathrm{cos}\\,\\theta \\hfill \\\\ \\hfill & \\mathrm{sin}\\,\\theta =\\frac{y}{r}\\to y=r\\mathrm{sin}\\,\\theta \\hfill \\\\ \\hfill & \\,\\,\\,\\,\\,\\,\\,{r}^{2}={x}^{2}+{y}^{2}\\hfill \\\\ \\hfill & \\mathrm{tan}\\,\\theta =\\frac{y}{x}\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1165137707236\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165137707243\">\n<li>The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.<\/li>\n<li>To plot a point in the form[latex]\\,\\left(r,\\theta \\right),\\,\\theta >0,\\,[\/latex]move in a counterclockwise direction from the polar axis by an angle of [latex]\\,\\theta ,\\,[\/latex]and then extend a directed line segment from the pole the length of [latex]\\,r\\,[\/latex] in the direction of [latex]\\,\\theta .\\,[\/latex]If[latex]\\,\\theta \\,[\/latex]is negative, move in a clockwise direction, and extend a directed line segment the length of [latex]\\,r\\,[\/latex] in the direction of[latex]\\,\\theta .[\/latex] See <a class=\"autogenerated-content\" href=\"#Example_08_03_01\">(Figure)<\/a>.<\/li>\n<li>If[latex]\\,r\\,[\/latex]is negative, extend the directed line segment in the opposite direction of[latex]\\,\\theta .\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_03_02\">(Figure)<\/a>.<\/li>\n<li>To convert from polar coordinates to rectangular coordinates, use the formulas[latex]\\,x=r\\mathrm{cos}\\,\\theta \\,[\/latex]and[latex]\\,y=r\\mathrm{sin}\\,\\theta .\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_03_03\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_03_04\">(Figure)<\/a>.<\/li>\n<li>To convert from rectangular coordinates to polar coordinates, use one or more of the formulas:[latex]\\,\\mathrm{cos}\\,\\theta =\\frac{x}{r},\\mathrm{sin}\\,\\theta =\\frac{y}{r},\\mathrm{tan}\\,\\theta =\\frac{y}{x},\\,[\/latex]and[latex]\\,r=\\sqrt{{x}^{2}+{y}^{2}}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_03_05\">(Figure)<\/a>.<\/li>\n<li>Transforming equations between polar and rectangular forms means making the appropriate substitutions based on the available formulas, together with algebraic manipulations. See <a class=\"autogenerated-content\" href=\"#Example_08_03_06\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_08_03_07\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_08_03_08\">(Figure)<\/a>.<\/li>\n<li>Using the appropriate substitutions makes it possible to rewrite a polar equation as a rectangular equation, and then graph it in the rectangular plane. See <a class=\"autogenerated-content\" href=\"#Example_08_03_09\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_08_03_10\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_08_03_11\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165134113788\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165134113791\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165134113796\">\n<div id=\"fs-id1165135481176\">\n<p id=\"fs-id1165135481179\">How are polar coordinates different from rectangular coordinates?<\/p>\n<\/div>\n<div id=\"fs-id1165135481183\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135481185\">For polar coordinates, the point in the plane depends on the angle from the positive <em>x-<\/em>axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137634226\">\n<div id=\"fs-id1165137634229\">\n<p id=\"fs-id1165137634231\">How are the polar axes different from the <em>x<\/em>&#8211; and <em>y<\/em>-axes of the Cartesian plane?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133075606\">\n<div id=\"fs-id1165133075608\">\n<p id=\"fs-id1165134193488\">Explain how polar coordinates are graphed.<\/p>\n<\/div>\n<div id=\"fs-id1165134193492\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134193494\">Determine[latex]\\,\\theta \\,[\/latex]for the point, then move[latex]\\,r\\,[\/latex]units from the pole to plot the point. If[latex]\\,r\\,[\/latex]is negative, move[latex]\\,r\\,[\/latex]units from the pole in the opposite direction but along the same angle. The point is a distance of[latex]\\,r\\,[\/latex]away from the origin at an angle of[latex]\\,\\theta \\,[\/latex]from the polar axis.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135316122\">\n<div id=\"fs-id1165135316124\">\n<p id=\"fs-id1165135316126\">How are the points[latex]\\,\\left(3,\\frac{\\pi }{2}\\right)\\,[\/latex]and[latex]\\,\\left(-3,\\frac{\\pi }{2}\\right)\\,[\/latex]related?