{"id":158,"date":"2019-08-20T17:03:29","date_gmt":"2019-08-20T21:03:29","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/polar-form-of-complex-numbers\/"},"modified":"2022-06-01T10:39:34","modified_gmt":"2022-06-01T14:39:34","slug":"polar-form-of-complex-numbers","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/polar-form-of-complex-numbers\/","title":{"raw":"Polar Form of Complex Numbers","rendered":"Polar Form of Complex Numbers"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Plot complex numbers in the complex plane.<\/li>\n \t<li>Find the absolute value of a complex number.<\/li>\n \t<li>Write complex numbers in polar form.<\/li>\n \t<li>Convert a complex number from polar to rectangular form.<\/li>\n \t<li>Find products of complex numbers in polar form.<\/li>\n \t<li>Find quotients of complex numbers in polar form.<\/li>\n \t<li>Find powers of complex numbers in polar form.<\/li>\n \t<li>Find roots of complex numbers in polar form.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137543411\">\u201cGod made the integers; all else is the work of man.\u201d This rather famous quote by nineteenth-century German mathematician Leopold <span class=\"no-emphasis\">Kronecker<\/span> sets the stage for this section on the polar form of a complex number. Complex numbers were invented by people and represent over a thousand years of continuous investigation and struggle by mathematicians such as <span class=\"no-emphasis\">Pythagoras<\/span>, <span class=\"no-emphasis\">Descartes<\/span>, De Moivre, <span class=\"no-emphasis\">Euler<\/span>, <span class=\"no-emphasis\">Gauss<\/span>, and others. Complex numbers answered questions that for centuries had puzzled the greatest minds in science.<\/p>\n<p id=\"fs-id1165134035955\">We first encountered complex numbers in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/complex-numbers\/\">Complex Numbers<\/a>. In this section, we will focus on the mechanics of working with complex numbers: translation of complex numbers from polar form to rectangular form and vice versa, interpretation of complex numbers in the scheme of applications, and application of De Moivre\u2019s Theorem.<\/p>\n\n<div id=\"fs-id1165133355948\" class=\"bc-section section\">\n<h3>Plotting Complex Numbers in the Complex Plane<\/h3>\n<p id=\"fs-id1165134037549\">Plotting a <span class=\"no-emphasis\">complex number<\/span>[latex]\\,a+bi\\,[\/latex]is similar to plotting a real number, except that the horizontal axis represents the real part of the number,[latex]\\,a,\\,[\/latex]and the vertical axis represents the imaginary part of the number,[latex]\\,bi.[\/latex]<\/p>\n\n<div id=\"fs-id1165137938490\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165134079788\"><strong>Given a complex number[latex]\\,a+bi,\\,[\/latex]plot it in the complex plane.<\/strong><\/p>\n\n<ol id=\"fs-id1165133192906\" type=\"1\">\n \t<li>Label the horizontal axis as the <em>real<\/em> axis and the vertical axis as the <em>imaginary axis.<\/em><\/li>\n \t<li>Plot the point in the complex plane by moving[latex]\\,a\\,[\/latex]units in the horizontal direction and[latex]\\,b\\,[\/latex]units in the vertical direction.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_08_05_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137628795\">\n<div id=\"fs-id1165137465062\">\n<h3>Plotting a Complex Number in the Complex Plane<\/h3>\n<p id=\"fs-id1165134331107\">Plot the complex number [latex]\\,2-3i\\,[\/latex]in the <span class=\"no-emphasis\">complex plane<\/span>.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137834397\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137834397\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137834397\"]\n<p id=\"fs-id1165137704829\">From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. See <a class=\"autogenerated-content\" href=\"#Figure_08_05_001\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_05_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\/19152553\/CNX_Precalc_Figure_08_05_001.jpg\" alt=\"Plot of 2-3i in the complex plane (2 along the real axis, -3 along the imaginary axis).\" width=\"487\" height=\"331\"> <strong>Figure 1.<\/strong>[\/caption]\n\n[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135581219\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_01\">\n<div id=\"fs-id1165135586379\">\n<p id=\"fs-id1165135514726\">Plot the point[latex]\\,1+5i\\,[\/latex]in the complex plane.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134167302\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134167302\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134167302\"]<span id=\"fs-id1165137938804\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152606\/CNX_Precalc_Figure_08_05_002-1.jpg\" alt=\"Plot of 1+5i in the complex plane (1 along the real axis, 5 along the imaginary axis).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137542524\" class=\"bc-section section\">\n<h3>Finding the Absolute Value of a Complex Number<\/h3>\n<p id=\"fs-id1165134325179\">The first step toward working with a complex number in <span class=\"no-emphasis\">polar form<\/span> is to find the absolute value. The absolute value of a complex number is the same as its <span class=\"no-emphasis\">magnitude<\/span>, or[latex]\\,|z|.\\,[\/latex]It measures the distance from the origin to a point in the plane. For example, the graph of[latex]\\,z=2+4i,\\,[\/latex]in <a class=\"autogenerated-content\" href=\"#Figure_08_05_003\">(Figure)<\/a>, shows[latex]\\,|z|.[\/latex]<\/p>\n\n<div id=\"Figure_08_05_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\/19152610\/CNX_Precalc_Figure_08_05_003.jpg\" alt=\"Plot of 2+4i in the complex plane and its magnitude, |z| = rad 20.\" width=\"487\" height=\"368\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1165134541127\" class=\"textbox key-takeaways\">\n<h3>Absolute Value of a Complex Number<\/h3>\n<p id=\"fs-id1165133353946\">Given[latex]\\,z=x+yi,\\,[\/latex]a complex number, the absolute value of[latex]\\,z\\,[\/latex]is defined as<\/p>\n\n<div id=\"fs-id1165135340598\" class=\"unnumbered aligncenter\">[latex]|z|=\\sqrt{{x}^{2}+{y}^{2}}[\/latex]<\/div>\n<p id=\"fs-id1165134199373\">It is the distance from the origin to the point[latex]\\,\\left(x,y\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135203416\">Notice that the absolute value of a real number gives the distance of the number from 0, while the absolute value of a complex number gives the distance of the number from the origin,[latex]\\,\\left(0,\\text{ }0\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"Example_08_05_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137921556\">\n<div id=\"fs-id1165135190837\">\n<h3>Finding the Absolute Value of a Complex Number with a Radical<\/h3>\n<p id=\"fs-id1165134232230\">Find the absolute value of[latex]\\,z=\\sqrt{5}-i.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133024224\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133024224\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133024224\"]\n<p id=\"fs-id1165137464603\">Using the formula, we have<\/p>\n\n<div id=\"fs-id1165137602050\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}|z|=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\\\ |z|=\\sqrt{{\\left(\\sqrt{5}\\right)}^{2}+{\\left(-1\\right)}^{2}}\\hfill \\\\ |z|=\\sqrt{5+1}\\hfill \\\\ |z|=\\sqrt{6}\\hfill \\end{array}[\/latex]<\/div>\nSee <a class=\"autogenerated-content\" href=\"#Figure_08_05_004\">(Figure)<\/a>.\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\/19152612\/CNX_Precalc_Figure_08_05_004.jpg\" alt=\"Plot of z=(rad5 - i) in the complex plane and its magnitude rad6.\" width=\"487\" height=\"331\"> <strong>Figure 3.<\/strong>[\/caption]\n<p id=\"fs-id1165133324050\">[\/hidden-answer]<span id=\"fs-id1165135333398\"><\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137461107\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_02\">\n<div id=\"fs-id1165137553007\">\n<p id=\"fs-id1165134401602\">Find the absolute value of the complex number[latex]\\,z=12-5i.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137771102\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137771102\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137771102\"]\n<p id=\"fs-id1165135403265\">13<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_05_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137454990\">\n<div id=\"fs-id1165137645660\">\n<h3>Finding the Absolute Value of a Complex Number<\/h3>\n<p id=\"fs-id1165137662370\">Given[latex]\\,z=3-4i,\\,[\/latex]find[latex]\\,|z|.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137419461\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137419461\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137419461\"]\n<p id=\"fs-id1165133233012\">Using the formula, we have<\/p>\n\n<div id=\"fs-id1165137410180\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}|z|=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\\\ |z|=\\sqrt{{\\left(3\\right)}^{2}+{\\left(-4\\right)}^{2}}\\hfill \\\\ |z|=\\sqrt{9+16}\\hfill \\\\ \\begin{array}{l}|z|=\\sqrt{25}\\\\ |z|=5\\end{array}\\hfill \\end{array}[\/latex]<\/div>\nThe absolute value[latex]\\,z\\,[\/latex]is 5. See <a class=\"autogenerated-content\" href=\"#Figure_08_05_005\">(Figure)<\/a>.\n\n[caption id=\"\" align=\"aligncenter\" width=\"488\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152620\/CNX_Precalc_Figure_08_05_005_Errata.jpg\" alt=\"Plot of (3-4i) in the complex plane and its magnitude |z| =5.\" width=\"488\" height=\"331\"> <strong>Figure 4.<\/strong>[\/caption]\n<p id=\"fs-id1165135553504\">[\/hidden-answer]<span id=\"fs-id1165135194751\"><\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137911592\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_03\">\n<div>\n<p id=\"fs-id1165137812758\">Given[latex]\\,z=1-7i,\\,[\/latex]find[latex]\\,|z|.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137643164\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137643164\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137643164\"]\n<p id=\"fs-id1165135237187\">[latex]|z|=\\sqrt{50}=5\\sqrt{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135530150\" class=\"bc-section section\">\n<h3>Writing Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165132928617\">The polar form of a complex number expresses a number in terms of an angle[latex]\\,\\theta \\,[\/latex]and its distance from the origin[latex]\\,r.\\,[\/latex]Given a complex number in <span class=\"no-emphasis\">rectangular form<\/span> expressed as[latex]\\,z=x+yi,\\,[\/latex]we use the same conversion formulas as we do to write the number in trigonometric form:<\/p>\n\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\,x=r\\mathrm{cos}\\,\\theta \\hfill \\\\ \\,y=r\\mathrm{sin}\\,\\theta \\hfill \\\\ \\,\\,r=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135161485\">We review these relationships in <a class=\"autogenerated-content\" href=\"#Figure_08_05_006\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_08_05_006\" 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\/19152623\/CNX_Precalc_Figure_08_05_006.jpg\" alt=\"Triangle plotted in the complex plane (x axis is real, y axis is imaginary). Base is along the x\/real axis, height is some y\/imaginary value in Q 1, and hypotenuse r extends from origin to that point (x+yi) in Q 1. The angle at the origin is theta. There is an arc going through (x+yi).\" width=\"487\" height=\"331\"> <strong>Figure 5.<\/strong>[\/caption]\n\n<\/div>\nWe use the term <strong>modulus<\/strong> to represent the absolute value of a complex number, or the distance from the origin to the point[latex]\\,\\left(x,y\\right).\\,[\/latex]The modulus, then, is the same as[latex]\\,r,\\,[\/latex]the radius in polar form. We use[latex]\\,\\theta \\,[\/latex]to indicate the angle of direction (just as with polar coordinates). Substituting, we have\n<div id=\"fs-id1165137835830\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}z=x+yi\\hfill \\\\ z=r\\mathrm{cos}\\,\\theta +\\left(r\\mathrm{sin}\\,\\theta \\right)i\\hfill \\\\ z=r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right)\\hfill \\end{array}[\/latex]<\/div>\n<div>\n<h3>Polar Form of a Complex Number<\/h3>\n<p id=\"fs-id1165137643584\">Writing a complex number in polar form involves the following conversion formulas:<\/p>\n\n<div id=\"fs-id1165134566530\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ x=r\\mathrm{cos}\\,\\theta \\hfill \\\\ y=r\\mathrm{sin}\\,\\theta \\hfill \\\\ r=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134073108\">Making a direct substitution, we have<\/p>\n\n<div id=\"fs-id1165135613632\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}z=x+yi\\hfill \\\\ z=\\left(r\\mathrm{cos}\\,\\theta \\right)+i\\left(r\\mathrm{sin}\\,\\theta \\right)\\hfill \\\\ z=r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right)\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137628482\">where[latex]\\,r\\,[\/latex]is the modulus and [latex]\\theta [\/latex] is the argument. We often use the abbreviation[latex]\\,r\\text{cis}\\,\\theta \\,[\/latex]to represent[latex]\\,r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"Example_08_05_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137454742\">\n<div id=\"fs-id1165132959045\">\n<h3>Expressing a Complex Number Using Polar Coordinates<\/h3>\n<p id=\"fs-id1165137804036\">Express the complex number[latex]\\,4i\\,[\/latex]using polar coordinates.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137595455\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137595455\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137595455\"]\n<p id=\"fs-id1165137803541\">On the complex plane, the number[latex]\\,z=4i\\,[\/latex]is the same as[latex]\\,z=0+4i.\\,[\/latex]Writing it in polar form, we have to calculate[latex]\\,r\\,[\/latex]first.<\/p>\n\n<div id=\"fs-id1165137483073\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}r=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\\\ r=\\sqrt{{0}^{2}+{4}^{2}}\\hfill \\\\ r=\\sqrt{16}\\hfill \\\\ r=4\\hfill \\end{array}[\/latex]<\/div>\nNext, we look at[latex]\\,x.\\,[\/latex]If[latex]\\,x=r\\mathrm{cos}\\,\\theta ,\\,[\/latex]and[latex]\\,x=0,\\,[\/latex]then[latex]\\,\\theta =\\frac{\\pi }{2}.\\,[\/latex]In polar coordinates, the complex number[latex]\\,z=0+4i\\,[\/latex]can be written as[latex]\\,z=4\\left(\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)+i\\mathrm{sin}\\left(\\frac{\\pi }{2}\\right)\\right)\\,[\/latex]or[latex]\\,4\\text{cis}\\left(\\,\\frac{\\pi }{2}\\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_08_05_007\">(Figure)<\/a>.\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\/19152626\/CNX_Precalc_Figure_08_05_007.jpg\" alt=\"Plot of z=4i in the complex plane, also shows that the in polar coordinate it would be (4,pi\/2).\" width=\"487\" height=\"294\"> <strong>Figure 6.<\/strong>[\/caption]\n<p id=\"fs-id1165137540200\">[\/hidden-answer]<span id=\"fs-id1165137460482\"><\/span><\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135175110\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_04\">\n<div id=\"fs-id1165134550614\">\n<p id=\"fs-id1165134550616\">Express[latex]\\,z=3i\\,[\/latex] as [latex]\\,r\\,\\text{cis}\\,\\theta \\,[\/latex] in polar form.