{"id":141,"date":"2019-08-20T17:03:11","date_gmt":"2019-08-20T21:03:11","guid":{"rendered":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/double-angle-half-angle-and-reduction-formulas\/"},"modified":"2022-06-01T10:39:32","modified_gmt":"2022-06-01T14:39:32","slug":"double-angle-half-angle-and-reduction-formulas","status":"publish","type":"chapter","link":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/chapter\/double-angle-half-angle-and-reduction-formulas\/","title":{"raw":"Double-Angle, Half-Angle, and Reduction Formulas","rendered":"Double-Angle, Half-Angle, and Reduction Formulas"},"content":{"raw":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\nIn this section, you will:\n<ul>\n \t<li>Use double-angle formulas to find exact values.<\/li>\n \t<li>Use double-angle formulas to verify identities.<\/li>\n \t<li>Use reduction formulas to simplify an expression.<\/li>\n \t<li>Use half-angle formulas to find exact values.<\/li>\n<\/ul>\n<\/div>\n<div class=\"wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"979\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144031\/CNX_Precalc_Figure_07_03_001.jpg\" alt=\"Picture of two bicycle ramps, one with a steep slope and one with a gentle slope.\" width=\"979\" height=\"287\"> <strong>Figure 1. <\/strong>Bicycle ramps for advanced riders have a steeper incline than those designed for novices.[\/caption]\n\n<\/div>\n<p id=\"fs-id1368511\">Bicycle ramps made for competition (see <a class=\"autogenerated-content\" href=\"#Figure_07_03_001\">(Figure)<\/a>) must vary in height depending on the skill level of the competitors. For advanced competitors, the angle formed by the ramp and the ground should be[latex]\\,\\theta \\,[\/latex]such that[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{5}{3}.\\,[\/latex]The angle is divided in half for novices. What is the steepness of the ramp for novices? In this section, we will investigate three additional categories of identities that we can use to answer questions such as this one.<\/p>\n\n<div id=\"fs-id1944514\" class=\"bc-section section\">\n<h3>Using Double-Angle Formulas to Find Exact Values<\/h3>\nIn the previous section, we used addition and subtraction formulas for trigonometric functions. Now, we take another look at those same formulas. The <span class=\"no-emphasis\">double-angle formulas<\/span> are a special case of the sum formulas, where[latex]\\,\\alpha =\\beta .\\,[\/latex]Deriving the double-angle formula for sine begins with the sum formula,\n<div id=\"fs-id1388755\" class=\"unnumbered aligncenter\">[latex]\\mathrm{sin}\\left(\\alpha +\\beta \\right)=\\mathrm{sin}\\,\\alpha \\,\\mathrm{cos}\\,\\beta +\\mathrm{cos}\\,\\alpha \\,\\mathrm{sin}\\,\\beta [\/latex]<\/div>\n<p id=\"fs-id1017955\">If we let[latex]\\,\\alpha =\\beta =\\theta ,[\/latex]then we have<\/p>\n\n<div id=\"fs-id1958300\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\left(\\theta +\\theta \\right)&amp; =&amp; \\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta +\\mathrm{cos}\\,\\theta \\,\\mathrm{sin}\\,\\theta \\hfill \\\\ \\hfill \\mathrm{sin}\\left(2\\theta \\right)&amp; =&amp; 2\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1691770\">Deriving the double-angle for cosine gives us three options. First, starting from the sum formula,[latex]\\,\\mathrm{cos}\\left(\\alpha +\\beta \\right)=\\mathrm{cos}\\,\\alpha \\,\\mathrm{cos}\\,\\beta -\\mathrm{sin}\\,\\alpha \\,\\mathrm{sin}\\,\\beta ,[\/latex]and letting[latex]\\,\\alpha =\\beta =\\theta ,[\/latex]we have<\/p>\n\n<div id=\"fs-id2073447\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(\\theta +\\theta \\right)&amp; =&amp; \\mathrm{cos}\\,\\theta \\,\\mathrm{cos}\\,\\theta -\\mathrm{sin}\\,\\theta \\,\\mathrm{sin}\\,\\theta \\hfill \\\\ \\hfill \\mathrm{cos}\\left(2\\theta \\right)&amp; =&amp; {\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1679855\">Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more variations. The first variation is:<\/p>\n\n<div id=\"fs-id1082227\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(2\\theta \\right)&amp; =&amp; {\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ &amp; =&amp; \\left(1-{\\mathrm{sin}}^{2}\\theta \\right)-{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ &amp; =&amp; 1-2{\\mathrm{sin}}^{2}\\theta \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1874226\">The second variation is:<\/p>\n\n<div id=\"fs-id1154694\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(2\\theta \\right)&amp; =&amp; {\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ &amp; =&amp; {\\mathrm{cos}}^{2}\\theta -\\left(1-{\\mathrm{cos}}^{2}\\theta \\right)\\hfill \\\\ &amp; =&amp; 2\\,{\\mathrm{cos}}^{2}\\theta -1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1541852\">Similarly, to derive the double-angle formula for tangent, replacing[latex]\\,\\alpha =\\beta =\\theta \\,[\/latex]in the sum formula gives<\/p>\n\n<div id=\"fs-id1688220\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{tan}\\left(\\alpha +\\beta \\right)&amp; =&amp; \\frac{\\mathrm{tan}\\,\\alpha +\\mathrm{tan}\\,\\beta }{1-\\mathrm{tan}\\,\\alpha \\,\\mathrm{tan}\\,\\beta }\\hfill \\\\ \\hfill \\mathrm{tan}\\left(\\theta +\\theta \\right)&amp; =&amp; \\frac{\\mathrm{tan}\\,\\theta +\\mathrm{tan}\\,\\theta }{1-\\mathrm{tan}\\,\\theta \\,\\mathrm{tan}\\,\\theta }\\hfill \\\\ \\hfill \\mathrm{tan}\\left(2\\theta \\right)&amp; =&amp; \\frac{2\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1992296\" class=\"textbox key-takeaways\">\n<h3>Double-Angle Formulas<\/h3>\n<p id=\"fs-id1681037\">The double-angle formulas are summarized as follows:<\/p>\n\n<div id=\"Eq_07_03_01\">[latex]\\begin{array}{ccc}\\hfill \\phantom{\\rule{.45em}{0ex}}\\mathrm{sin}\\left(2\\theta \\right)&amp; =&amp; 2\\,\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta \\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<div id=\"Eq_07_03_02\">[latex]\\begin{array}{ccc}\\hfill \\phantom{\\rule{1.5em}{0ex}}\\mathrm{cos}\\left(2\\theta \\right)&amp; =&amp; {\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ &amp; =&amp; 1-2\\,{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ &amp; =&amp; 2\\,{\\mathrm{cos}}^{2}\\theta -1\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<div id=\"Eq_07_03_03\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{tan}\\left(2\\theta \\right)&amp; =&amp; \\frac{2\\,\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1578259\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1342627\"><strong>Given the tangent of an angle and the quadrant in which it is located, use the double-angle formulas to find the exact value.\n<\/strong><\/p>\n\n<ol id=\"fs-id2248451\" type=\"1\">\n \t<li>Draw a triangle to reflect the given information.<\/li>\n \t<li>Determine the correct double-angle formula.<\/li>\n \t<li>Substitute values into the formula based on the triangle.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1764242\">\n<div id=\"fs-id1256954\">\n<h3>Using a Double-Angle Formula to Find the Exact Value Involving Tangent<\/h3>\n<p id=\"fs-id1491583\">Given that[latex]\\,\\mathrm{tan}\\,\\theta =-\\frac{3}{4}\\,[\/latex]and[latex]\\,\\theta \\,[\/latex]is in quadrant II, find the following:<\/p>\n\n<ol id=\"fs-id2182373\" type=\"a\">\n \t<li>[latex]\\mathrm{sin}\\left(2\\theta \\right)[\/latex]<\/li>\n \t<li>[latex]\\mathrm{cos}\\left(2\\theta \\right)[\/latex]<\/li>\n \t<li>[latex]\\mathrm{tan}\\left(2\\theta \\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1418852\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1418852\"]\n<p id=\"fs-id1418852\">If we draw a triangle to reflect the information given, we can find the values needed to solve the problems on the image. We are given[latex]\\,\\mathrm{tan}\\,\\theta =-\\frac{3}{4},[\/latex]such that[latex]\\,\\theta \\,[\/latex]is in quadrant II. The tangent of an angle is equal to the opposite side over the adjacent side, and because[latex]\\,\\theta \\,[\/latex]is in the second quadrant, the adjacent side is on the <em>x<\/em>-axis and is negative. Use the <span class=\"no-emphasis\">Pythagorean Theorem<\/span> to find the length of the hypotenuse:<\/p>\n\n<div id=\"fs-id2303132\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {\\left(-4\\right)}^{2}+{\\left(3\\right)}^{2}&amp; =&amp; {c}^{2}\\hfill \\\\ \\hfill 16+9&amp; =&amp; {c}^{2}\\hfill \\\\ \\hfill 25&amp; =&amp; {c}^{2}\\hfill \\\\ \\hfill c&amp; =&amp; 5\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1208430\">Now we can draw a triangle similar to the one shown in <a class=\"autogenerated-content\" href=\"#Figure_07_03_02\">(Figure)<\/a>.