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135403191\">\n<div id=\"fs-id1165135403193\">\n<p id=\"fs-id1165135403195\">Explain why the points[latex]\\,\\left(-3,\\frac{\\pi }{2}\\right)\\,[\/latex]and[latex]\\,\\left(3,-\\frac{\\pi }{2}\\right)\\,[\/latex]are the same.<\/p>\n<\/div>\n<div id=\"fs-id1165135319445\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135319448\">The point[latex]\\,\\left(-3,\\frac{\\pi }{2}\\right)\\,[\/latex]has a positive angle but a negative radius and is plotted by moving to an angle of[latex]\\,\\frac{\\pi }{2}\\,[\/latex]and then moving 3 units in the negative direction. This places the point 3 units down the negative <em>y<\/em>-axis. The point[latex]\\,\\left(3,-\\frac{\\pi }{2}\\right)\\,[\/latex]has a negative angle and a positive radius and is plotted by first moving to an angle of[latex]\\,-\\frac{\\pi }{2}\\,[\/latex]and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative <em>y<\/em>-axis.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135329737\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165133143104\">For the following exercises, convert the given polar coordinates to Cartesian coordinates with[latex]\\,r>0\\,[\/latex]and[latex]\\,0\\le \\theta \\le 2\\pi .\\,[\/latex]Remember to consider the quadrant in which the given point is located when determining[latex]\\,\\theta \\,[\/latex]for the point.<\/p>\n<div id=\"fs-id1165131986178\">\n<div id=\"fs-id1165131986180\">\n<p id=\"fs-id1165131986182\">[latex]\\left(7,\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134312184\">\n<div id=\"fs-id1165134312186\">\n<p id=\"fs-id1165134312188\">[latex]\\left(5,\\pi \\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135616318\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135616321\">[latex]\\left(-5,0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134384438\">\n<div id=\"fs-id1165134384440\">[latex]\\left(6,-\\frac{\\pi }{4}\\right)[\/latex]<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134196150\">\n<p id=\"fs-id1165134196152\">[latex]\\left(-3,\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134383794\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134383796\">[latex]\\left(-\\frac{3\\sqrt{3}}{2},-\\frac{3}{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134177068\">\n<div id=\"fs-id1165134177070\">\n<p id=\"fs-id1165134177072\">[latex]\\left(4,\\frac{7\\pi }{4}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165131961716\">For the following exercises, convert the given Cartesian coordinates to polar coordinates with[latex]\\,r>0,\\,\\,0\\le \\theta <2\\pi .\\,[\/latex]Remember to consider the quadrant in which the given point is located.<\/p>\n<div id=\"fs-id1165134312061\">\n<div id=\"fs-id1165134312063\">\n<p id=\"fs-id1165134312065\">[latex]\\left(4,2\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134279740\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134279743\">[latex]\\left(2\\sqrt{5}, 0.464\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137696692\">\n<div id=\"fs-id1165137696694\">[latex]\\left(-4,6\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134113855\">\n<div id=\"fs-id1165134113857\">\n<p id=\"fs-id1165134113859\">[latex]\\left(3,-5\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135407446\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135407448\">[latex]\\left(\\sqrt{34},5.253\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135434706\">\n<div id=\"fs-id1165135434708\">\n<p id=\"fs-id1165134380127\">[latex]\\left(-10,-13\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135390847\">\n<div id=\"fs-id1165135390849\">\n<p id=\"fs-id1165135254614\">[latex]\\left(8,8\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137890658\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137890660\">[latex]\\left(8\\sqrt{2},\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134153314\">For the following exercises, convert the given Cartesian equation to a polar equation.