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137397894\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137397894\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137397894\"]\n<p id=\"fs-id1165137397895\">[latex]z=3\\left(\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)+i\\mathrm{sin}\\left(\\frac{\\pi }{2}\\right)\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_05_05\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137851363\">\n<h3>Finding the Polar Form of a Complex Number<\/h3>\n<p id=\"fs-id1165133097232\">Find the polar form of[latex]\\,-4+4i.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134077346\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134077346\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134077346\"]\n<p id=\"fs-id1165135499750\">First, find the value of[latex]\\,r.[\/latex]<\/p>\n\n<div id=\"fs-id1165137405565\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}r=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\\\ r=\\sqrt{{\\left(-4\\right)}^{2}+\\left({4}^{2}\\right)}\\hfill \\\\ r=\\sqrt{32}\\hfill \\\\ r=4\\sqrt{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165133141427\">Find the angle[latex]\\,\\theta \\,[\/latex]using the formula:<\/p>\n\n<div id=\"fs-id1165133155817\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{cos}\\,\\theta =\\frac{x}{r}\\hfill \\\\ \\mathrm{cos}\\,\\theta =\\frac{-4}{4\\sqrt{2}}\\hfill \\\\ \\mathrm{cos}\\,\\theta =-\\frac{1}{\\sqrt{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\theta ={\\mathrm{cos}}^{-1}\\left(-\\frac{1}{\\sqrt{2}}\\right)=\\frac{3\\pi }{4}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134184438\">Thus, the solution is[latex]\\,4\\sqrt{2}\\text{cis}\\left(\\frac{3\\pi }{4}\\right).[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135475905\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_05\">\n<div id=\"fs-id1165137451880\">\n\nWrite[latex]\\,z=\\sqrt{3}+i\\,[\/latex]in polar form.\n\n<\/div>\n<div id=\"fs-id1165137603699\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137603699\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137603699\"]\n<p id=\"fs-id1165137603700\">[latex]z=2\\left(\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)+i\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135311667\" class=\"bc-section section\">\n<h3>Converting a Complex Number from Polar to Rectangular Form<\/h3>\n<p id=\"fs-id1165135664797\">Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. In other words, given[latex]\\,z=r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right),\\,[\/latex]first evaluate the trigonometric functions[latex]\\,\\mathrm{cos}\\,\\theta \\,[\/latex]and[latex]\\,\\mathrm{sin}\\,\\theta .\\,[\/latex]Then, multiply through by[latex]\\,r.[\/latex]<\/p>\n\n<div id=\"Example_08_05_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135187864\">\n<div id=\"fs-id1165135187866\">\n<h3>Converting from Polar to Rectangular Form<\/h3>\n<p id=\"fs-id1165137732869\">Convert the polar form of the given complex number to rectangular form:<\/p>\n\n<div id=\"fs-id1165137732872\" class=\"unnumbered aligncenter\">[latex]z=12\\left(\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)+i\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137827978\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137827978\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137827978\"]\n<p id=\"fs-id1165134279359\">We begin by evaluating the trigonometric expressions.<\/p>\n\n<div id=\"eip-id811719\" class=\"unnumbered\">[latex]\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)=\\frac{\\sqrt{3}}{2}\\,\\text{and}\\,\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)=\\frac{1}{2}\\,[\/latex]<\/div>\n<p id=\"eip-id1472898\">After substitution, the complex number is<\/p>\n\n<div id=\"fs-id1165137723445\" class=\"unnumbered aligncenter\">[latex]z=12\\left(\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i\\right)[\/latex]<\/div>\n<p id=\"fs-id1165135517158\">We apply the distributive property:<\/p>\n\n<div class=\"unnumbered\">[latex]\\begin{array}{l}z=12\\left(\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i\\right)\\hfill \\\\ \\text{ }=\\left(12\\right)\\frac{\\sqrt{3}}{2}+\\left(12\\right)\\frac{1}{2}i\\hfill \\\\ \\text{ }=6\\sqrt{3}+6i\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137733561\">The rectangular form of the given point in complex form is[latex]\\,6\\sqrt{3}+6i.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_05_07\" class=\"textbox examples\">\n<div id=\"fs-id1165135571758\">\n<div id=\"fs-id1165135571760\">\n<h3>Finding the Rectangular Form of a Complex Number<\/h3>\n<p id=\"fs-id1165137844019\">Find the rectangular form of the complex number given[latex]\\,r=13\\,[\/latex]and[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{5}{12}.[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133210162\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133210162\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133210162\"]\n<p id=\"fs-id1165134537174\">If[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{5}{12},\\,[\/latex]and[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{y}{x},\\,[\/latex]we first determine[latex]\\,r=\\sqrt{{x}^{2}+{y}^{2}}=\\sqrt{{12}^{2}+{5}^{2}}=13\\text{.}[\/latex] We then find[latex]\\,\\mathrm{cos}\\,\\theta =\\frac{x}{r}\\,[\/latex]and[latex]\\,\\mathrm{sin}\\,\\theta =\\frac{y}{r}.[\/latex]<\/p>\n\n<div id=\"fs-id1165137443057\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}z=13\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right)\\hfill \\\\ \\,\\,\\,=13\\left(\\frac{12}{13}+\\frac{5}{13}i\\right)\\hfill \\\\ \\,\\,\\,=12+5i\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135160119\">The rectangular form of the given number in complex form is[latex]\\,12+5i.[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135511380\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_06\">\n<div id=\"fs-id1165137571081\">\n<p id=\"fs-id1165137571082\">Convert the complex number to rectangular form:<\/p>\n\n<div id=\"fs-id1165137571086\" class=\"unnumbered aligncenter\">[latex]z=4\\left(\\mathrm{cos}\\frac{11\\pi }{6}+i\\mathrm{sin}\\frac{11\\pi }{6}\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137444688\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137444688\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137444688\"]\n<p id=\"fs-id1165137444689\">[latex]z=2\\sqrt{3}-2i[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134324933\" class=\"bc-section section\">\n<h3>Finding Products of Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165137660156\">Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. For the rest of this section, we will work with formulas developed by French mathematician Abraham <span class=\"no-emphasis\">de Moivre<\/span> (1667-1754). These formulas have made working with products, quotients, powers, and roots of complex numbers much simpler than they appear. The rules are based on multiplying the moduli and adding the arguments.<\/p>\n\n<div id=\"fs-id1165137473607\" class=\"textbox key-takeaways\">\n<h3>Products of Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165135178402\">If[latex]\\,{z}_{1}={r}_{1}\\left(\\mathrm{cos}\\,{\\theta }_{1}+i\\mathrm{sin}\\,{\\theta }_{1}\\right)\\,[\/latex]and[latex]\\,{z}_{2}={r}_{2}\\left(\\mathrm{cos}\\,{\\theta }_{2}+i\\mathrm{sin}\\,{\\theta }_{2}\\right),[\/latex] then the product of these numbers is given as:<\/p>\n\n<div id=\"fs-id1165137844291\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}{z}_{1}{z}_{2}={r}_{1}{r}_{2}\\left[\\mathrm{cos}\\left({\\theta }_{1}+{\\theta }_{2}\\right)+i\\mathrm{sin}\\left({\\theta }_{1}+{\\theta }_{2}\\right)\\right]\\hfill \\\\ {z}_{1}{z}_{2}={r}_{1}{r}_{2}\\text{cis}\\left({\\theta }_{1}+{\\theta }_{2}\\right)\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137738246\">Notice that the product calls for multiplying the moduli and adding the angles.<\/p>\n\n<\/div>\n<div id=\"Example_08_05_08\" class=\"textbox examples\">\n<div id=\"fs-id1165137665678\">\n<div id=\"fs-id1165137665680\">\n<h3>Finding the Product of Two Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165137583978\">Find the product of[latex]\\,{z}_{1}{z}_{2},\\,[\/latex]given[latex]\\,{z}_{1}=4\\left(\\mathrm{cos}\\left(80\u00b0\\right)+i\\mathrm{sin}\\left(80\u00b0\\right)\\right)\\,[\/latex]and[latex]\\,{z}_{2}=2\\left(\\mathrm{cos}\\left(145\u00b0\\right)+i\\mathrm{sin}\\left(145\u00b0\\right)\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135369384\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135369384\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135369384\"]\n<p id=\"fs-id1165137531259\">Follow the formula<\/p>\n\n<div id=\"fs-id1165137531262\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{z}_{1}{z}_{2}=4\\cdot 2\\left[\\mathrm{cos}\\left(80\u00b0+145\u00b0\\right)+i\\mathrm{sin}\\left(80\u00b0+145\u00b0\\right)\\right]\\hfill \\\\ {z}_{1}{z}_{2}=8\\left[\\mathrm{cos}\\left(225\u00b0\\right)+i\\mathrm{sin}\\left(225\u00b0\\right)\\right]\\hfill \\\\ {z}_{1}{z}_{2}=8\\left[\\mathrm{cos}\\left(\\frac{5\\pi }{4}\\right)+i\\mathrm{sin}\\left(\\frac{5\\pi }{4}\\right)\\right]\\hfill \\\\ {z}_{1}{z}_{2}=8\\left[-\\frac{\\sqrt{2}}{2}+i\\left(-\\frac{\\sqrt{2}}{2}\\right)\\right]\\hfill \\\\ {z}_{1}{z}_{2}=-4\\sqrt{2}-4i\\sqrt{2}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736474\" class=\"bc-section section\">\n<h3>Finding Quotients of Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165134389944\">The quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two arguments.<\/p>\n\n<div id=\"fs-id1165137892433\" class=\"textbox key-takeaways\">\n<h3>Quotients of Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165135541727\">If[latex]\\,{z}_{1}={r}_{1}\\left(\\mathrm{cos}\\,{\\theta }_{1}+i\\mathrm{sin}\\,{\\theta }_{1}\\right)\\,[\/latex]and[latex]\\,{z}_{2}={r}_{2}\\left(\\mathrm{cos}\\,{\\theta }_{2}+i\\mathrm{sin}\\,{\\theta }_{2}\\right),[\/latex] then the quotient of these numbers is<\/p>\n\n<div id=\"fs-id1165135316034\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{{z}_{1}}{{z}_{2}}=\\frac{{r}_{1}}{{r}_{2}}\\left[\\mathrm{cos}\\left({\\theta }_{1}-{\\theta }_{2}\\right)+i\\mathrm{sin}\\left({\\theta }_{1}-{\\theta }_{2}\\right)\\right],\\,\\,{z}_{2}\\ne 0\\\\ \\frac{{z}_{1}}{{z}_{2}}=\\frac{{r}_{1}}{{r}_{2}}\\text{cis}\\left({\\theta }_{1}-{\\theta }_{2}\\right),\\,\\,{z}_{2}\\ne 0\\,\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135332827\">Notice that the moduli are divided, and the angles are subtracted.<\/p>\n\n<\/div>\n<div id=\"fs-id1165137897028\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135189946\"><strong>Given two complex numbers in polar form, find the quotient.\n<\/strong><\/p>\n\n<ol id=\"fs-id1165135260707\" type=\"1\">\n \t<li>Divide[latex]\\,\\frac{{r}_{1}}{{r}_{2}}.[\/latex]<\/li>\n \t<li>Find[latex]\\,{\\theta }_{1}-{\\theta }_{2}.[\/latex]<\/li>\n \t<li>Substitute the results into the formula:[latex]\\,z=r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right).\\,[\/latex]Replace[latex]\\,r\\,[\/latex]with[latex]\\,\\frac{{r}_{1}}{{r}_{2}},\\,[\/latex]and replace[latex]\\,\\theta \\,[\/latex]with[latex]\\,{\\theta }_{1}-{\\theta }_{2}.[\/latex]<\/li>\n \t<li>Calculate the new trigonometric expressions and multiply through by[latex]\\,r.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_08_05_09\" class=\"textbox examples\">\n<div id=\"fs-id1165134329605\">\n<div id=\"fs-id1165134329607\">\n<h3>Finding the Quotient of Two Complex Numbers<\/h3>\n<p id=\"fs-id1165135436548\">Find the quotient of[latex]\\,{z}_{1}=2\\left(\\mathrm{cos}\\left(213\u00b0\\right)+i\\mathrm{sin}\\left(213\u00b0\\right)\\right)\\,[\/latex]and[latex]\\,{z}_{2}=4\\left(\\mathrm{cos}\\left(33\u00b0\\right)+i\\mathrm{sin}\\left(33\u00b0\\right)\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133318744\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133318744\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133318744\"]\n<p id=\"fs-id1165134129652\">Using the formula, we have<\/p>\n\n<div id=\"fs-id1165134129655\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{{z}_{1}}{{z}_{2}}=\\frac{2}{4}\\left[\\mathrm{cos}\\left(213\u00b0-33\u00b0\\right)+i\\mathrm{sin}\\left(213\u00b0-33\u00b0\\right)\\right]\\hfill \\\\ \\frac{{z}_{1}}{{z}_{2}}=\\frac{1}{2}\\left[\\mathrm{cos}\\left(180\u00b0\\right)+i\\mathrm{sin}\\left(180\u00b0\\right)\\right]\\hfill \\\\ \\frac{{z}_{1}}{{z}_{2}}=\\frac{1}{2}\\left[-1+0i\\right]\\hfill \\\\ \\frac{{z}_{1}}{{z}_{2}}=-\\frac{1}{2}+0i\\hfill \\\\ \\frac{{z}_{1}}{{z}_{2}}=-\\frac{1}{2}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135593394\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_07\">\n<div id=\"fs-id1165134568987\">\n<p id=\"fs-id1165134568988\">Find the product and the quotient of[latex]\\,{z}_{1}=2\\sqrt{3}\\left(\\mathrm{cos}\\left(150\u00b0\\right)+i\\mathrm{sin}\\left(150\u00b0\\right)\\right)\\,[\/latex]and[latex]\\,{z}_{2}=2\\left(\\mathrm{cos}\\left(30\u00b0\\right)+i\\mathrm{sin}\\left(30\u00b0\\right)\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135707943\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135707943\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135707943\"]\n<p id=\"fs-id1165135707944\">[latex]\\,{z}_{1}{z}_{2}=-4\\sqrt{3};\\frac{{z}_{1}}{{z}_{2}}=-\\frac{\\sqrt{3}}{2}+\\frac{3}{2}i\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137673550\" class=\"bc-section section\">\n<h3>Finding Powers of Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165137673556\">Finding powers of complex numbers is greatly simplified using De Moivre\u2019s Theorem. It states that, for a positive integer[latex]\\,n,{z}^{n}\\,[\/latex]is found by raising the modulus to the[latex]\\,n\\text{th}\\,[\/latex]power and multiplying the argument by[latex]\\,n.\\,[\/latex]It is the standard method used in modern mathematics.<\/p>\n\n<div id=\"fs-id1165135388434\" class=\"textbox key-takeaways\">\n<h3>De Moivre\u2019s Theorem<\/h3>\n<p id=\"fs-id1165133349328\">If[latex]\\,z=r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right)\\,[\/latex]is a complex number, then<\/p>\n\n<div id=\"fs-id1165135299806\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{z}^{n}={r}^{n}\\left[\\mathrm{cos}\\left(n\\theta \\right)+i\\mathrm{sin}\\left(n\\theta \\right)\\right]\\\\ {z}^{n}={r}^{n}\\text{cis}\\left(n\\theta \\right)\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137846221\">where[latex]\\,n\\,[\/latex]\nis a positive integer.<\/p>\n\n<\/div>\n<div id=\"Example_08_05_10\" class=\"textbox examples\">\n<div id=\"fs-id1165133103896\">\n<div id=\"fs-id1165133103898\">\n<h3>Evaluating an Expression Using De Moivre\u2019s Theorem<\/h3>\n<p id=\"fs-id1165131884600\">Evaluate the expression[latex]\\,{\\left(1+i\\right)}^{5}\\,[\/latex]using De Moivre\u2019s Theorem.<\/p>\n\n<\/div>\n<div id=\"fs-id1165132959150\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165132959150\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165132959150\"]\n<p id=\"fs-id1165132959153\">Since De Moivre\u2019s Theorem applies to complex numbers written in polar form, we must first write[latex]\\,\\left(1+i\\right)\\,[\/latex]in polar form. Let us find[latex]\\,r.[\/latex]<\/p>\n\n<div id=\"fs-id1165135365743\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}r=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\\\ r=\\sqrt{{\\left(1\\right)}^{2}+{\\left(1\\right)}^{2}}\\hfill \\\\ r=\\sqrt{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135200427\">Then we find[latex]\\,\\theta .