<\/p>\n\n<div id=\"Figure_07_03_02\" 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\/19144034\/CNX_Precalc_Figure_07_03_002.jpg\" alt=\"Diagram of a triangle in the x,y-plane. The vertices are at the origin, (-4,0), and (-4,3). The angle at the origin is theta. The angle formed by the side (-4,3) to (-4,0) forms a right angle with the x axis. The hypotenuse across from the right angle is length 5.\" width=\"487\" height=\"251\"> <strong>Figure 2.<\/strong>[\/caption]\n\n<\/div>\n<ol id=\"fs-id1754093\" type=\"a\">\n \t<li>Let\u2019s begin by writing the double-angle formula for sine.\n<div id=\"fs-id1298188\" class=\"unnumbered aligncenter\">[latex]\\mathrm{sin}\\left(2\\theta \\right)=2\\,\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta [\/latex]<\/div>\n<p id=\"fs-id948689\">We see that we to need to find[latex]\\,\\mathrm{sin}\\,\\theta \\,[\/latex]and[latex]\\,\\mathrm{cos}\\,\\theta .\\,[\/latex]Based on <a class=\"autogenerated-content\" href=\"#Figure_07_03_02\">(Figure)<\/a>, we see that the hypotenuse equals 5, so[latex]\\,\\mathrm{sin}\\,\\theta =\\frac{3}{5},[\/latex]and[latex]\\,\\mathrm{cos}\\,\\theta =-\\frac{4}{5}.\\,[\/latex]Substitute these values into the equation, and simplify.<\/p>\n<p id=\"fs-id1494112\">Thus,<\/p>\n\n<div id=\"fs-id1316292\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\left(2\\theta \\right)&amp; =&amp; 2\\left(\\frac{3}{5}\\right)\\left(-\\frac{4}{5}\\right)\\hfill \\\\ &amp; =&amp; -\\frac{24}{25}\\hfill \\end{array}[\/latex]<\/div><\/li>\n \t<li>Write the double-angle formula for cosine.\n<div id=\"fs-id2347943\" class=\"unnumbered aligncenter\">[latex]\\mathrm{cos}\\left(2\\theta \\right)={\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta [\/latex]<\/div>\n<p id=\"fs-id1793867\">Again, substitute the values of the sine and cosine into the equation, and simplify.<\/p>\n\n<div id=\"fs-id1530574\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(2\\theta \\right)&amp; =&amp; {\\left(-\\frac{4}{5}\\right)}^{2}-{\\left(\\frac{3}{5}\\right)}^{2}\\hfill \\\\ &amp; =&amp; \\frac{16}{25}-\\frac{9}{25}\\hfill \\\\ &amp; =&amp; \\frac{7}{25}\\hfill \\end{array}[\/latex]<\/div><\/li>\n \t<li>Write the double-angle formula for tangent.\n<div id=\"fs-id2078890\" class=\"unnumbered aligncenter\">[latex]\\mathrm{tan}\\left(2\\theta \\right)=\\frac{2\\,\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }[\/latex]<\/div>\n<p id=\"fs-id2130326\">In this formula, we need the tangent, which we were given as[latex]\\,\\mathrm{tan}\\,\\theta =-\\frac{3}{4}.\\,[\/latex]Substitute this value into the equation, and simplify.<\/p>\n\n<div id=\"fs-id2121569\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{tan}\\left(2\\theta \\right)&amp; =&amp; \\frac{2\\left(-\\frac{3}{4}\\right)}{1-{\\left(-\\frac{3}{4}\\right)}^{2}}\\hfill \\\\ &amp; =&amp; \\frac{-\\frac{3}{2}}{1-\\frac{9}{16}}\\hfill \\\\ &amp; =&amp; -\\frac{3}{2}\\left(\\frac{16}{7}\\right)\\hfill \\\\ &amp; =&amp; -\\frac{24}{7}\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div><\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1864098\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id2336091\">\n<p id=\"fs-id1506209\">Given[latex]\\,\\mathrm{sin}\\,\\alpha =\\frac{5}{8},[\/latex]with[latex]\\,\\theta \\,[\/latex]in quadrant I, find[latex]\\,\\mathrm{cos}\\left(2\\alpha \\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1338288\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1338288\"]\n<p id=\"fs-id1338288\">[latex]\\mathrm{cos}\\left(2\\alpha \\right)=\\frac{7}{32}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1400362\">\n<div id=\"fs-id2327940\">\n<h3>Using the Double-Angle Formula for Cosine without Exact Values<\/h3>\n<p id=\"fs-id2303223\">Use the double-angle formula for cosine to write[latex]\\,\\mathrm{cos}\\left(6x\\right)\\,[\/latex]in terms of[latex]\\,\\mathrm{cos}\\left(3x\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">\n\n[reveal-answer q=\"1928247\"]Show Solution[\/reveal-answer][hidden-answer