<\/p>\n<div id=\"fs-id1165135390660\">\n<div id=\"fs-id1165135390663\">\n<p id=\"fs-id1165135390665\">[latex]x=3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134237043\">\n<div id=\"fs-id1165134237046\">\n<p id=\"fs-id1165134237048\">[latex]y=4[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134174890\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134174892\">[latex]r=4\\mathrm{csc}\\theta[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137896831\">\n<div id=\"fs-id1165137896834\">\n<p id=\"fs-id1165133277552\">[latex]y=4{x}^{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135339520\">\n<div id=\"fs-id1165135339523\">\n<p id=\"fs-id1165135339525\">[latex]y=2{x}^{4}[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135404270\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135698548\">[latex]r=\\sqrt[3]{\\frac{sin\\theta }{2co{s}^{4}\\theta }}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135538875\">\n<div id=\"fs-id1165135538877\">\n<p id=\"fs-id1165135538879\">[latex]{x}^{2}+{y}^{2}=4y[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134129902\">[latex]{x}^{2}+{y}^{2}=3x[\/latex]<\/div>\n<div id=\"fs-id1165132912574\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165132912576\">[latex]r=3\\mathrm{cos}\\theta[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134279393\">\n<div id=\"fs-id1165134279395\">\n<p id=\"fs-id1165134279397\">[latex]{x}^{2}-{y}^{2}=x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135531341\">\n<div id=\"fs-id1165134047499\">\n<p id=\"fs-id1165134047501\">[latex]{x}^{2}-{y}^{2}=3y[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135512559\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135512561\">[latex]r=\\frac{3\\mathrm{sin}\\theta }{\\mathrm{cos}\\left(2\\theta \\right)}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135397902\">\n<div id=\"fs-id1165135397905\">\n<p id=\"fs-id1165135674058\">[latex]{x}^{2}+{y}^{2}=9[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135680173\">\n<div id=\"fs-id1165135680176\">\n<p id=\"fs-id1165135680178\">[latex]{x}^{2}=9y[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134325175\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134325177\">[latex]r=\\frac{9\\mathrm{sin}\\theta }{{\\mathrm{cos}}^{2}\\theta }[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135570196\">\n<div id=\"fs-id1165135570198\">\n<p id=\"fs-id1165135570200\">[latex]{y}^{2}=9x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134039275\">\n<div id=\"fs-id1165134039278\">\n<p id=\"fs-id1165131895989\">[latex]9xy=1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134116891\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134116893\">[latex]r=\\sqrt{\\frac{1}{9\\mathrm{cos}\\theta \\mathrm{sin}\\theta }}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165137619903\">For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.<\/p>\n<div id=\"fs-id1165135639210\">\n<div id=\"fs-id1165135639212\">\n<p id=\"fs-id1165135639214\">[latex]r=3\\mathrm{sin}\\,\\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137590001\">\n<div id=\"fs-id1165137590003\">\n<p id=\"fs-id1165137590005\">[latex]r=4\\mathrm{cos}\\,\\theta[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137810361\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137810363\">[latex]{x}^{2}+{y}^{2}=4x\\,[\/latex]or[latex]\\,\\frac{{\\left(x-2\\right)}^{2}}{4}+\\frac{{y}^{2}}{4}=1;[\/latex]circle<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134237180\">\n<p id=\"fs-id1165134081878\">[latex]r=\\frac{4}{\\mathrm{sin}\\,\\theta +7\\mathrm{cos}\\,\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131893786\">\n<div id=\"fs-id1165131893788\">\n<p id=\"fs-id1165131893790\">[latex]r=\\frac{6}{\\mathrm{cos}\\,\\theta +3\\mathrm{sin}\\,\\theta }[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134393823\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134393825\">[latex]3y+x=6;\\,[\/latex]line<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135351455\">\n<div id=\"fs-id1165135351457\">\n<p id=\"fs-id1165137731259\">[latex]r=2\\mathrm{sec}\\,\\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133035956\">\n<div id=\"fs-id1165133035958\">\n<p id=\"fs-id1165133035961\">[latex]r=3\\mathrm{csc}\\,\\theta[\/latex]<\/p>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id1165133035956\">\n<div>\n<p id=\"fs-id1165137628658\">[latex]y=3;\\,[\/latex]line<\/p>\n<\/div>\n<\/div>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135665421\">\n<div id=\"fs-id1165135665423\">\n<p id=\"fs-id1165135665425\">[latex]r=\\sqrt{r\\mathrm{cos}\\,\\theta +2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134418774\">\n<div id=\"fs-id1165134418776\">\n<p id=\"fs-id1165134418778\">[latex]{r}^{2}=4\\mathrm{sec}\\,\\theta \\,\\mathrm{csc}\\,\\theta[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135555427\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137642992\">[latex]xy=4;\\,[\/latex]hyperbola<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134356832\">\n<div