\\,[\/latex]Using the formula[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{y}{x}\\,[\/latex]gives<\/p>\n\n<div id=\"fs-id1165134130860\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{tan}\\,\\theta =\\frac{1}{1}\\hfill \\\\ \\mathrm{tan}\\,\\theta =1\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\theta =\\frac{\\pi }{4}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135415590\">Use De Moivre\u2019s Theorem to evaluate the expression.<\/p>\n\n<div id=\"fs-id1165135415594\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{\\left(a+bi\\right)}^{n}={r}^{n}\\left[\\mathrm{cos}\\left(n\\theta \\right)+i\\mathrm{sin}\\left(n\\theta \\right)\\right]\\hfill \\\\ \\,\\,\\,\\,{\\left(1+i\\right)}^{5}={\\left(\\sqrt{2}\\right)}^{5}\\left[\\mathrm{cos}\\left(5\\cdot \\frac{\\pi }{4}\\right)+i\\mathrm{sin}\\left(5\\cdot \\frac{\\pi }{4}\\right)\\right]\\hfill \\\\ \\,\\,\\,\\,{\\left(1+i\\right)}^{5}=4\\sqrt{2}\\left[\\mathrm{cos}\\left(\\frac{5\\pi }{4}\\right)+i\\mathrm{sin}\\left(\\frac{5\\pi }{4}\\right)\\right]\\hfill \\\\ \\,\\,\\,\\,{\\left(1+i\\right)}^{5}=4\\sqrt{2}\\left[-\\frac{\\sqrt{2}}{2}+i\\left(-\\frac{\\sqrt{2}}{2}\\right)\\right]\\hfill \\\\ \\,\\,\\,\\,{\\left(1+i\\right)}^{5}=-4-4i\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135204415\" class=\"bc-section section\">\n<h3>Finding Roots of Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165135204421\">To find the <span class=\"no-emphasis\"><em>n<\/em>th root of a complex number<\/span> in polar form, we use the[latex]\\,n\\text{th}\\,[\/latex]Root Theorem or <span class=\"no-emphasis\">De Moivre\u2019s Theorem<\/span> and raise the complex number to a power with a rational exponent. There are several ways to represent a formula for finding[latex]\\,n\\text{th}\\,[\/latex]roots of complex numbers in polar form.<\/p>\n\n<div id=\"fs-id1165135420398\">\n<h3>The <em>n<\/em>th Root Theorem<\/h3>\n<p id=\"fs-id1165135436403\">To find the[latex]\\,n\\text{th}\\,[\/latex]root of a complex number in polar form, use the formula given as<\/p>\n\n<div id=\"fs-id1165134281475\" class=\"unnumbered aligncenter\">[latex]{z}^{\\frac{1}{n}}={r}^{\\frac{1}{n}}\\left[\\mathrm{cos}\\left(\\frac{\\theta }{n}+\\frac{2k\\pi }{n}\\right)+i\\mathrm{sin}\\left(\\frac{\\theta }{n}+\\frac{2k\\pi }{n}\\right)\\right][\/latex]<\/div>\n<p id=\"fs-id1165134485535\">where[latex]\\,k=0,\\,\\,1,\\,\\,2,\\,\\,3,\\,.\\,\\,.\\,\\,.\\,\\,,\\,\\,n-1.\\,[\/latex]We add [latex]\\,\\frac{2k\\pi }{n}\\,\\,[\/latex]to[latex]\\,\\frac{\\theta }{n}\\,[\/latex]in order to obtain the periodic roots.<\/p>\n\n<\/div>\n<div id=\"Example_08_05_11\" class=\"textbox examples\">\n<div id=\"fs-id1165133341021\">\n<div id=\"fs-id1165133341023\">\n<h3>Finding the <em>n<\/em>th Root of a Complex Number<\/h3>\n<p id=\"fs-id1165133046793\">Evaluate the cube roots of[latex]\\,z=8\\left(\\mathrm{cos}\\left(\\frac{2\\pi }{3}\\right)+i\\mathrm{sin}\\left(\\frac{2\\pi }{3}\\right)\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137656965\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137656965\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137656965\"]\n<p id=\"fs-id1165137656967\">We have<\/p>\n\n<div id=\"fs-id1165137656970\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{z}^{\\frac{1}{3}}={8}^{\\frac{1}{3}}\\left[\\mathrm{cos}\\left(\\frac{\\frac{2\\pi }{3}}{3}+\\frac{2k\\pi }{3}\\right)+i\\mathrm{sin}\\left(\\frac{\\frac{2\\pi }{3}}{3}+\\frac{2k\\pi }{3}\\right)\\right]\\hfill \\\\ {z}^{\\frac{1}{3}}=2\\left[\\mathrm{cos}\\left(\\frac{2\\pi }{9}+\\frac{2k\\pi }{3}\\right)+i\\mathrm{sin}\\left(\\frac{2\\pi }{9}+\\frac{2k\\pi }{3}\\right)\\right]\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134060257\">There will be three roots:[latex]\\,k=0,\\,\\,1,\\,\\,2.\\,[\/latex]When[latex]\\,k=0,\\,[\/latex]we have<\/p>\n\n<div id=\"fs-id1165135365831\" class=\"unnumbered aligncenter\">[latex]{z}^{\\frac{1}{3}}=2\\left(\\mathrm{cos}\\left(\\frac{2\\pi }{9}\\right)+i\\mathrm{sin}\\left(\\frac{2\\pi }{9}\\right)\\right)[\/latex]<\/div>\n<p id=\"fs-id1165135510902\">When[latex]\\,k=1,\\,[\/latex]we have<\/p>\n\n<div id=\"fs-id1165134258392\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{z}^{\\frac{1}{3}}=2\\left[\\mathrm{cos}\\left(\\frac{2\\pi }{9}+\\frac{6\\pi }{9}\\right)+i\\mathrm{sin}\\left(\\frac{2\\pi }{9}+\\frac{6\\pi }{9}\\right)\\right]\\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\text{ Add }\\frac{2\\left(1\\right)\\pi }{3}\\text{ to each angle.}\\hfill \\\\ {z}^{\\frac{1}{3}}=2\\left(\\mathrm{cos}\\left(\\frac{8\\pi }{9}\\right)+i\\mathrm{sin}\\left(\\frac{8\\pi }{9}\\right)\\right)\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134311190\">When[latex]\\,k=2,\\,[\/latex] we have<\/p>\n\n<div id=\"fs-id1165133221800\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}{z}^{\\frac{1}{3}}=2\\left[\\mathrm{cos}\\left(\\frac{2\\pi }{9}+\\frac{12\\pi }{9}\\right)+i\\mathrm{sin}\\left(\\frac{2\\pi }{9}+\\frac{12\\pi }{9}\\right)\\right]\\begin{array}{cccc}&amp; &amp; &amp; \\end{array}\\hfill &amp; \\text{Add }\\frac{2\\left(2\\right)\\pi }{3}\\text{ to each angle.}\\hfill \\\\ {z}^{\\frac{1}{3}}=2\\left(\\mathrm{cos}\\left(\\frac{14\\pi }{9}\\right)+i\\mathrm{sin}\\left(\\frac{14\\pi }{9}\\right)\\right)\\hfill &amp; \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165133102510\">Remember to find the common denominator to simplify fractions in situations like this one. For[latex]\\,k=1,\\,[\/latex]the angle simplification is<\/p>\n\n<div id=\"fs-id1165134192876\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{\\frac{2\\pi }{3}}{3}+\\frac{2\\left(1\\right)\\pi }{3}=\\frac{2\\pi }{3}\\left(\\frac{1}{3}\\right)+\\frac{2\\left(1\\right)\\pi }{3}\\left(\\frac{3}{3}\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{2\\pi }{9}+\\frac{6\\pi }{9}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{8\\pi }{9}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135404709\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165135404715\">\n<div id=\"fs-id1165135404718\">\n<p id=\"fs-id1165135329823\">Find the four fourth roots of[latex]\\,16\\left(\\mathrm{cos}\\left(120\u00b0\\right)+i\\mathrm{sin}\\left(120\u00b0\\right)\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134356866\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134356866\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134356866\"]\n<p id=\"fs-id1165133111152\">[latex]{z}_{0}=2\\left(\\mathrm{cos}\\left(30\u00b0\\right)+i\\mathrm{sin}\\left(30\u00b0\\right)\\right)[\/latex]<\/p>\n<p id=\"fs-id1165134234229\">[latex]{z}_{1}=2\\left(\\mathrm{cos}\\left(120\u00b0\\right)+i\\mathrm{sin}\\left(120\u00b0\\right)\\right)[\/latex]<\/p>\n<p id=\"fs-id1165137428218\">[latex]{z}_{2}=2\\left(\\mathrm{cos}\\left(210\u00b0\\right)+i\\mathrm{sin}\\left(210\u00b0\\right)\\right)[\/latex]<\/p>\n<p id=\"fs-id1165134166590\">[latex]{z}_{3}=2\\left(\\mathrm{cos}\\left(300\u00b0\\right)+i\\mathrm{sin}\\left(300\u00b0\\right)\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133361907\" class=\"precalculus media\">\n<p id=\"fs-id1165133361914\">Access these online resources for additional instruction and practice with polar forms of complex numbers.<\/p>\n\n<ul id=\"fs-id1165135329691\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/prodquocomplex\">The Product and Quotient of Complex Numbers in Trigonometric Form<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/demoivre\">De Moivre\u2019s Theorem<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165134129908\">\n \t<li>Complex numbers in the form[latex]\\,a+bi\\,[\/latex]are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. Label the <em>x-<\/em>axis as the <em>real <\/em>axis and the <em>y-<\/em>axis as the <em>imaginary<\/em> axis. See <a class=\"autogenerated-content\" href=\"#Example_08_05_01\">(Figure)<\/a>.<\/li>\n \t<li>The absolute value of a complex number is the same as its magnitude. It is the distance from the origin to the point:[latex]\\,|z|=\\sqrt{{a}^{2}+{b}^{2}}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_05_02\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_05_03\">(Figure)<\/a>.<\/li>\n \t<li>To write complex numbers in polar form, we use the formulas[latex]\\,x=r\\mathrm{cos}\\,\\theta ,y=r\\mathrm{sin}\\,\\theta ,\\,[\/latex]and [latex]\\,r=\\sqrt{{x}^{2}+{y}^{2}}.\\,[\/latex]Then,[latex]\\,z=r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_05_04\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_05_05\">(Figure)<\/a>.<\/li>\n \t<li>To convert from polar form to rectangular form, first evaluate the trigonometric functions. Then, multiply through by[latex]\\,r.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_05_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_05_07\">(Figure)<\/a>.<\/li>\n \t<li>To find the product of two complex numbers, multiply the two moduli and add the two angles. Evaluate the trigonometric functions, and multiply using the distributive property. See <a class=\"autogenerated-content\" href=\"#Example_08_05_08\">(Figure)<\/a>.<\/li>\n \t<li>To find the quotient of two complex numbers in polar form, find the quotient of the two moduli and the difference of the two angles. See <a class=\"autogenerated-content\" href=\"#Example_08_05_09\">(Figure)<\/a>.<\/li>\n \t<li>To find the power of a complex number[latex]\\,{z}^{n},\\,[\/latex]raise [latex]\\,r\\,[\/latex] to the power [latex]\\,n,[\/latex] and multiply [latex]\\,\\theta \\,[\/latex] by [latex]\\,n.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_05_10\">(Figure)<\/a>.<\/li>\n \t<li>Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. See <a class=\"autogenerated-content\" href=\"#Example_08_05_11\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165134370081\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135613677\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165135613682\">\n<div id=\"fs-id1165135613684\">\n<p id=\"fs-id1165135613686\">A complex number is[latex]\\,a+bi.\\,[\/latex]Explain each part.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134395224\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134395224\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134395224\"]\n<p id=\"fs-id1165134395227\"><em>a<\/em> is the real part, <em>b<\/em> is the imaginary part, and[latex]\\,i=\\sqrt{-1}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135586313\">\n<div id=\"fs-id1165131926307\">\n<p id=\"fs-id1165131926309\">What does the absolute value of a complex number represent?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165131926314\">\n<div id=\"fs-id1165131926316\">\n<p id=\"fs-id1165131926319\">How is a complex number converted to polar form?<\/p>\n\n<\/div>\n<div id=\"fs-id1165131926323\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165131926323\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165131926323\"]\n<p id=\"fs-id1165131926325\">Polar form converts the real and imaginary part of the complex number in polar form using[latex]\\,x=r\\mathrm{cos}\\theta \\,[\/latex] and [latex]\\,y=r\\mathrm{sin}\\theta .[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134205897\">\n<div id=\"fs-id1165134205899\">\n<p id=\"fs-id1165134205901\">How do we find the product of two complex numbers?<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134205906\">\n<div id=\"fs-id1165133309248\">\n<p id=\"fs-id1165133309250\">What is De Moivre\u2019s Theorem and what is it used for?<\/p>\n\n<\/div>\n<div id=\"fs-id1165133309255\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133309255\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133309255\"]\n<p id=\"fs-id1165133309257\">[latex]{z}^{n}={r}^{n}\\left(\\mathrm{cos}\\left(n\\theta \\right)+i\\mathrm{sin}\\left(n\\theta \\right)\\right)\\,[\/latex]It is used to simplify polar form when a number has been raised to a power.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134183779\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165135665413\">For the following exercises, find the absolute value of the given complex number.<\/p>\n\n<div id=\"fs-id1165135665416\">\n<div id=\"fs-id1165135665418\">\n<p id=\"fs-id1165135665420\">[latex]5+\\text{\u200b}3i[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134385720\">\n<div id=\"fs-id1165134385722\">\n<p id=\"fs-id1165134385725\">[latex]-7+\\text{\u200b}i[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135499578\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135499578\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135499578\"]\n<p id=\"fs-id1165135499580\">[latex]5\\sqrt{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541528\">\n<div id=\"fs-id1165134481951\">\n<p id=\"fs-id1165134481953\">[latex]-3-3i[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135633877\">\n<div id=\"fs-id1165135633879\">\n<p id=\"fs-id1165135633881\">[latex]\\sqrt{2}-6i[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133213891\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133213891\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133213891\"]\n<p id=\"fs-id1165133213893\">[latex]\\sqrt{38}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135694472\">\n<div id=\"fs-id1165135694474\">\n<p id=\"fs-id1165137843211\">[latex]2i[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137843226\">\n<div id=\"fs-id1165137843228\">\n<p id=\"fs-id1165134116954\">[latex]2.2-3.1i[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134116972\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134116972\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134116972\"]\n<p id=\"fs-id1165137843105\">[latex]\\sqrt{14.45}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165134386618\">For the following exercises, write the complex number in polar form.<\/p>\n\n<div id=\"fs-id1165134386621\">\n<div id=\"fs-id1165134386623\">\n<p id=\"fs-id1165134386625\">[latex]2+2i[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134178500\">\n<div id=\"fs-id1165134178502\">\n<p id=\"fs-id1165134178504\">[latex]8-4i[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134129997\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134129997\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134129997\"]\n<p id=\"fs-id1165134129999\">[latex]4\\sqrt{5}\\mathrm{cis}\\left(333.4\u00b0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132959023\">\n<div id=\"fs-id1165132959025\">\n<p id=\"fs-id1165132959027\">[latex]-\\frac{1}{2}-\\frac{1}{2}\\text{\u200b}i[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135633990\">\n<div id=\"fs-id1165135633992\">\n<p id=\"fs-id1165135317569\">[latex]\\sqrt{3}+i[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165131993571\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165131993571\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165131993571\"]\n<p id=\"fs-id1165131993573\">[latex]2\\mathrm{cis}\\left(\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135207298\">\n<div id=\"fs-id1165135207300\">\n<p id=\"fs-id1165135207302\">[latex]3i[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135499587\">For the following exercises, convert the complex number from polar to rectangular form.<\/p>\n\n<div id=\"fs-id1165135499590\">\n<div id=\"fs-id1165135499592\">\n<p id=\"fs-id1165135499594\">[latex]z=7\\mathrm{cis}\\left(\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135367688\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135367688\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135367688\"]\n<p id=\"fs-id1165135367691\">[latex]\\frac{7\\sqrt{3}}{2}+i\\frac{7}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135532374\">\n<div>\n<p id=\"fs-id1165135407393\">[latex]z=2\\mathrm{cis}\\left(\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134089467\">\n<div id=\"fs-id1165134089469\">\n<p id=\"fs-id1165134089471\">[latex]z=4\\mathrm{cis}\\left(\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134573209\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134573209\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134573209\"]\n<p id=\"fs-id1165134573211\">[latex]-2\\sqrt{3}-2i[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134190676\">\n<div id=\"fs-id1165134190678\">\n<p id=\"fs-id1165134190680\">[latex]z=7\\mathrm{cis}\\left(25\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135388835\">\n<div id=\"fs-id1165135388838\">\n<p id=\"fs-id1165135388840\">[latex]z=3\\mathrm{cis}\\left(240\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135640462\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135640462\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135640462\"]\n<p id=\"fs-id1165135640464\">[latex]-1.5-i\\frac{3\\sqrt{3}}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135519200\">\n<div id=\"fs-id1165135519202\">\n<p id=\"fs-id1165135519205\">[latex]z=\\sqrt{2}\\mathrm{cis}\\left(100\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165134357546\">For the following exercises, find[latex]\\,{z}_{1}{z}_{2}\\,[\/latex]in polar form.<\/p>\n\n<div id=\"fs-id1165133078076\">\n<div id=\"fs-id1165133078079\">\n<p id=\"fs-id1165134130033\">[latex]{z}_{1}=2\\sqrt{3}\\mathrm{cis}\\left(116\u00b0\\right);\\,\\text{ }{z}_{2}=2\\mathrm{cis}\\left(82\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135419732\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135419732\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135419732\"]\n<p id=\"fs-id1165135419734\">[latex]4\\sqrt{3}\\mathrm{cis}\\left(198\u00b0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134430434\">\n<div id=\"fs-id1165134430436\">\n<p id=\"fs-id1165134430438\">[latex]{z}_{1}=\\sqrt{2}\\mathrm{cis}\\left(205\u00b0\\right);\\text{ }{z}_{2}=2\\sqrt{2}\\mathrm{cis}\\left(118\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137756875\">\n<div>[latex]{z}_{1}=3\\mathrm{cis}\\left(120\u00b0\\right);\\text{ }{z}_{2}=\\frac{1}{4}\\mathrm{cis}\\left(60\u00b0\\right)[\/latex]<\/div>\n<div id=\"fs-id1165135472921\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135472921\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135472921\"]\n<p id=\"fs-id1165135472923\">[latex]\\frac{3}{4}\\mathrm{cis}\\left(180\u00b0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134093149\">\n<div id=\"fs-id1165134093151\">\n<p id=\"fs-id1165133023575\">[latex]{z}_{1}=3\\mathrm{cis}\\left(\\frac{\\pi }{4}\\right);\\text{ }{z}_{2}=5\\mathrm{cis}\\left(\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165131863233\">\n<div id=\"fs-id1165131863235\">\n<p id=\"fs-id1165131863237\">[latex]{z}_{1}=\\sqrt{5}\\mathrm{cis}\\left(\\frac{5\\pi }{8}\\right);\\text{ }{z}_{2}=\\sqrt{15}\\mathrm{cis}\\left(\\frac{\\pi }{12}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133077994\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133077994\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133077994\"]\n<p id=\"fs-id1165135518168\">[latex]5\\sqrt{3}\\mathrm{cis}\\left(\\frac{17\\pi }{24}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134088751\">\n<div id=\"fs-id1165134088753\">\n<p id=\"fs-id1165134088755\">[latex]{z}_{1}=4\\mathrm{cis}\\left(\\frac{\\pi }{2}\\right);\\text{ }{z}_{2}=2\\mathrm{cis}\\left(\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135662421\">For the following exercises, find[latex]\\,\\frac{{z}_{1}}{{z}_{2}}\\,[\/latex]in polar form.<\/p>\n\n<div id=\"fs-id1165134160326\">\n<div id=\"fs-id1165134160328\">\n<p id=\"fs-id1165134160330\">[latex]{z}_{1}=21\\mathrm{cis}\\left(135\u00b0\\right);\\text{ }{z}_{2}=3\\mathrm{cis}\\left(65\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134174904\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134174904\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134174904\"]\n<p id=\"fs-id1165134174906\">[latex]7\\mathrm{cis}\\left(70\u00b0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134221768\">\n<div id=\"fs-id1165134221770\">\n<p id=\"fs-id1165134221772\">[latex]{z}_{1}=\\sqrt{2}\\mathrm{cis}\\left(90\u00b0\\right);\\text{ }{z}_{2}=2\\mathrm{cis}\\left(60\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134325223\">\n<div id=\"fs-id1165134325225\">\n<p id=\"fs-id1165131863137\">[latex]{z}_{1}=15\\mathrm{cis}\\left(120\u00b0\\right);\\text{ }{z}_{2}=3\\mathrm{cis}\\left(40\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137480080\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137480080\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137480080\"]\n<p id=\"fs-id1165137480082\">[latex]5\\mathrm{cis}\\left(80\u00b0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134279277\">\n<div id=\"fs-id1165134279279\">\n<p id=\"fs-id1165134279281\">[latex]{z}_{1}=6\\mathrm{cis}\\left(\\frac{\\pi }{3}\\right);\\text{ }{z}_{2}=2\\mathrm{cis}\\left(\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135652331\">\n<div id=\"fs-id1165134401618\">\n<p id=\"fs-id1165134401620\">[latex]{z}_{1}=5\\sqrt{2}\\mathrm{cis}\\left(\\pi \\right);\\text{ }{z}_{2}=\\sqrt{2}\\mathrm{cis}\\left(\\frac{2\\pi }{3}\\right)[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134388962\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134388962\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134388962\"]\n<p id=\"fs-id1165134388964\">[latex]5\\mathrm{cis}\\left(\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134534047\">\n<div id=\"fs-id1165134534049\">\n<p id=\"fs-id1165134534051\">[latex]{z}_{1}=2\\mathrm{cis}\\left(\\frac{3\\pi }{5}\\right);\\text{ }{z}_{2}=3\\mathrm{cis}\\left(\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165133028482\">For the following exercises, find the powers of each complex number in polar form.<\/p>\n\n<div id=\"fs-id1165133028485\">\n<div id=\"fs-id1165133028487\">\n<p id=\"fs-id1165133028489\">Find[latex]\\,{z}^{3}\\,[\/latex]when[latex]\\,z=5\\mathrm{cis}\\left(45\u00b0\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133349423\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133349423\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133349423\"]\n<p id=\"fs-id1165133349425\">[latex]125\\mathrm{cis}\\left(135\u00b0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135523252\">\n<div id=\"fs-id1165135523254\">\n<p id=\"fs-id1165135523256\">Find[latex]\\,{z}^{4}\\,[\/latex]when[latex]\\,z=2\\mathrm{cis}\\left(70\u00b0\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440037\">\n<div id=\"fs-id1165135440039\">\n<p id=\"fs-id1165135440041\">Find[latex]\\,{z}^{2}\\,[\/latex]when[latex]\\,z=3\\mathrm{cis}\\left(120\u00b0\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135512799\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135512799\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135512799\"]\n<p id=\"fs-id1165135512801\">[latex]9\\mathrm{cis}\\left(240\u00b0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134113850\">\n<div id=\"fs-id1165134113852\">\n<p id=\"fs-id1165135619413\">Find[latex]\\,{z}^{2}\\,[\/latex]when[latex]\\,z=4\\mathrm{cis}\\left(\\frac{\\pi }{4}\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134070876\">\n<div id=\"fs-id1165134070879\">\n<p id=\"fs-id1165134070881\">Find[latex]\\,{z}^{4}\\,[\/latex]when[latex]\\,z=\\mathrm{cis}\\left(\\frac{3\\pi }{16}\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165134177544\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134177544\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134177544\"]\n<p id=\"fs-id1165134177546\">[latex]\\mathrm{cis}\\left(\\frac{3\\pi }{4}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134128415\">\n<div id=\"fs-id1165134128417\">\n<p id=\"fs-id1165134128419\">Find[latex]\\,{z}^{3}\\,[\/latex]when[latex]\\,z=3\\mathrm{cis}\\left(\\frac{5\\pi }{3}\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id1165135352989\">For the following exercises, evaluate each root.<\/p>\n\n<div id=\"fs-id1165135352992\">\n<div id=\"fs-id1165134039259\">\n<p id=\"fs-id1165134039261\">Evaluate the cube root of[latex]\\,z\\,[\/latex]when[latex]\\,z=27\\mathrm{cis}\\left(240\u00b0\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135609212\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135609212\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135609212\"]\n<p id=\"fs-id1165135609214\">[latex]\\,3\\mathrm{cis}\\left(80\u00b0\\right),3\\mathrm{cis}\\left(200\u00b0\\right),3\\mathrm{cis}\\left(320\u00b0\\right)\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135170392\">\n<div id=\"fs-id1165135170394\">\n<p id=\"fs-id1165135170396\">Evaluate the square root of[latex]\\,z\\,[\/latex]when[latex]\\,z=16\\mathrm{cis}\\left(100\u00b0\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134534293\">\n<div id=\"fs-id1165133243591\">\n<p id=\"fs-id1165133243593\">Evaluate the cube root of[latex]\\,z\\,[\/latex]when[latex]\\,z=32\\mathrm{cis}\\left(\\frac{2\\pi }{3}\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135238453\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135238453\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135238453\"]\n<p id=\"fs-id1165137780013\">[latex]\\,2\\sqrt[3]{4}\\mathrm{cis}\\left(\\frac{2\\pi }{9}\\right),2\\sqrt[3]{4}\\mathrm{cis}\\left(\\frac{8\\pi }{9}\\right),2\\sqrt[3]{4}\\mathrm{cis}\\left(\\frac{14\\pi }{9}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165134378615\">Evaluate the square root of[latex]\\,z\\,[\/latex]when[latex]\\,z=32\\text{cis}\\left(\\pi \\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134356902\">\n<div id=\"fs-id1165134356904\">\n<p id=\"fs-id1165134356906\">Evaluate the cube root of[latex]\\,z\\,[\/latex]when[latex]\\,z=8\\mathrm{cis}\\left(\\frac{7\\pi }{4}\\right).[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165131907304\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165131907304\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165131907304\"]\n<p id=\"fs-id1165131907306\">[latex]2\\sqrt{2}\\mathrm{cis}\\left(\\frac{7\\pi }{8}\\right),2\\sqrt{2}\\mathrm{cis}\\left(\\frac{15\\pi }{8}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135586834\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165137895099\">For the following exercises, plot the complex number in the complex plane.<\/p>\n\n<div id=\"fs-id1165137895102\">\n<div id=\"fs-id1165137895104\">\n<p id=\"fs-id1165137895107\">[latex]2+4i[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135621933\">\n<div id=\"fs-id1165135621935\">\n<p id=\"fs-id1165135621938\">[latex]-3-3i[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133221775\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133221775\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133221775\"]<span id=\"fs-id1165133221780\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152636\/CNX_Precalc_Figure_08_05_202.jpg\" alt=\"Plot of -3 -3i in the complex plane (-3 along real axis, -3 along imaginary axis).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135457049\">\n<div id=\"fs-id1165135457051\">\n<p id=\"fs-id1165135457054\">[latex]5-4i[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165137835713\">\n<div id=\"fs-id1165137835715\">\n<p id=\"fs-id1165137835717\">[latex]-1-5i[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165133260388\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165133260388\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165133260388\"]<span id=\"fs-id1165133260393\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152645\/CNX_Precalc_Figure_08_05_204.jpg\" alt=\"Plot of -1 -5i in the complex plane (-1 along real axis, -5 along imaginary axis).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135664876\">\n<div id=\"fs-id1165135664878\">\n<p id=\"fs-id1165135664880\">[latex]3+2i[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135251281\">\n<div id=\"fs-id1165135251283\">\n<p id=\"fs-id1165135251285\">[latex]2i[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165135702570\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135702570\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135702570\"]<span id=\"fs-id1165135702576\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152648\/CNX_Precalc_Figure_08_05_206.jpg\" alt=\"Plot of 2i in the complex plane (0 along the real axis, 2 along the imaginary axis).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165134254413\">\n<div id=\"fs-id1165134254415\">\n<p id=\"fs-id1165134254418\">[latex]-4[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165134254432\">\n<div id=\"fs-id1165135160374\">[latex]6-2i[\/latex]<\/div>\n<div id=\"fs-id1165135672734\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135672734\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135672734\"]<span id=\"fs-id1165135672739\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152651\/CNX_Precalc_Figure_08_05_208.jpg\" alt=\"Plot of 6-2i in the complex plane (6 along the real axis, -2 along the imaginary axis).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1165135672751\">\n<div id=\"fs-id1165135672753\">\n<p id=\"fs-id1165133359361\">[latex]-2+i[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133359380\">\n<div id=\"fs-id1165137580684\">\n<p id=\"fs-id1165137580686\">[latex]1-4i[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id1165137580704\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165137580704\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165137580704\"]<span id=\"fs-id1165132949966\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152654\/CNX_Precalc_Figure_08_05_210.jpg\" alt=\"Plot of 1-4i in the complex plane (1 along the real axis, -4 along the imaginary axis).\"><\/span>[\/hidden-answer]<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132949979\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165137647538\">For the following exercises, find all answers rounded to the nearest hundredth.<\/p>\n\n<div id=\"fs-id1165137647541\">\n<div id=\"fs-id1165137647543\">\n<p id=\"fs-id1165137647545\">Use the rectangular to polar feature on the graphing calculator to change[latex]\\,5+5i\\,[\/latex]to polar form.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135186955\">\n<div id=\"fs-id1165135186958\">\n<p id=\"fs-id1165135186960\">Use the rectangular to polar feature on the graphing calculator to change[latex]\\,3-2i\\,[\/latex]\nto polar form.<\/p>\n\n<\/div>\n<div id=\"fs-id1165134329643\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165134329643\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165134329643\"]\n<p id=\"fs-id1165132972772\">[latex]\\,3.61{e}^{-0.59i}\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165133243512\">\n<div id=\"fs-id1165133243514\">\n<p id=\"fs-id1165133243516\">Use the rectangular to polar feature on the graphing calculator to change [latex]-3-8i\\,[\/latex]\nto polar form.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135189784\">\n<div id=\"fs-id1165135189786\">\n<p id=\"fs-id1165135189788\">Use the polar to rectangular feature on the graphing calculator to change[latex]\\,4\\mathrm{cis}\\left(120\u00b0\\right)\\,[\/latex]to rectangular form.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135694966\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135694966\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135694966\"]\n<p id=\"fs-id1165135694969\">[latex]\\,-2+3.46i\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165132945471\">\n<div>\n<p id=\"fs-id1165132945475\">Use the polar to rectangular feature on the graphing calculator to change[latex]\\,2\\mathrm{cis}\\left(45\u00b0\\right)\\,[\/latex]to rectangular form.