id=\"fs-id1165135546060\">[latex]r=4[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165134358530\">\n<div id=\"fs-id1165134358532\">\n<p id=\"fs-id1165134358534\">[latex]{r}^{2}=4[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135702647\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135702649\">[latex]{x}^{2}+{y}^{2}=4;\\,[\/latex]circle<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137838193\">\n<div id=\"fs-id1165137838195\">\n<p id=\"fs-id1165137838198\">[latex]r=\\frac{1}{4\\mathrm{cos}\\,\\theta -3\\mathrm{sin}\\,\\theta }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135381198\">\n<div id=\"fs-id1165135381200\">\n<p id=\"fs-id1165135381203\">[latex]r=\\frac{3}{\\mathrm{cos}\\,\\theta -5\\mathrm{sin}\\,\\theta }[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133318753\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133318755\">[latex]x-5y=3;\\,[\/latex]line<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135202388\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165135202394\">For the following exercises, find the polar coordinates of the point.<\/p>\n<div id=\"fs-id1165135202397\">\n<div id=\"fs-id1165135202399\"><span id=\"fs-id1165133267806\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150215\/CNX_Precalc_Figure_08_03_201n.jpg\" alt=\"Polar coordinate system with a point located on the third concentric circle and pi\/2.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165134108566\">\n<div id=\"fs-id1165134108568\"><span id=\"fs-id1165134108574\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150230\/CNX_Precalc_Figure_08_03_202n.jpg\" alt=\"Polar coordinate system with a point located on the third concentric circle and midway between pi\/2 and pi in the second quadrant.\" \/><\/span><\/div>\n<div id=\"fs-id1165135203456\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135203458\">[latex]\\left(3,\\frac{3\\pi }{4}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137900018\">\n<div id=\"fs-id1165137900019\"><span id=\"fs-id1165135445695\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150232\/CNX_Precalc_Figure_08_03_203n.jpg\" alt=\"Polar coordinate system with a point located midway between the first and second concentric circles and a third of the way between pi and 3pi\/2 (closer to pi).\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1165135397089\">\n<div id=\"fs-id1165135397091\"><span id=\"fs-id1165135397098\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150243\/CNX_Precalc_Figure_08_03_204n.jpg\" alt=\"Polar coordinate system with a point located on the fifth concentric circle and pi.\" \/><\/span><\/div>\n<div id=\"fs-id1165137445164\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137445166\">[latex]\\left(5,\\pi \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137652655\">\n<div id=\"fs-id1165135468162\"><span id=\"fs-id1165135468169\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150257\/CNX_Precalc_Figure_08_03_205n.jpg\" alt=\"Polar coordinate system with a point located on the fourth concentric circle and a third of the way between 3pi\/2 and 2pi (closer to 3pi\/2).\" \/><\/span><\/div>\n<\/div>\n<p id=\"fs-id1165137554410\">For the following exercises, plot the points.<\/p>\n<div id=\"fs-id1165137554413\">\n<div id=\"fs-id1165137554415\">\n<p id=\"fs-id1165137554418\">[latex]\\left(-2,\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137653604\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165133318711\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150301\/CNX_Precalc_Figure_08_03_206.jpg\" alt=\"Polar coordinate system with a point located on the second concentric circle and two-thirds of the way between pi and 3pi\/2 (closer to 3pi\/2).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135494475\">\n<p id=\"fs-id1165135494477\">[latex]\\left(-1,-\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133178305\">\n<div id=\"fs-id1165133178307\">[latex]\\left(3.5,\\frac{7\\pi }{4}\\right)[\/latex]<\/div>\n<div id=\"fs-id1165137812544\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135528493\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150303\/CNX_Precalc_Figure_08_03_208.jpg\" alt=\"Polar coordinate system with a point located midway between the third and fourth concentric circles and midway between 3pi\/2 and 2pi.