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1165135546054\">\n<div id=\"fs-id1165135546056\">\n<p id=\"fs-id1165135546058\">Use the polar to rectangular feature on the graphing calculator to change[latex]\\,5\\mathrm{cis}\\left(210\u00b0\\right)\\,[\/latex]to rectangular form.<\/p>\n\n<\/div>\n<div id=\"fs-id1165135252140\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id1165135252140\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1165135252140\"]\n<p id=\"fs-id1165135252142\">[latex]\\,-4.33-2.50i\\,[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165135344835\">\n \t<dt>argument<\/dt>\n \t<dd id=\"fs-id1165135344841\">the angle associated with a complex number; the angle between the line from the origin to the point and the positive real axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135344846\">\n \t<dt>De Moivre\u2019s Theorem<\/dt>\n \t<dd id=\"fs-id1165134357535\">formula used to find the[latex]\\,n\\text{th}\\,[\/latex]power or <em>n<\/em>th roots of a complex number; states that, for a positive integer[latex]\\,n,{z}^{n}\\,[\/latex]is found by raising the modulus to the[latex]\\,n\\text{th}\\,[\/latex]power and multiplying the angles by[latex]\\,n\\,[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134279306\">\n \t<dt>modulus<\/dt>\n \t<dd id=\"fs-id1165134279312\">the absolute value of a complex number, or the distance from the origin to the point[latex]\\,\\left(x,y\\right);\\,[\/latex]also called the amplitude<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133162992\">\n \t<dt>polar form of a complex number<\/dt>\n \t<dd id=\"fs-id1165133162998\">a complex number expressed in terms of an angle [latex]\\theta [\/latex] and its distance from the origin[latex]\\,r;\\,[\/latex]can be found by using conversion formulas[latex]\\,x=r\\mathrm{cos}\\,\\theta ,\\,\\,y=r\\mathrm{sin}\\,\\theta ,\\,\\,[\/latex]and[latex]\\,r=\\sqrt{{x}^{2}+{y}^{2}}[\/latex]<\/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 complex numbers in the complex plane.<\/li>\n<li>Find the absolute value of a complex number.<\/li>\n<li>Write complex numbers in polar form.<\/li>\n<li>Convert a complex number from polar to rectangular form.<\/li>\n<li>Find products of complex numbers in polar form.<\/li>\n<li>Find quotients of complex numbers in polar form.<\/li>\n<li>Find powers of complex numbers in polar form.<\/li>\n<li>Find roots of complex numbers in polar form.<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1165137543411\">\u201cGod made the integers; all else is the work of man.\u201d This rather famous quote by nineteenth-century German mathematician Leopold <span class=\"no-emphasis\">Kronecker<\/span> sets the stage for this section on the polar form of a complex number. Complex numbers were invented by people and represent over a thousand years of continuous investigation and struggle by mathematicians such as <span class=\"no-emphasis\">Pythagoras<\/span>, <span class=\"no-emphasis\">Descartes<\/span>, De Moivre, <span class=\"no-emphasis\">Euler<\/span>, <span class=\"no-emphasis\">Gauss<\/span>, and others. Complex numbers answered questions that for centuries had puzzled the greatest minds in science.<\/p>\n<p id=\"fs-id1165134035955\">We first encountered complex numbers in <a class=\"target-chapter\" href=\"https:\/\/courses.lumenlearning.com\/suny-osalgebratrig\/chapter\/complex-numbers\/\">Complex Numbers<\/a>. In this section, we will focus on the mechanics of working with complex numbers: translation of complex numbers from polar form to rectangular form and vice versa, interpretation of complex numbers in the scheme of applications, and application of De Moivre\u2019s Theorem.<\/p>\n<div id=\"fs-id1165133355948\" class=\"bc-section section\">\n<h3>Plotting Complex Numbers in the Complex Plane<\/h3>\n<p id=\"fs-id1165134037549\">Plotting a <span class=\"no-emphasis\">complex number<\/span>[latex]\\,a+bi\\,[\/latex]is similar to plotting a real number, except that the horizontal axis represents the real part of the number,[latex]\\,a,\\,[\/latex]and the vertical axis represents the imaginary part of the number,[latex]\\,bi.[\/latex]<\/p>\n<div id=\"fs-id1165137938490\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165134079788\"><strong>Given a complex number[latex]\\,a+bi,\\,[\/latex]plot it in the complex plane.<\/strong><\/p>\n<ol id=\"fs-id1165133192906\" type=\"1\">\n<li>Label the horizontal axis as the <em>real<\/em> axis and the vertical axis as the <em>imaginary axis.<\/em><\/li>\n<li>Plot the point in the complex plane by moving[latex]\\,a\\,[\/latex]units in the horizontal direction and[latex]\\,b\\,[\/latex]units in the vertical direction.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_08_05_01\" class=\"textbox examples\">\n<div id=\"fs-id1165137628795\">\n<div id=\"fs-id1165137465062\">\n<h3>Plotting a Complex Number in the Complex Plane<\/h3>\n<p id=\"fs-id1165134331107\">Plot the complex number [latex]\\,2-3i\\,[\/latex]in the <span class=\"no-emphasis\">complex plane<\/span>.<\/p>\n<\/div>\n<div id=\"fs-id1165137834397\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137704829\">From the origin, move two units in the positive horizontal direction and three units in the negative vertical direction. See <a class=\"autogenerated-content\" href=\"#Figure_08_05_001\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_05_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\/19152553\/CNX_Precalc_Figure_08_05_001.jpg\" alt=\"Plot of 2-3i in the complex plane (2 along the real axis, -3 along the imaginary axis).\" width=\"487\" height=\"331\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1.<\/strong><\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135581219\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_01\">\n<div id=\"fs-id1165135586379\">\n<p id=\"fs-id1165135514726\">Plot the point[latex]\\,1+5i\\,[\/latex]in the complex plane.<\/p>\n<\/div>\n<div id=\"fs-id1165134167302\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165137938804\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152606\/CNX_Precalc_Figure_08_05_002-1.jpg\" alt=\"Plot of 1+5i in the complex plane (1 along the real axis, 5 along the imaginary axis).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137542524\" class=\"bc-section section\">\n<h3>Finding the Absolute Value of a Complex Number<\/h3>\n<p id=\"fs-id1165134325179\">The first step toward working with a complex number in <span class=\"no-emphasis\">polar form<\/span> is to find the absolute value. The absolute value of a complex number is the same as its <span class=\"no-emphasis\">magnitude<\/span>, or[latex]\\,|z|.\\,[\/latex]It measures the distance from the origin to a point in the plane. For example, the graph of[latex]\\,z=2+4i,\\,[\/latex]in <a class=\"autogenerated-content\" href=\"#Figure_08_05_003\">(Figure)<\/a>, shows[latex]\\,|z|.[\/latex]<\/p>\n<div id=\"Figure_08_05_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\/19152610\/CNX_Precalc_Figure_08_05_003.jpg\" alt=\"Plot of 2+4i in the complex plane and its magnitude, |z| = rad 20.\" width=\"487\" height=\"368\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1165134541127\" class=\"textbox key-takeaways\">\n<h3>Absolute Value of a Complex Number<\/h3>\n<p id=\"fs-id1165133353946\">Given[latex]\\,z=x+yi,\\,[\/latex]a complex number, the absolute value of[latex]\\,z\\,[\/latex]is defined as<\/p>\n<div id=\"fs-id1165135340598\" class=\"unnumbered aligncenter\">[latex]|z|=\\sqrt{{x}^{2}+{y}^{2}}[\/latex]<\/div>\n<p id=\"fs-id1165134199373\">It is the distance from the origin to the point[latex]\\,\\left(x,y\\right).[\/latex]<\/p>\n<p id=\"fs-id1165135203416\">Notice that the absolute value of a real number gives the distance of the number from 0, while the absolute value of a complex number gives the distance of the number from the origin,[latex]\\,\\left(0,\\text{ }0\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"Example_08_05_02\" class=\"textbox examples\">\n<div id=\"fs-id1165137921556\">\n<div id=\"fs-id1165135190837\">\n<h3>Finding the Absolute Value of a Complex Number with a Radical<\/h3>\n<p id=\"fs-id1165134232230\">Find the absolute value of[latex]\\,z=\\sqrt{5}-i.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133024224\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137464603\">Using the formula, we have<\/p>\n<div id=\"fs-id1165137602050\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}|z|=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\\\ |z|=\\sqrt{{\\left(\\sqrt{5}\\right)}^{2}+{\\left(-1\\right)}^{2}}\\hfill \\\\ |z|=\\sqrt{5+1}\\hfill \\\\ |z|=\\sqrt{6}\\hfill \\end{array}[\/latex]<\/div>\n<p>See <a class=\"autogenerated-content\" href=\"#Figure_08_05_004\">(Figure)<\/a>.<\/p>\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\/19152612\/CNX_Precalc_Figure_08_05_004.jpg\" alt=\"Plot of z=(rad5 - i) in the complex plane and its magnitude rad6.\" width=\"487\" height=\"331\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165133324050\"><\/details>\n<p><span id=\"fs-id1165135333398\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137461107\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_02\">\n<div id=\"fs-id1165137553007\">\n<p id=\"fs-id1165134401602\">Find the absolute value of the complex number[latex]\\,z=12-5i.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137771102\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135403265\">13<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_05_03\" class=\"textbox examples\">\n<div id=\"fs-id1165137454990\">\n<div id=\"fs-id1165137645660\">\n<h3>Finding the Absolute Value of a Complex Number<\/h3>\n<p id=\"fs-id1165137662370\">Given[latex]\\,z=3-4i,\\,[\/latex]find[latex]\\,|z|.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137419461\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133233012\">Using the formula, we have<\/p>\n<div id=\"fs-id1165137410180\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}|z|=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\\\ |z|=\\sqrt{{\\left(3\\right)}^{2}+{\\left(-4\\right)}^{2}}\\hfill \\\\ |z|=\\sqrt{9+16}\\hfill \\\\ \\begin{array}{l}|z|=\\sqrt{25}\\\\ |z|=5\\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p>The absolute value[latex]\\,z\\,[\/latex]is 5. See <a class=\"autogenerated-content\" href=\"#Figure_08_05_005\">(Figure)<\/a>.<\/p>\n<figure style=\"width: 488px\" 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\/19152620\/CNX_Precalc_Figure_08_05_005_Errata.jpg\" alt=\"Plot of (3-4i) in the complex plane and its magnitude |z| =5.\" width=\"488\" height=\"331\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165135553504\"><\/details>\n<p><span id=\"fs-id1165135194751\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137911592\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_03\">\n<div>\n<p id=\"fs-id1165137812758\">Given[latex]\\,z=1-7i,\\,[\/latex]find[latex]\\,|z|.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137643164\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135237187\">[latex]|z|=\\sqrt{50}=5\\sqrt{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135530150\" class=\"bc-section section\">\n<h3>Writing Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165132928617\">The polar form of a complex number expresses a number in terms of an angle[latex]\\,\\theta \\,[\/latex]and its distance from the origin[latex]\\,r.\\,[\/latex]Given a complex number in <span class=\"no-emphasis\">rectangular form<\/span> expressed as[latex]\\,z=x+yi,\\,[\/latex]we use the same conversion formulas as we do to write the number in trigonometric form:<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}\\,x=r\\mathrm{cos}\\,\\theta \\hfill \\\\ \\,y=r\\mathrm{sin}\\,\\theta \\hfill \\\\ \\,\\,r=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135161485\">We review these relationships in <a class=\"autogenerated-content\" href=\"#Figure_08_05_006\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_08_05_006\" 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\/19152623\/CNX_Precalc_Figure_08_05_006.jpg\" alt=\"Triangle plotted in the complex plane (x axis is real, y axis is imaginary). Base is along the x\/real axis, height is some y\/imaginary value in Q 1, and hypotenuse r extends from origin to that point (x+yi) in Q 1. The angle at the origin is theta. There is an arc going through (x+yi).\" width=\"487\" height=\"331\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p>We use the term <strong>modulus<\/strong> to represent the absolute value of a complex number, or the distance from the origin to the point[latex]\\,\\left(x,y\\right).\\,[\/latex]The modulus, then, is the same as[latex]\\,r,\\,[\/latex]the radius in polar form. We use[latex]\\,\\theta \\,[\/latex]to indicate the angle of direction (just as with polar coordinates). Substituting, we have<\/p>\n<div id=\"fs-id1165137835830\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}z=x+yi\\hfill \\\\ z=r\\mathrm{cos}\\,\\theta +\\left(r\\mathrm{sin}\\,\\theta \\right)i\\hfill \\\\ z=r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right)\\hfill \\end{array}[\/latex]<\/div>\n<div>\n<h3>Polar Form of a Complex Number<\/h3>\n<p id=\"fs-id1165137643584\">Writing a complex number in polar form involves the following conversion formulas:<\/p>\n<div id=\"fs-id1165134566530\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ x=r\\mathrm{cos}\\,\\theta \\hfill \\\\ y=r\\mathrm{sin}\\,\\theta \\hfill \\\\ r=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134073108\">Making a direct substitution, we have<\/p>\n<div id=\"fs-id1165135613632\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}z=x+yi\\hfill \\\\ z=\\left(r\\mathrm{cos}\\,\\theta \\right)+i\\left(r\\mathrm{sin}\\,\\theta \\right)\\hfill \\\\ z=r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right)\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137628482\">where[latex]\\,r\\,[\/latex]is the modulus and [latex]\\theta[\/latex] is the argument. We often use the abbreviation[latex]\\,r\\text{cis}\\,\\theta \\,[\/latex]to represent[latex]\\,r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right).[\/latex]<\/p>\n<\/div>\n<div id=\"Example_08_05_04\" class=\"textbox examples\">\n<div id=\"fs-id1165137454742\">\n<div id=\"fs-id1165132959045\">\n<h3>Expressing a Complex Number Using Polar Coordinates<\/h3>\n<p id=\"fs-id1165137804036\">Express the complex number[latex]\\,4i\\,[\/latex]using polar coordinates.<\/p>\n<\/div>\n<div id=\"fs-id1165137595455\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137803541\">On the complex plane, the number[latex]\\,z=4i\\,[\/latex]is the same as[latex]\\,z=0+4i.\\,[\/latex]Writing it in polar form, we have to calculate[latex]\\,r\\,[\/latex]first.<\/p>\n<div id=\"fs-id1165137483073\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}r=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\\\ r=\\sqrt{{0}^{2}+{4}^{2}}\\hfill \\\\ r=\\sqrt{16}\\hfill \\\\ r=4\\hfill \\end{array}[\/latex]<\/div>\n<p>Next, we look at[latex]\\,x.\\,[\/latex]If[latex]\\,x=r\\mathrm{cos}\\,\\theta ,\\,[\/latex]and[latex]\\,x=0,\\,[\/latex]then[latex]\\,\\theta =\\frac{\\pi }{2}.\\,[\/latex]In polar coordinates, the complex number[latex]\\,z=0+4i\\,[\/latex]can be written as[latex]\\,z=4\\left(\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)+i\\mathrm{sin}\\left(\\frac{\\pi }{2}\\right)\\right)\\,[\/latex]or[latex]\\,4\\text{cis}\\left(\\,\\frac{\\pi }{2}\\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Figure_08_05_007\">(Figure)<\/a>.<\/p>\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\/19152626\/CNX_Precalc_Figure_08_05_007.jpg\" alt=\"Plot of z=4i in the complex plane, also shows that the in polar coordinate it would be (4,pi\/2).\" width=\"487\" height=\"294\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 6.<\/strong><\/figcaption><\/figure>\n<p id=\"fs-id1165137540200\"><\/details>\n<p><span id=\"fs-id1165137460482\"><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135175110\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_04\">\n<div id=\"fs-id1165134550614\">\n<p id=\"fs-id1165134550616\">Express[latex]\\,z=3i\\,[\/latex] as [latex]\\,r\\,\\text{cis}\\,\\theta \\,[\/latex] in polar form.<\/p>\n<\/div>\n<div id=\"fs-id1165137397894\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137397895\">[latex]z=3\\left(\\mathrm{cos}\\left(\\frac{\\pi }{2}\\right)+i\\mathrm{sin}\\left(\\frac{\\pi }{2}\\right)\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_05_05\" class=\"textbox examples\">\n<div>\n<div id=\"fs-id1165137851363\">\n<h3>Finding the Polar Form of a Complex Number<\/h3>\n<p id=\"fs-id1165133097232\">Find the polar form of[latex]\\,-4+4i.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134077346\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135499750\">First, find the value of[latex]\\,r.[\/latex]<\/p>\n<div id=\"fs-id1165137405565\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}r=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\\\ r=\\sqrt{{\\left(-4\\right)}^{2}+\\left({4}^{2}\\right)}\\hfill \\\\ r=\\sqrt{32}\\hfill \\\\ r=4\\sqrt{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165133141427\">Find the angle[latex]\\,\\theta \\,[\/latex]using the formula:<\/p>\n<div id=\"fs-id1165133155817\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{cos}\\,\\theta =\\frac{x}{r}\\hfill \\\\ \\mathrm{cos}\\,\\theta =\\frac{-4}{4\\sqrt{2}}\\hfill \\\\ \\mathrm{cos}\\,\\theta =-\\frac{1}{\\sqrt{2}}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\theta ={\\mathrm{cos}}^{-1}\\left(-\\frac{1}{\\sqrt{2}}\\right)=\\frac{3\\pi }{4}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134184438\">Thus, the solution is[latex]\\,4\\sqrt{2}\\text{cis}\\left(\\frac{3\\pi }{4}\\right).[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135475905\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_05\">\n<div id=\"fs-id1165137451880\">\n<p>Write[latex]\\,z=\\sqrt{3}+i\\,[\/latex]in polar form.<\/p>\n<\/div>\n<div id=\"fs-id1165137603699\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137603700\">[latex]z=2\\left(\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)+i\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135311667\" class=\"bc-section section\">\n<h3>Converting a Complex Number from Polar to Rectangular Form<\/h3>\n<p id=\"fs-id1165135664797\">Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. In other words, given[latex]\\,z=r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right),\\,[\/latex]first evaluate the trigonometric functions[latex]\\,\\mathrm{cos}\\,\\theta \\,[\/latex]and[latex]\\,\\mathrm{sin}\\,\\theta .\\,[\/latex]Then, multiply through by[latex]\\,r.[\/latex]<\/p>\n<div id=\"Example_08_05_06\" class=\"textbox examples\">\n<div id=\"fs-id1165135187864\">\n<div id=\"fs-id1165135187866\">\n<h3>Converting from Polar to Rectangular Form<\/h3>\n<p id=\"fs-id1165137732869\">Convert the polar form of the given complex number to rectangular form:<\/p>\n<div id=\"fs-id1165137732872\" class=\"unnumbered aligncenter\">[latex]z=12\\left(\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)+i\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137827978\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134279359\">We begin by evaluating the trigonometric expressions.<\/p>\n<div id=\"eip-id811719\" class=\"unnumbered\">[latex]\\mathrm{cos}\\left(\\frac{\\pi }{6}\\right)=\\frac{\\sqrt{3}}{2}\\,\\text{and}\\,\\mathrm{sin}\\left(\\frac{\\pi }{6}\\right)=\\frac{1}{2}\\,[\/latex]<\/div>\n<p id=\"eip-id1472898\">After substitution, the complex number is<\/p>\n<div id=\"fs-id1165137723445\" class=\"unnumbered aligncenter\">[latex]z=12\\left(\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i\\right)[\/latex]<\/div>\n<p id=\"fs-id1165135517158\">We apply the distributive property:<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{l}z=12\\left(\\frac{\\sqrt{3}}{2}+\\frac{1}{2}i\\right)\\hfill \\\\ \\text{ }=\\left(12\\right)\\frac{\\sqrt{3}}{2}+\\left(12\\right)\\frac{1}{2}i\\hfill \\\\ \\text{ }=6\\sqrt{3}+6i\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137733561\">The rectangular form of the given point in complex form is[latex]\\,6\\sqrt{3}+6i.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_08_05_07\" class=\"textbox examples\">\n<div id=\"fs-id1165135571758\">\n<div id=\"fs-id1165135571760\">\n<h3>Finding the Rectangular Form of a Complex Number<\/h3>\n<p id=\"fs-id1165137844019\">Find the rectangular form of the complex number given[latex]\\,r=13\\,[\/latex]and[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{5}{12}.[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133210162\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134537174\">If[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{5}{12},\\,[\/latex]and[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{y}{x},\\,[\/latex]we first determine[latex]\\,r=\\sqrt{{x}^{2}+{y}^{2}}=\\sqrt{{12}^{2}+{5}^{2}}=13\\text{.}[\/latex] We then find[latex]\\,\\mathrm{cos}\\,\\theta =\\frac{x}{r}\\,[\/latex]and[latex]\\,\\mathrm{sin}\\,\\theta =\\frac{y}{r}.[\/latex]<\/p>\n<div id=\"fs-id1165137443057\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}z=13\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right)\\hfill \\\\ \\,\\,\\,=13\\left(\\frac{12}{13}+\\frac{5}{13}i\\right)\\hfill \\\\ \\,\\,\\,=12+5i\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135160119\">The rectangular form of the given number in complex form is[latex]\\,12+5i.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135511380\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_06\">\n<div id=\"fs-id1165137571081\">\n<p id=\"fs-id1165137571082\">Convert the complex number to rectangular form:<\/p>\n<div id=\"fs-id1165137571086\" class=\"unnumbered aligncenter\">[latex]z=4\\left(\\mathrm{cos}\\frac{11\\pi }{6}+i\\mathrm{sin}\\frac{11\\pi }{6}\\right)[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1165137444688\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137444689\">[latex]z=2\\sqrt{3}-2i[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134324933\" class=\"bc-section section\">\n<h3>Finding Products of Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165137660156\">Now that we can convert complex numbers to polar form we will learn how to perform operations on complex numbers in polar form. For the rest of this section, we will work with formulas developed by French mathematician Abraham <span class=\"no-emphasis\">de Moivre<\/span> (1667-1754). These formulas have made working with products, quotients, powers, and roots of complex numbers much simpler than they appear. The rules are based on multiplying the moduli and adding the arguments.<\/p>\n<div id=\"fs-id1165137473607\" class=\"textbox key-takeaways\">\n<h3>Products of Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165135178402\">If[latex]\\,{z}_{1}={r}_{1}\\left(\\mathrm{cos}\\,{\\theta }_{1}+i\\mathrm{sin}\\,{\\theta }_{1}\\right)\\,[\/latex]and[latex]\\,{z}_{2}={r}_{2}\\left(\\mathrm{cos}\\,{\\theta }_{2}+i\\mathrm{sin}\\,{\\theta }_{2}\\right),[\/latex] then the product of these numbers is given as:<\/p>\n<div id=\"fs-id1165137844291\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\hfill \\\\ \\begin{array}{l}{z}_{1}{z}_{2}={r}_{1}{r}_{2}\\left[\\mathrm{cos}\\left({\\theta }_{1}+{\\theta }_{2}\\right)+i\\mathrm{sin}\\left({\\theta }_{1}+{\\theta }_{2}\\right)\\right]\\hfill \\\\ {z}_{1}{z}_{2}={r}_{1}{r}_{2}\\text{cis}\\left({\\theta }_{1}+{\\theta }_{2}\\right)\\hfill \\end{array}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137738246\">Notice that the product calls for multiplying the moduli and adding the angles.<\/p>\n<\/div>\n<div id=\"Example_08_05_08\" class=\"textbox examples\">\n<div id=\"fs-id1165137665678\">\n<div id=\"fs-id1165137665680\">\n<h3>Finding the Product of Two Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165137583978\">Find the product of[latex]\\,{z}_{1}{z}_{2},\\,[\/latex]given[latex]\\,{z}_{1}=4\\left(\\mathrm{cos}\\left(80\u00b0\\right)+i\\mathrm{sin}\\left(80\u00b0\\right)\\right)\\,[\/latex]and[latex]\\,{z}_{2}=2\\left(\\mathrm{cos}\\left(145\u00b0\\right)+i\\mathrm{sin}\\left(145\u00b0\\right)\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135369384\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137531259\">Follow the formula<\/p>\n<div id=\"fs-id1165137531262\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{z}_{1}{z}_{2}=4\\cdot 2\\left[\\mathrm{cos}\\left(80\u00b0+145\u00b0\\right)+i\\mathrm{sin}\\left(80\u00b0+145\u00b0\\right)\\right]\\hfill \\\\ {z}_{1}{z}_{2}=8\\left[\\mathrm{cos}\\left(225\u00b0\\right)+i\\mathrm{sin}\\left(225\u00b0\\right)\\right]\\hfill \\\\ {z}_{1}{z}_{2}=8\\left[\\mathrm{cos}\\left(\\frac{5\\pi }{4}\\right)+i\\mathrm{sin}\\left(\\frac{5\\pi }{4}\\right)\\right]\\hfill \\\\ {z}_{1}{z}_{2}=8\\left[-\\frac{\\sqrt{2}}{2}+i\\left(-\\frac{\\sqrt{2}}{2}\\right)\\right]\\hfill \\\\ {z}_{1}{z}_{2}=-4\\sqrt{2}-4i\\sqrt{2}\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137736474\" class=\"bc-section section\">\n<h3>Finding Quotients of Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165134389944\">The quotient of two complex numbers in polar form is the quotient of the two moduli and the difference of the two arguments.<\/p>\n<div id=\"fs-id1165137892433\" class=\"textbox key-takeaways\">\n<h3>Quotients of Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165135541727\">If[latex]\\,{z}_{1}={r}_{1}\\left(\\mathrm{cos}\\,{\\theta }_{1}+i\\mathrm{sin}\\,{\\theta }_{1}\\right)\\,[\/latex]and[latex]\\,{z}_{2}={r}_{2}\\left(\\mathrm{cos}\\,{\\theta }_{2}+i\\mathrm{sin}\\,{\\theta }_{2}\\right),[\/latex] then the quotient of these numbers is<\/p>\n<div id=\"fs-id1165135316034\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{{z}_{1}}{{z}_{2}}=\\frac{{r}_{1}}{{r}_{2}}\\left[\\mathrm{cos}\\left({\\theta }_{1}-{\\theta }_{2}\\right)+i\\mathrm{sin}\\left({\\theta }_{1}-{\\theta }_{2}\\right)\\right],\\,\\,{z}_{2}\\ne 0\\\\ \\frac{{z}_{1}}{{z}_{2}}=\\frac{{r}_{1}}{{r}_{2}}\\text{cis}\\left({\\theta }_{1}-{\\theta }_{2}\\right),\\,\\,{z}_{2}\\ne 0\\,\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135332827\">Notice that the moduli are divided, and the angles are subtracted.<\/p>\n<\/div>\n<div id=\"fs-id1165137897028\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1165135189946\"><strong>Given two complex numbers in polar form, find the quotient.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id1165135260707\" type=\"1\">\n<li>Divide[latex]\\,\\frac{{r}_{1}}{{r}_{2}}.[\/latex]<\/li>\n<li>Find[latex]\\,{\\theta }_{1}-{\\theta }_{2}.[\/latex]<\/li>\n<li>Substitute the results into the formula:[latex]\\,z=r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right).\\,[\/latex]Replace[latex]\\,r\\,[\/latex]with[latex]\\,\\frac{{r}_{1}}{{r}_{2}},\\,[\/latex]and replace[latex]\\,\\theta \\,[\/latex]with[latex]\\,{\\theta }_{1}-{\\theta }_{2}.[\/latex]<\/li>\n<li>Calculate the new trigonometric expressions and multiply through by[latex]\\,r.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_08_05_09\" class=\"textbox examples\">\n<div id=\"fs-id1165134329605\">\n<div id=\"fs-id1165134329607\">\n<h3>Finding the Quotient of Two Complex Numbers<\/h3>\n<p id=\"fs-id1165135436548\">Find the quotient of[latex]\\,{z}_{1}=2\\left(\\mathrm{cos}\\left(213\u00b0\\right)+i\\mathrm{sin}\\left(213\u00b0\\right)\\right)\\,[\/latex]and[latex]\\,{z}_{2}=4\\left(\\mathrm{cos}\\left(33\u00b0\\right)+i\\mathrm{sin}\\left(33\u00b0\\right)\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133318744\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134129652\">Using the formula, we have<\/p>\n<div id=\"fs-id1165134129655\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{{z}_{1}}{{z}_{2}}=\\frac{2}{4}\\left[\\mathrm{cos}\\left(213\u00b0-33\u00b0\\right)+i\\mathrm{sin}\\left(213\u00b0-33\u00b0\\right)\\right]\\hfill \\\\ \\frac{{z}_{1}}{{z}_{2}}=\\frac{1}{2}\\left[\\mathrm{cos}\\left(180\u00b0\\right)+i\\mathrm{sin}\\left(180\u00b0\\right)\\right]\\hfill \\\\ \\frac{{z}_{1}}{{z}_{2}}=\\frac{1}{2}\\left[-1+0i\\right]\\hfill \\\\ \\frac{{z}_{1}}{{z}_{2}}=-\\frac{1}{2}+0i\\hfill \\\\ \\frac{{z}_{1}}{{z}_{2}}=-\\frac{1}{2}\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135593394\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"ti_08_05_07\">\n<div id=\"fs-id1165134568987\">\n<p id=\"fs-id1165134568988\">Find the product and the quotient of[latex]\\,{z}_{1}=2\\sqrt{3}\\left(\\mathrm{cos}\\left(150\u00b0\\right)+i\\mathrm{sin}\\left(150\u00b0\\right)\\right)\\,[\/latex]and[latex]\\,{z}_{2}=2\\left(\\mathrm{cos}\\left(30\u00b0\\right)+i\\mathrm{sin}\\left(30\u00b0\\right)\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135707943\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135707944\">[latex]\\,{z}_{1}{z}_{2}=-4\\sqrt{3};\\frac{{z}_{1}}{{z}_{2}}=-\\frac{\\sqrt{3}}{2}+\\frac{3}{2}i\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137673550\" class=\"bc-section section\">\n<h3>Finding Powers of Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165137673556\">Finding powers of complex numbers is greatly simplified using De Moivre\u2019s Theorem. It states that, for a positive integer[latex]\\,n,{z}^{n}\\,[\/latex]is found by raising the modulus to the[latex]\\,n\\text{th}\\,[\/latex]power and multiplying the argument by[latex]\\,n.\\,[\/latex]It is the standard method used in modern mathematics.<\/p>\n<div id=\"fs-id1165135388434\" class=\"textbox key-takeaways\">\n<h3>De Moivre\u2019s Theorem<\/h3>\n<p id=\"fs-id1165133349328\">If[latex]\\,z=r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right)\\,[\/latex]is a complex number, then<\/p>\n<div id=\"fs-id1165135299806\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{z}^{n}={r}^{n}\\left[\\mathrm{cos}\\left(n\\theta \\right)+i\\mathrm{sin}\\left(n\\theta \\right)\\right]\\\\ {z}^{n}={r}^{n}\\text{cis}\\left(n\\theta \\right)\\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165137846221\">where[latex]\\,n\\,[\/latex]<br \/>\nis a positive integer.<\/p>\n<\/div>\n<div id=\"Example_08_05_10\" class=\"textbox examples\">\n<div id=\"fs-id1165133103896\">\n<div id=\"fs-id1165133103898\">\n<h3>Evaluating an Expression Using De Moivre\u2019s Theorem<\/h3>\n<p id=\"fs-id1165131884600\">Evaluate the expression[latex]\\,{\\left(1+i\\right)}^{5}\\,[\/latex]using De Moivre\u2019s Theorem.<\/p>\n<\/div>\n<div id=\"fs-id1165132959150\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165132959153\">Since De Moivre\u2019s Theorem applies to complex numbers written in polar form, we must first write[latex]\\,\\left(1+i\\right)\\,[\/latex]in polar form. Let us find[latex]\\,r.[\/latex]<\/p>\n<div id=\"fs-id1165135365743\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}r=\\sqrt{{x}^{2}+{y}^{2}}\\hfill \\\\ r=\\sqrt{{\\left(1\\right)}^{2}+{\\left(1\\right)}^{2}}\\hfill \\\\ r=\\sqrt{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135200427\">Then we find[latex]\\,\\theta .