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137398634\">\n<div id=\"fs-id1165137398637\">\n<p id=\"fs-id1165137398639\">[latex]\\left(-4,\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137862022\">\n<div id=\"fs-id1165137862024\">\n<p id=\"fs-id1165137862026\">[latex]\\left(5,\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134081844\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165134081850\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150306\/CNX_Precalc_Figure_08_03_210.jpg\" alt=\"Polar coordinate system with a point located on the fifth concentric circle and pi\/2.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134167348\">\n<div id=\"fs-id1165134167350\">\n<p id=\"fs-id1165134167352\">[latex]\\left(4,\\frac{-5\\pi }{4}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134394439\">\n<div id=\"fs-id1165135648722\">\n<p id=\"fs-id1165135430921\">[latex]\\left(3,\\frac{5\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137897886\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137897892\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150308\/CNX_Precalc_Figure_08_03_212.jpg\" alt=\"Polar coordinate system with a point located on the third concentric circle and 2\/3 of the way between pi\/2 and pi (closer to pi).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165137911004\">\n<p id=\"fs-id1165137911006\">[latex]\\left(-1.5,\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133104562\">\n<div id=\"fs-id1165133104564\">\n<p id=\"fs-id1165133104566\">[latex]\\left(-2,\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135414186\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135414193\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150313\/CNX_Precalc_Figure_08_03_214.jpg\" alt=\"Polar coordinate system with a point located on the second concentric circle and midway between pi and 3pi\/2.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134033194\">\n<div id=\"fs-id1165134033196\">\n<p id=\"fs-id1165134033198\">[latex]\\left(1,\\frac{3\\pi }{2}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134081104\">For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.<\/p>\n<div>\n<div>[latex]5x-y=6[\/latex]<\/div>\n<div id=\"fs-id1165133306819\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133306821\">[latex]r=\\frac{6}{5\\mathrm{cos}\\theta -\\mathrm{sin}\\theta }[\/latex]<\/p>\n<p><span id=\"fs-id1165134070906\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150317\/CNX_Precalc_Figure_08_03_222.jpg\" alt=\"Plot of given line in the polar coordinate grid\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135559505\">\n<div>\n<p id=\"fs-id1165135559509\">[latex]2x+7y=-3[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135349442\">\n<p id=\"fs-id1165135349444\">[latex]{x}^{2}+{\\left(y-1\\right)}^{2}=1[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137771343\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137771345\">[latex]r=2\\mathrm{sin}\\theta[\/latex]<\/p>\n<p><span id=\"fs-id1165135407253\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150319\/CNX_Precalc_Figure_08_03_224.jpg\" alt=\"Plot of given circle in the polar coordinate grid\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134068940\">\n<div id=\"fs-id1165134068943\">\n<p id=\"fs-id1165134068945\">[latex]{\\left(x+2\\right)}^{2}+{\\left(y+3\\right)}^{2}=13[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135299829\">\n<div id=\"fs-id1165135299831\">\n<p id=\"fs-id1165135694348\">[latex]x=2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134279766\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134279768\">[latex]r=\\frac{2}{\\mathrm{cos}\\theta }[\/latex]<\/p>\n<p><span id=\"fs-id1165131991391\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150322\/CNX_Precalc_Figure_08_03_226.jpg\" alt=\"Plot of given circle in the polar coordinate grid\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133077985\">\n<div id=\"fs-id1165133077987\">\n<p id=\"fs-id1165134224066\">[latex]{x}^{2}+{y}^{2}=5y[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133388197\">\n<div id=\"fs-id1165133388199\">\n<p id=\"fs-id1165133388201\">[latex]{x}^{2}+{y}^{2}=3x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134185761\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134185763\">[latex]r=3\\mathrm{cos}\\theta[\/latex]<\/p>\n<p><span id=\"fs-id1165137844424\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150328\/CNX_Precalc_Figure_08_03_228.jpg\" alt=\"Plot of given circle in the polar coordinate grid.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135547602\">For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.