\\,[\/latex]Using the formula[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{y}{x}\\,[\/latex]gives<\/p>\n<div id=\"fs-id1165134130860\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\mathrm{tan}\\,\\theta =\\frac{1}{1}\\hfill \\\\ \\mathrm{tan}\\,\\theta =1\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\theta =\\frac{\\pi }{4}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165135415590\">Use De Moivre\u2019s Theorem to evaluate the expression.<\/p>\n<div id=\"fs-id1165135415594\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{\\left(a+bi\\right)}^{n}={r}^{n}\\left[\\mathrm{cos}\\left(n\\theta \\right)+i\\mathrm{sin}\\left(n\\theta \\right)\\right]\\hfill \\\\ \\,\\,\\,\\,{\\left(1+i\\right)}^{5}={\\left(\\sqrt{2}\\right)}^{5}\\left[\\mathrm{cos}\\left(5\\cdot \\frac{\\pi }{4}\\right)+i\\mathrm{sin}\\left(5\\cdot \\frac{\\pi }{4}\\right)\\right]\\hfill \\\\ \\,\\,\\,\\,{\\left(1+i\\right)}^{5}=4\\sqrt{2}\\left[\\mathrm{cos}\\left(\\frac{5\\pi }{4}\\right)+i\\mathrm{sin}\\left(\\frac{5\\pi }{4}\\right)\\right]\\hfill \\\\ \\,\\,\\,\\,{\\left(1+i\\right)}^{5}=4\\sqrt{2}\\left[-\\frac{\\sqrt{2}}{2}+i\\left(-\\frac{\\sqrt{2}}{2}\\right)\\right]\\hfill \\\\ \\,\\,\\,\\,{\\left(1+i\\right)}^{5}=-4-4i\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135204415\" class=\"bc-section section\">\n<h3>Finding Roots of Complex Numbers in Polar Form<\/h3>\n<p id=\"fs-id1165135204421\">To find the <span class=\"no-emphasis\"><em>n<\/em>th root of a complex number<\/span> in polar form, we use the[latex]\\,n\\text{th}\\,[\/latex]Root Theorem or <span class=\"no-emphasis\">De Moivre\u2019s Theorem<\/span> and raise the complex number to a power with a rational exponent. There are several ways to represent a formula for finding[latex]\\,n\\text{th}\\,[\/latex]roots of complex numbers in polar form.<\/p>\n<div id=\"fs-id1165135420398\">\n<h3>The <em>n<\/em>th Root Theorem<\/h3>\n<p id=\"fs-id1165135436403\">To find the[latex]\\,n\\text{th}\\,[\/latex]root of a complex number in polar form, use the formula given as<\/p>\n<div id=\"fs-id1165134281475\" class=\"unnumbered aligncenter\">[latex]{z}^{\\frac{1}{n}}={r}^{\\frac{1}{n}}\\left[\\mathrm{cos}\\left(\\frac{\\theta }{n}+\\frac{2k\\pi }{n}\\right)+i\\mathrm{sin}\\left(\\frac{\\theta }{n}+\\frac{2k\\pi }{n}\\right)\\right][\/latex]<\/div>\n<p id=\"fs-id1165134485535\">where[latex]\\,k=0,\\,\\,1,\\,\\,2,\\,\\,3,\\,.\\,\\,.\\,\\,.\\,\\,,\\,\\,n-1.\\,[\/latex]We add [latex]\\,\\frac{2k\\pi }{n}\\,\\,[\/latex]to[latex]\\,\\frac{\\theta }{n}\\,[\/latex]in order to obtain the periodic roots.<\/p>\n<\/div>\n<div id=\"Example_08_05_11\" class=\"textbox examples\">\n<div id=\"fs-id1165133341021\">\n<div id=\"fs-id1165133341023\">\n<h3>Finding the <em>n<\/em>th Root of a Complex Number<\/h3>\n<p id=\"fs-id1165133046793\">Evaluate the cube roots of[latex]\\,z=8\\left(\\mathrm{cos}\\left(\\frac{2\\pi }{3}\\right)+i\\mathrm{sin}\\left(\\frac{2\\pi }{3}\\right)\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137656965\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137656967\">We have<\/p>\n<div id=\"fs-id1165137656970\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{z}^{\\frac{1}{3}}={8}^{\\frac{1}{3}}\\left[\\mathrm{cos}\\left(\\frac{\\frac{2\\pi }{3}}{3}+\\frac{2k\\pi }{3}\\right)+i\\mathrm{sin}\\left(\\frac{\\frac{2\\pi }{3}}{3}+\\frac{2k\\pi }{3}\\right)\\right]\\hfill \\\\ {z}^{\\frac{1}{3}}=2\\left[\\mathrm{cos}\\left(\\frac{2\\pi }{9}+\\frac{2k\\pi }{3}\\right)+i\\mathrm{sin}\\left(\\frac{2\\pi }{9}+\\frac{2k\\pi }{3}\\right)\\right]\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134060257\">There will be three roots:[latex]\\,k=0,\\,\\,1,\\,\\,2.\\,[\/latex]When[latex]\\,k=0,\\,[\/latex]we have<\/p>\n<div id=\"fs-id1165135365831\" class=\"unnumbered aligncenter\">[latex]{z}^{\\frac{1}{3}}=2\\left(\\mathrm{cos}\\left(\\frac{2\\pi }{9}\\right)+i\\mathrm{sin}\\left(\\frac{2\\pi }{9}\\right)\\right)[\/latex]<\/div>\n<p id=\"fs-id1165135510902\">When[latex]\\,k=1,\\,[\/latex]we have<\/p>\n<div id=\"fs-id1165134258392\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}{z}^{\\frac{1}{3}}=2\\left[\\mathrm{cos}\\left(\\frac{2\\pi }{9}+\\frac{6\\pi }{9}\\right)+i\\mathrm{sin}\\left(\\frac{2\\pi }{9}+\\frac{6\\pi }{9}\\right)\\right]\\begin{array}{cccc}& & & \\end{array}\\text{ Add }\\frac{2\\left(1\\right)\\pi }{3}\\text{ to each angle.}\\hfill \\\\ {z}^{\\frac{1}{3}}=2\\left(\\mathrm{cos}\\left(\\frac{8\\pi }{9}\\right)+i\\mathrm{sin}\\left(\\frac{8\\pi }{9}\\right)\\right)\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165134311190\">When[latex]\\,k=2,\\,[\/latex] we have<\/p>\n<div id=\"fs-id1165133221800\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ll}{z}^{\\frac{1}{3}}=2\\left[\\mathrm{cos}\\left(\\frac{2\\pi }{9}+\\frac{12\\pi }{9}\\right)+i\\mathrm{sin}\\left(\\frac{2\\pi }{9}+\\frac{12\\pi }{9}\\right)\\right]\\begin{array}{cccc}& & & \\end{array}\\hfill & \\text{Add }\\frac{2\\left(2\\right)\\pi }{3}\\text{ to each angle.}\\hfill \\\\ {z}^{\\frac{1}{3}}=2\\left(\\mathrm{cos}\\left(\\frac{14\\pi }{9}\\right)+i\\mathrm{sin}\\left(\\frac{14\\pi }{9}\\right)\\right)\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1165133102510\">Remember to find the common denominator to simplify fractions in situations like this one. For[latex]\\,k=1,\\,[\/latex]the angle simplification is<\/p>\n<div id=\"fs-id1165134192876\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{l}\\frac{\\frac{2\\pi }{3}}{3}+\\frac{2\\left(1\\right)\\pi }{3}=\\frac{2\\pi }{3}\\left(\\frac{1}{3}\\right)+\\frac{2\\left(1\\right)\\pi }{3}\\left(\\frac{3}{3}\\right)\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{2\\pi }{9}+\\frac{6\\pi }{9}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{8\\pi }{9}\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135404709\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div id=\"fs-id1165135404715\">\n<div id=\"fs-id1165135404718\">\n<p id=\"fs-id1165135329823\">Find the four fourth roots of[latex]\\,16\\left(\\mathrm{cos}\\left(120\u00b0\\right)+i\\mathrm{sin}\\left(120\u00b0\\right)\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134356866\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133111152\">[latex]{z}_{0}=2\\left(\\mathrm{cos}\\left(30\u00b0\\right)+i\\mathrm{sin}\\left(30\u00b0\\right)\\right)[\/latex]<\/p>\n<p id=\"fs-id1165134234229\">[latex]{z}_{1}=2\\left(\\mathrm{cos}\\left(120\u00b0\\right)+i\\mathrm{sin}\\left(120\u00b0\\right)\\right)[\/latex]<\/p>\n<p id=\"fs-id1165137428218\">[latex]{z}_{2}=2\\left(\\mathrm{cos}\\left(210\u00b0\\right)+i\\mathrm{sin}\\left(210\u00b0\\right)\\right)[\/latex]<\/p>\n<p id=\"fs-id1165134166590\">[latex]{z}_{3}=2\\left(\\mathrm{cos}\\left(300\u00b0\\right)+i\\mathrm{sin}\\left(300\u00b0\\right)\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133361907\" class=\"precalculus media\">\n<p id=\"fs-id1165133361914\">Access these online resources for additional instruction and practice with polar forms of complex numbers.<\/p>\n<ul id=\"fs-id1165135329691\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/prodquocomplex\">The Product and Quotient of Complex Numbers in Trigonometric Form<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/demoivre\">De Moivre\u2019s Theorem<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1165134129908\">\n<li>Complex numbers in the form[latex]\\,a+bi\\,[\/latex]are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. Label the <em>x-<\/em>axis as the <em>real <\/em>axis and the <em>y-<\/em>axis as the <em>imaginary<\/em> axis. See <a class=\"autogenerated-content\" href=\"#Example_08_05_01\">(Figure)<\/a>.<\/li>\n<li>The absolute value of a complex number is the same as its magnitude. It is the distance from the origin to the point:[latex]\\,|z|=\\sqrt{{a}^{2}+{b}^{2}}.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_05_02\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_05_03\">(Figure)<\/a>.<\/li>\n<li>To write complex numbers in polar form, we use the formulas[latex]\\,x=r\\mathrm{cos}\\,\\theta ,y=r\\mathrm{sin}\\,\\theta ,\\,[\/latex]and [latex]\\,r=\\sqrt{{x}^{2}+{y}^{2}}.\\,[\/latex]Then,[latex]\\,z=r\\left(\\mathrm{cos}\\,\\theta +i\\mathrm{sin}\\,\\theta \\right).\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_05_04\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_05_05\">(Figure)<\/a>.<\/li>\n<li>To convert from polar form to rectangular form, first evaluate the trigonometric functions. Then, multiply through by[latex]\\,r.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_05_06\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_08_05_07\">(Figure)<\/a>.<\/li>\n<li>To find the product of two complex numbers, multiply the two moduli and add the two angles. Evaluate the trigonometric functions, and multiply using the distributive property. See <a class=\"autogenerated-content\" href=\"#Example_08_05_08\">(Figure)<\/a>.<\/li>\n<li>To find the quotient of two complex numbers in polar form, find the quotient of the two moduli and the difference of the two angles. See <a class=\"autogenerated-content\" href=\"#Example_08_05_09\">(Figure)<\/a>.<\/li>\n<li>To find the power of a complex number[latex]\\,{z}^{n},\\,[\/latex]raise [latex]\\,r\\,[\/latex] to the power [latex]\\,n,[\/latex] and multiply [latex]\\,\\theta \\,[\/latex] by [latex]\\,n.\\,[\/latex]See <a class=\"autogenerated-content\" href=\"#Example_08_05_10\">(Figure)<\/a>.<\/li>\n<li>Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. See <a class=\"autogenerated-content\" href=\"#Example_08_05_11\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1165134370081\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1165135613677\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1165135613682\">\n<div id=\"fs-id1165135613684\">\n<p id=\"fs-id1165135613686\">A complex number is[latex]\\,a+bi.\\,[\/latex]Explain each part.<\/p>\n<\/div>\n<div id=\"fs-id1165134395224\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134395227\"><em>a<\/em> is the real part, <em>b<\/em> is the imaginary part, and[latex]\\,i=\\sqrt{-1}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135586313\">\n<div id=\"fs-id1165131926307\">\n<p id=\"fs-id1165131926309\">What does the absolute value of a complex number represent?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131926314\">\n<div id=\"fs-id1165131926316\">\n<p id=\"fs-id1165131926319\">How is a complex number converted to polar form?<\/p>\n<\/div>\n<div id=\"fs-id1165131926323\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165131926325\">Polar form converts the real and imaginary part of the complex number in polar form using[latex]\\,x=r\\mathrm{cos}\\theta \\,[\/latex] and [latex]\\,y=r\\mathrm{sin}\\theta .[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134205897\">\n<div id=\"fs-id1165134205899\">\n<p id=\"fs-id1165134205901\">How do we find the product of two complex numbers?<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134205906\">\n<div id=\"fs-id1165133309248\">\n<p id=\"fs-id1165133309250\">What is De Moivre\u2019s Theorem and what is it used for?<\/p>\n<\/div>\n<div id=\"fs-id1165133309255\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133309257\">[latex]{z}^{n}={r}^{n}\\left(\\mathrm{cos}\\left(n\\theta \\right)+i\\mathrm{sin}\\left(n\\theta \\right)\\right)\\,[\/latex]It is used to simplify polar form when a number has been raised to a power.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134183779\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id1165135665413\">For the following exercises, find the absolute value of the given complex number.<\/p>\n<div id=\"fs-id1165135665416\">\n<div id=\"fs-id1165135665418\">\n<p id=\"fs-id1165135665420\">[latex]5+\\text{\u200b}3i[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134385720\">\n<div id=\"fs-id1165134385722\">\n<p id=\"fs-id1165134385725\">[latex]-7+\\text{\u200b}i[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135499578\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135499580\">[latex]5\\sqrt{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135541528\">\n<div id=\"fs-id1165134481951\">\n<p id=\"fs-id1165134481953\">[latex]-3-3i[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135633877\">\n<div id=\"fs-id1165135633879\">\n<p id=\"fs-id1165135633881\">[latex]\\sqrt{2}-6i[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133213891\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133213893\">[latex]\\sqrt{38}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135694472\">\n<div id=\"fs-id1165135694474\">\n<p id=\"fs-id1165137843211\">[latex]2i[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137843226\">\n<div id=\"fs-id1165137843228\">\n<p id=\"fs-id1165134116954\">[latex]2.2-3.1i[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134116972\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137843105\">[latex]\\sqrt{14.45}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134386618\">For the following exercises, write the complex number in polar form.<\/p>\n<div id=\"fs-id1165134386621\">\n<div id=\"fs-id1165134386623\">\n<p id=\"fs-id1165134386625\">[latex]2+2i[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134178500\">\n<div id=\"fs-id1165134178502\">\n<p id=\"fs-id1165134178504\">[latex]8-4i[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134129997\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134129999\">[latex]4\\sqrt{5}\\mathrm{cis}\\left(333.4\u00b0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132959023\">\n<div id=\"fs-id1165132959025\">\n<p id=\"fs-id1165132959027\">[latex]-\\frac{1}{2}-\\frac{1}{2}\\text{\u200b}i[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135633990\">\n<div id=\"fs-id1165135633992\">\n<p id=\"fs-id1165135317569\">[latex]\\sqrt{3}+i[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165131993571\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165131993573\">[latex]2\\mathrm{cis}\\left(\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135207298\">\n<div id=\"fs-id1165135207300\">\n<p id=\"fs-id1165135207302\">[latex]3i[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135499587\">For the following exercises, convert the complex number from polar to rectangular form.<\/p>\n<div id=\"fs-id1165135499590\">\n<div id=\"fs-id1165135499592\">\n<p id=\"fs-id1165135499594\">[latex]z=7\\mathrm{cis}\\left(\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135367688\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135367691\">[latex]\\frac{7\\sqrt{3}}{2}+i\\frac{7}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135532374\">\n<div>\n<p id=\"fs-id1165135407393\">[latex]z=2\\mathrm{cis}\\left(\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134089467\">\n<div id=\"fs-id1165134089469\">\n<p id=\"fs-id1165134089471\">[latex]z=4\\mathrm{cis}\\left(\\frac{7\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134573209\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134573211\">[latex]-2\\sqrt{3}-2i[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134190676\">\n<div id=\"fs-id1165134190678\">\n<p id=\"fs-id1165134190680\">[latex]z=7\\mathrm{cis}\\left(25\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135388835\">\n<div id=\"fs-id1165135388838\">\n<p id=\"fs-id1165135388840\">[latex]z=3\\mathrm{cis}\\left(240\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135640462\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135640464\">[latex]-1.5-i\\frac{3\\sqrt{3}}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135519200\">\n<div id=\"fs-id1165135519202\">\n<p id=\"fs-id1165135519205\">[latex]z=\\sqrt{2}\\mathrm{cis}\\left(100\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165134357546\">For the following exercises, find[latex]\\,{z}_{1}{z}_{2}\\,[\/latex]in polar form.<\/p>\n<div id=\"fs-id1165133078076\">\n<div id=\"fs-id1165133078079\">\n<p id=\"fs-id1165134130033\">[latex]{z}_{1}=2\\sqrt{3}\\mathrm{cis}\\left(116\u00b0\\right);\\,\\text{ }{z}_{2}=2\\mathrm{cis}\\left(82\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135419732\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135419734\">[latex]4\\sqrt{3}\\mathrm{cis}\\left(198\u00b0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134430434\">\n<div id=\"fs-id1165134430436\">\n<p id=\"fs-id1165134430438\">[latex]{z}_{1}=\\sqrt{2}\\mathrm{cis}\\left(205\u00b0\\right);\\text{ }{z}_{2}=2\\sqrt{2}\\mathrm{cis}\\left(118\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137756875\">\n<div>[latex]{z}_{1}=3\\mathrm{cis}\\left(120\u00b0\\right);\\text{ }{z}_{2}=\\frac{1}{4}\\mathrm{cis}\\left(60\u00b0\\right)[\/latex]<\/div>\n<div id=\"fs-id1165135472921\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135472923\">[latex]\\frac{3}{4}\\mathrm{cis}\\left(180\u00b0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134093149\">\n<div id=\"fs-id1165134093151\">\n<p id=\"fs-id1165133023575\">[latex]{z}_{1}=3\\mathrm{cis}\\left(\\frac{\\pi }{4}\\right);\\text{ }{z}_{2}=5\\mathrm{cis}\\left(\\frac{\\pi }{6}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165131863233\">\n<div id=\"fs-id1165131863235\">\n<p id=\"fs-id1165131863237\">[latex]{z}_{1}=\\sqrt{5}\\mathrm{cis}\\left(\\frac{5\\pi }{8}\\right);\\text{ }{z}_{2}=\\sqrt{15}\\mathrm{cis}\\left(\\frac{\\pi }{12}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133077994\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135518168\">[latex]5\\sqrt{3}\\mathrm{cis}\\left(\\frac{17\\pi }{24}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134088751\">\n<div id=\"fs-id1165134088753\">\n<p id=\"fs-id1165134088755\">[latex]{z}_{1}=4\\mathrm{cis}\\left(\\frac{\\pi }{2}\\right);\\text{ }{z}_{2}=2\\mathrm{cis}\\left(\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135662421\">For the following exercises, find[latex]\\,\\frac{{z}_{1}}{{z}_{2}}\\,[\/latex]in polar form.<\/p>\n<div id=\"fs-id1165134160326\">\n<div id=\"fs-id1165134160328\">\n<p id=\"fs-id1165134160330\">[latex]{z}_{1}=21\\mathrm{cis}\\left(135\u00b0\\right);\\text{ }{z}_{2}=3\\mathrm{cis}\\left(65\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134174904\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134174906\">[latex]7\\mathrm{cis}\\left(70\u00b0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134221768\">\n<div id=\"fs-id1165134221770\">\n<p id=\"fs-id1165134221772\">[latex]{z}_{1}=\\sqrt{2}\\mathrm{cis}\\left(90\u00b0\\right);\\text{ }{z}_{2}=2\\mathrm{cis}\\left(60\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134325223\">\n<div id=\"fs-id1165134325225\">\n<p id=\"fs-id1165131863137\">[latex]{z}_{1}=15\\mathrm{cis}\\left(120\u00b0\\right);\\text{ }{z}_{2}=3\\mathrm{cis}\\left(40\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137480080\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137480082\">[latex]5\\mathrm{cis}\\left(80\u00b0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134279277\">\n<div id=\"fs-id1165134279279\">\n<p id=\"fs-id1165134279281\">[latex]{z}_{1}=6\\mathrm{cis}\\left(\\frac{\\pi }{3}\\right);\\text{ }{z}_{2}=2\\mathrm{cis}\\left(\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135652331\">\n<div id=\"fs-id1165134401618\">\n<p id=\"fs-id1165134401620\">[latex]{z}_{1}=5\\sqrt{2}\\mathrm{cis}\\left(\\pi \\right);\\text{ }{z}_{2}=\\sqrt{2}\\mathrm{cis}\\left(\\frac{2\\pi }{3}\\right)[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134388962\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134388964\">[latex]5\\mathrm{cis}\\left(\\frac{\\pi }{3}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134534047\">\n<div id=\"fs-id1165134534049\">\n<p id=\"fs-id1165134534051\">[latex]{z}_{1}=2\\mathrm{cis}\\left(\\frac{3\\pi }{5}\\right);\\text{ }{z}_{2}=3\\mathrm{cis}\\left(\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165133028482\">For the following exercises, find the powers of each complex number in polar form.<\/p>\n<div id=\"fs-id1165133028485\">\n<div id=\"fs-id1165133028487\">\n<p id=\"fs-id1165133028489\">Find[latex]\\,{z}^{3}\\,[\/latex]when[latex]\\,z=5\\mathrm{cis}\\left(45\u00b0\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133349423\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165133349425\">[latex]125\\mathrm{cis}\\left(135\u00b0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135523252\">\n<div id=\"fs-id1165135523254\">\n<p id=\"fs-id1165135523256\">Find[latex]\\,{z}^{4}\\,[\/latex]when[latex]\\,z=2\\mathrm{cis}\\left(70\u00b0\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135440037\">\n<div id=\"fs-id1165135440039\">\n<p id=\"fs-id1165135440041\">Find[latex]\\,{z}^{2}\\,[\/latex]when[latex]\\,z=3\\mathrm{cis}\\left(120\u00b0\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135512799\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135512801\">[latex]9\\mathrm{cis}\\left(240\u00b0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134113850\">\n<div id=\"fs-id1165134113852\">\n<p id=\"fs-id1165135619413\">Find[latex]\\,{z}^{2}\\,[\/latex]when[latex]\\,z=4\\mathrm{cis}\\left(\\frac{\\pi }{4}\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134070876\">\n<div id=\"fs-id1165134070879\">\n<p id=\"fs-id1165134070881\">Find[latex]\\,{z}^{4}\\,[\/latex]when[latex]\\,z=\\mathrm{cis}\\left(\\frac{3\\pi }{16}\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165134177544\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165134177546\">[latex]\\mathrm{cis}\\left(\\frac{3\\pi }{4}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134128415\">\n<div id=\"fs-id1165134128417\">\n<p id=\"fs-id1165134128419\">Find[latex]\\,{z}^{3}\\,[\/latex]when[latex]\\,z=3\\mathrm{cis}\\left(\\frac{5\\pi }{3}\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135352989\">For the following exercises, evaluate each root.<\/p>\n<div id=\"fs-id1165135352992\">\n<div id=\"fs-id1165134039259\">\n<p id=\"fs-id1165134039261\">Evaluate the cube root of[latex]\\,z\\,[\/latex]when[latex]\\,z=27\\mathrm{cis}\\left(240\u00b0\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135609212\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135609214\">[latex]\\,3\\mathrm{cis}\\left(80\u00b0\\right),3\\mathrm{cis}\\left(200\u00b0\\right),3\\mathrm{cis}\\left(320\u00b0\\right)\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135170392\">\n<div id=\"fs-id1165135170394\">\n<p id=\"fs-id1165135170396\">Evaluate the square root of[latex]\\,z\\,[\/latex]when[latex]\\,z=16\\mathrm{cis}\\left(100\u00b0\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134534293\">\n<div id=\"fs-id1165133243591\">\n<p id=\"fs-id1165133243593\">Evaluate the cube root of[latex]\\,z\\,[\/latex]when[latex]\\,z=32\\mathrm{cis}\\left(\\frac{2\\pi }{3}\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135238453\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165137780013\">[latex]\\,2\\sqrt[3]{4}\\mathrm{cis}\\left(\\frac{2\\pi }{9}\\right),2\\sqrt[3]{4}\\mathrm{cis}\\left(\\frac{8\\pi }{9}\\right),2\\sqrt[3]{4}\\mathrm{cis}\\left(\\frac{14\\pi }{9}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1165134378615\">Evaluate the square root of[latex]\\,z\\,[\/latex]when[latex]\\,z=32\\text{cis}\\left(\\pi \\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134356902\">\n<div id=\"fs-id1165134356904\">\n<p id=\"fs-id1165134356906\">Evaluate the cube root of[latex]\\,z\\,[\/latex]when[latex]\\,z=8\\mathrm{cis}\\left(\\frac{7\\pi }{4}\\right).[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165131907304\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165131907306\">[latex]2\\sqrt{2}\\mathrm{cis}\\left(\\frac{7\\pi }{8}\\right),2\\sqrt{2}\\mathrm{cis}\\left(\\frac{15\\pi }{8}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135586834\" class=\"bc-section section\">\n<h4>Graphical<\/h4>\n<p id=\"fs-id1165137895099\">For the following exercises, plot the complex number in the complex plane.<\/p>\n<div id=\"fs-id1165137895102\">\n<div id=\"fs-id1165137895104\">\n<p id=\"fs-id1165137895107\">[latex]2+4i[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135621933\">\n<div id=\"fs-id1165135621935\">\n<p id=\"fs-id1165135621938\">[latex]-3-3i[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133221775\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165133221780\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152636\/CNX_Precalc_Figure_08_05_202.jpg\" alt=\"Plot of -3 -3i in the complex plane (-3 along real axis, -3 along imaginary axis).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135457049\">\n<div id=\"fs-id1165135457051\">\n<p id=\"fs-id1165135457054\">[latex]5-4i[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165137835713\">\n<div id=\"fs-id1165137835715\">\n<p id=\"fs-id1165137835717\">[latex]-1-5i[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165133260388\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165133260393\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152645\/CNX_Precalc_Figure_08_05_204.jpg\" alt=\"Plot of -1 -5i in the complex plane (-1 along real axis, -5 along imaginary axis).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135664876\">\n<div id=\"fs-id1165135664878\">\n<p id=\"fs-id1165135664880\">[latex]3+2i[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135251281\">\n<div id=\"fs-id1165135251283\">\n<p id=\"fs-id1165135251285\">[latex]2i[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165135702570\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135702576\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152648\/CNX_Precalc_Figure_08_05_206.jpg\" alt=\"Plot of 2i in the complex plane (0 along the real axis, 2 along the imaginary axis).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134254413\">\n<div id=\"fs-id1165134254415\">\n<p id=\"fs-id1165134254418\">[latex]-4[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165134254432\">\n<div id=\"fs-id1165135160374\">[latex]6-2i[\/latex]<\/div>\n<div id=\"fs-id1165135672734\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165135672739\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152651\/CNX_Precalc_Figure_08_05_208.jpg\" alt=\"Plot of 6-2i in the complex plane (6 along the real axis, -2 along the imaginary axis).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135672751\">\n<div id=\"fs-id1165135672753\">\n<p id=\"fs-id1165133359361\">[latex]-2+i[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133359380\">\n<div id=\"fs-id1165137580684\">\n<p id=\"fs-id1165137580686\">[latex]1-4i[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id1165137580704\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p><span id=\"fs-id1165132949966\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19152654\/CNX_Precalc_Figure_08_05_210.jpg\" alt=\"Plot of 1-4i in the complex plane (1 along the real axis, -4 along the imaginary axis).\" \/><\/span><\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132949979\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1165137647538\">For the following exercises, find all answers rounded to the nearest hundredth.<\/p>\n<div id=\"fs-id1165137647541\">\n<div id=\"fs-id1165137647543\">\n<p id=\"fs-id1165137647545\">Use the rectangular to polar feature on the graphing calculator to change[latex]\\,5+5i\\,[\/latex]to polar form.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135186955\">\n<div id=\"fs-id1165135186958\">\n<p id=\"fs-id1165135186960\">Use the rectangular to polar feature on the graphing calculator to change[latex]\\,3-2i\\,[\/latex]<br \/>\nto polar form.<\/p>\n<\/div>\n<div id=\"fs-id1165134329643\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165132972772\">[latex]\\,3.61{e}^{-0.59i}\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165133243512\">\n<div id=\"fs-id1165133243514\">\n<p id=\"fs-id1165133243516\">Use the rectangular to polar feature on the graphing calculator to change [latex]-3-8i\\,[\/latex]<br \/>\nto polar form.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135189784\">\n<div id=\"fs-id1165135189786\">\n<p id=\"fs-id1165135189788\">Use the polar to rectangular feature on the graphing calculator to change[latex]\\,4\\mathrm{cis}\\left(120\u00b0\\right)\\,[\/latex]to rectangular form.<\/p>\n<\/div>\n<div id=\"fs-id1165135694966\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135694969\">[latex]\\,-2+3.46i\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1165132945471\">\n<div>\n<p id=\"fs-id1165132945475\">Use the polar to rectangular feature on the graphing calculator to change[latex]\\,2\\mathrm{cis}\\left(45\u00b0\\right)\\,[\/latex]to rectangular form.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1165135546054\">\n<div id=\"fs-id1165135546056\">\n<p id=\"fs-id1165135546058\">Use the polar to rectangular feature on the graphing calculator to change[latex]\\,5\\mathrm{cis}\\left(210\u00b0\\right)\\,[\/latex]to rectangular form.<\/p>\n<\/div>\n<div id=\"fs-id1165135252140\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1165135252142\">[latex]\\,-4.33-2.50i\\,[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id1165135344835\">\n<dt>argument<\/dt>\n<dd id=\"fs-id1165135344841\">the angle associated with a complex number; the angle between the line from the origin to the point and the positive real axis<\/dd>\n<\/dl>\n<dl id=\"fs-id1165135344846\">\n<dt>De Moivre\u2019s Theorem<\/dt>\n<dd id=\"fs-id1165134357535\">formula used to find the[latex]\\,n\\text{th}\\,[\/latex]power or <em>n<\/em>th roots of a complex number; states that, for a positive integer[latex]\\,n,{z}^{n}\\,[\/latex]is found by raising the modulus to the[latex]\\,n\\text{th}\\,[\/latex]power and multiplying the angles by[latex]\\,n\\,[\/latex]<\/dd>\n<\/dl>\n<dl id=\"fs-id1165134279306\">\n<dt>modulus<\/dt>\n<dd id=\"fs-id1165134279312\">the absolute value of a complex number, or the distance from the origin to the point[latex]\\,\\left(x,y\\right);\\,[\/latex]also called the amplitude<\/dd>\n<\/dl>\n<dl id=\"fs-id1165133162992\">\n<dt>polar form of a complex number<\/dt>\n<dd id=\"fs-id1165133162998\">a complex number expressed in terms of an angle [latex]\\theta[\/latex] and its distance from the origin[latex]\\,r;\\,[\/latex]can be found by using conversion formulas[latex]\\,x=r\\mathrm{cos}\\,\\theta ,\\,\\,y=r\\mathrm{sin}\\,\\theta ,\\,\\,[\/latex]and[latex]\\,r=\\sqrt{{x}^{2}+{y}^{2}}[\/latex]<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":6,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-158","chapter","type-chapter","status-publish","hentry"],"part":147,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/158","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\/158\/revisions"}],"predecessor-version":[{"id":159,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/158\/revisions\/159"}],"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\/158\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=158"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=158"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=158"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=158"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}