<\/p>\n<div id=\"fs-id1165135547607\">\n<div id=\"fs-id1165135547609\">\n<p id=\"fs-id1165135547611\">[latex]r=6[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135634353\">\n<div id=\"fs-id1165135634355\">\n<p id=\"fs-id1165135634357\">[latex]r=-4[\/latex]<\/p>\n<\/div>\n<div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134212018\">[latex]{x}^{2}+{y}^{2}=16[\/latex]<\/p>\n<p><span id=\"fs-id1165137888093\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150344\/CNX_Precalc_Figure_08_03_230.jpg\" alt=\"Plot of circle with radius 4 centered at the origin in the rectangular coordinates grid.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137888104\">\n<div>[latex]\\theta =-\\frac{2\\pi }{3}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165135518106\">\n<div id=\"fs-id1165135518108\">[latex]\\theta =\\frac{\\pi }{4}[\/latex]<\/div>\n<div id=\"fs-id1165133092668\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133092670\">[latex]y=x[\/latex]<\/p>\n<p><span id=\"fs-id1165134037559\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150355\/CNX_Precalc_Figure_08_03_232.jpg\" alt=\"Plot of line y=x in the rectangular coordinates grid.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165135519143\">\n<p id=\"fs-id1165135519145\">[latex]r=\\mathrm{sec}\\,\\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134192812\">\n<div id=\"fs-id1165134192815\">\n<p id=\"fs-id1165134192817\">[latex]r=-10\\mathrm{sin}\\,\\theta[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134383713\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>[latex]{x}^{2}+{\\left(y+5\\right)}^{2}=25[\/latex]<span id=\"fs-id1165135255439\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150408\/CNX_Precalc_Figure_08_03_234.jpg\" alt=\"Plot of circle with radius 5 centered at (0,-5).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135452296\">\n<div id=\"fs-id1165135452299\">\n<p id=\"fs-id1165135452301\">[latex]r=3\\mathrm{cos}\\,\\theta[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h4>Technology<\/h4>\n<div id=\"fs-id1165134032293\">\n<div id=\"fs-id1165134032295\">\n<p>Use a graphing calculator to find the rectangular coordinates of[latex]\\,\\left(2,-\\frac{\\pi }{5}\\right).\\,[\/latex]Round to the nearest thousandth.<\/p>\n<\/div>\n<div id=\"fs-id1165135597693\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135597695\">[latex]\\left(1.618,-1.176\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135192524\">\n<div id=\"fs-id1165135192526\">\n<p>Use a graphing calculator to find the rectangular coordinates of[latex]\\,\\left(-3,\\frac{3\\pi }{7}\\right).\\,[\/latex]Round to the nearest thousandth.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135354958\">\n<div id=\"fs-id1165135354960\">\n<p id=\"fs-id1165135354962\">Use a graphing calculator to find the polar coordinates of[latex]\\,\\left(-7,8\\right)\\,[\/latex]in degrees. Round to the nearest thousandth.<\/p>\n<\/div>\n<div id=\"fs-id1165135538845\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135538847\">[latex]\\left(10.630,131.186\u00b0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134391107\">\n<div id=\"fs-id1165137938461\">\n<p id=\"fs-id1165137938463\">Use a graphing calculator to find the polar coordinates of[latex]\\,\\left(3,-4\\right)\\,[\/latex]in degrees. Round to the nearest hundredth.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132973458\">\n<div id=\"fs-id1165132973460\">\n<p id=\"fs-id1165132973463\">Use a graphing calculator to find the polar coordinates of[latex]\\,\\left(-2,0\\right)\\,[\/latex]in radians. Round to the nearest hundredth.<\/p>\n<\/div>\n<div id=\"fs-id1165133389118\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133389121\">[latex]\\,\\left(2,3.14\\right)or\\left(2,\\pi \\right)\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135424580\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<div id=\"fs-id1165135424585\">\n<div id=\"fs-id1165135424587\">\n<p id=\"fs-id1165135424589\">Describe the graph of[latex]\\,r=a\\mathrm{sec}\\,\\theta ;a>0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135622466\">\n<div id=\"fs-id1165135622468\">\n<p id=\"fs-id1165135622470\">Describe the graph of[latex]\\,r=a\\mathrm{sec}\\,\\theta ;a<0.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165132912586\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165132912588\">A vertical line with[latex]\\,a\\,[\/latex]units left of the <em>y<\/em>-axis.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135618284\">\n<div id=\"fs-id1165135618286\">\n<p id=\"fs-id1165131968582\">Describe the graph of[latex]\\,r=a\\mathrm{csc}\\,\\theta ;a>0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135538984\">\n<div id=\"fs-id1165135538986\">\n<p>Describe the graph of[latex]\\,r=a\\mathrm{csc}\\,\\theta ;a<0.[\/latex]\n\n<\/div>\n<div id=\"fs-id1165132011254\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165132011256\">A horizontal line with[latex]\\,a\\,[\/latex]units below the <em>x<\/em>-axis.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135208493\">\n<div id=\"fs-id1165135208495\">\n<p id=\"fs-id1165131963828\">What polar equations will give an oblique line?<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135208500\">For the following exercise, graph the polar inequality.<\/p>\n<div>\n<div id=\"fs-id1165135186214\">\n<p id=\"fs-id1165135186216\">[latex]r<4[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134177579\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165134177587\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150413\/CNX_Precalc_Figure_08_03_216.jpg\" alt=\"Graph of shaded circle of radius 4 with the edge not included (dotted line) - polar coordinate grid.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137734330\">\n<div id=\"fs-id1165137734332\">\n<p id=\"fs-id1165133310440\">[latex]0\\le \\theta \\le \\frac{\\pi }{4}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134086016\">\n<div id=\"fs-id1165134086018\">\n<p id=\"fs-id1165134086020\">[latex]\\theta =\\frac{\\pi }{4},\\,r\\,\\ge \\,2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135154583\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135154592\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150415\/CNX_Precalc_Figure_08_03_218.jpg\" alt=\"Graph of ray starting at (2, pi\/4) and extending in a positive direction along pi\/4 - polar coordinate grid.\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1165134378620\">[latex]\\theta =\\frac{\\pi }{4},\\,r\\,\\ge -3[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165131880477\">\n<div id=\"fs-id1165131880479\">\n<p id=\"fs-id1165131880481\">[latex]0\\le \\theta \\le \\frac{\\pi }{3},\\,r\\,<\\,2[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133199370\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135641642\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19150420\/CNX_Precalc_Figure_08_03_220.jpg\" alt=\"Graph of the shaded region 0 to pi\/3 from r=0 to 2 with the edge not included (dotted line) - polar coordinate grid\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137643355\">\n<div id=\"fs-id1165137643357\">\n<p id=\"fs-id1165137643360\">[latex]\\frac{-\\pi }{6}<\\theta \\le \\frac{\\pi }{3},-3<r\\,<\\,2[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165135439812\">\n<dt>polar axis<\/dt>\n<dd id=\"fs-id1165134391590\">on the polar grid, the equivalent of the positive <em>x-<\/em>axis on the rectangular grid<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134391599\">\n<dt>polar coordinates<\/dt>\n<dd id=\"fs-id1165134041274\">on the polar grid, the coordinates of a point labeled[latex]\\,\\left(r,\\theta \\right),\\,[\/latex]where[latex]\\,\\theta \\,[\/latex]indicates the angle of rotation from the polar axis and[latex]\\,r\\,[\/latex]represents the radius, or the distance of the point from the pole in the direction of[latex]\\,\\theta[\/latex]<\/dd>\n<\/dl>\n<dl>\n<dt>pole<\/dt>\n<dd id=\"fs-id1165133307606\">the origin of the polar grid<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":4,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-154","chapter","type-chapter","status-publish","hentry"],"part":147,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/154","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/users\/291"}],"version-history":[{"count":1,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/154\/revisions"}],"predecessor-version":[{"id":155,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/154\/revisions\/155"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/147"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/154\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=154"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=154"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=154"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=154"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}