a=\"1928247\"]\n<div id=\"fs-id2038748\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(6x\\right)&amp; =&amp; \\mathrm{cos}\\left(3x+3x\\right)\\hfill \\\\ &amp; =&amp; \\mathrm{cos}\\,3x\\,\\mathrm{cos}\\,3x-\\mathrm{sin}\\,3x\\,\\mathrm{sin}\\,3x\\hfill \\\\ &amp; =&amp; {\\mathrm{cos}}^{2}3x-{\\mathrm{sin}}^{2}3x\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1167807\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1514609\">This example illustrates that we can use the double-angle formula without having exact values. It emphasizes that the pattern is what we need to remember and that identities are true for all values in the domain of the trigonometric function.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2249050\" class=\"bc-section section\">\n<h3>Using Double-Angle Formulas to Verify Identities<\/h3>\n<p id=\"fs-id1313018\">Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.<\/p>\n\n<div class=\"textbox examples\">\n<div id=\"fs-id728598\">\n<div id=\"fs-id2712986\">\n<h3>Using the Double-Angle Formulas to Verify an Identity<\/h3>\n<p id=\"fs-id2067948\">Verify the following identity using double-angle formulas:<\/p>\n\n<div id=\"fs-id1169806\" class=\"unnumbered aligncenter\">[latex]1+\\mathrm{sin}\\left(2\\theta \\right)={\\left(\\mathrm{sin}\\,\\theta +\\mathrm{cos}\\,\\theta \\right)}^{2}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2208344\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2208344\"]\n<p id=\"fs-id2208344\">We will work on the right side of the equal sign and rewrite the expression until it matches the left side.<\/p>\n\n<div id=\"fs-id1478975\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {\\left(\\mathrm{sin}\\,\\theta +\\mathrm{cos}\\,\\theta \\right)}^{2}&amp; =&amp; {\\mathrm{sin}}^{2}\\theta +2\\,\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta +{\\mathrm{cos}}^{2}\\theta \\hfill \\\\ &amp; =&amp; \\left({\\mathrm{sin}}^{2}\\theta +{\\mathrm{cos}}^{2}\\theta \\right)+2\\,\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta \\hfill \\\\ &amp; =&amp; 1+2\\,\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta \\hfill \\\\ &amp; =&amp; 1+\\mathrm{sin}\\left(2\\theta \\right)\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1364527\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1351871\">This process is not complicated, as long as we recall the perfect square formula from algebra:<\/p>\n\n<div id=\"fs-id1205403\" class=\"unnumbered aligncenter\">[latex]{\\left(a\u00b1b\\right)}^{2}={a}^{2}\u00b12ab+{b}^{2}[\/latex]<\/div>\n<p id=\"fs-id2431590\">where[latex]\\,a=\\mathrm{sin}\\,\\theta \\,[\/latex]and[latex]\\,b=\\mathrm{cos}\\,\\theta .\\,[\/latex]Part of being successful in mathematics is the ability to recognize patterns. While the terms or symbols may change, the algebra remains consistent.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1897349\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id2301016\">\n<p id=\"fs-id1530074\">Verify the identity:[latex]\\,{\\mathrm{cos}}^{4}\\theta -{\\mathrm{sin}}^{4}\\theta =\\mathrm{cos}\\left(2\\theta \\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1313281\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1313281\"]\n<p id=\"fs-id1313281\">[latex]{\\mathrm{cos}}^{4}\\theta -{\\mathrm{sin}}^{4}\\theta =\\left({\\mathrm{cos}}^{2}\\theta +{\\mathrm{sin}}^{2}\\theta \\right)\\left({\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\right)=\\mathrm{cos}\\left(2\\theta \\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1524820\">\n<div id=\"fs-id1316219\">\n<h3>Verifying a Double-Angle Identity for Tangent<\/h3>\n<p id=\"fs-id1319888\">Verify the identity:<\/p>\n\n<div id=\"fs-id1486278\" class=\"unnumbered aligncenter\">[latex]\\mathrm{tan}\\left(2\\theta \\right)=\\frac{2}{\\mathrm{cot}\\,\\theta -\\mathrm{tan}\\,\\theta }[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1333908\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1333908\"]\n<p id=\"fs-id1333908\">In this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the equation.<\/p>\n\n<div id=\"fs-id1199236\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill \\mathrm{tan}\\left(2\\theta \\right)&amp; =&amp; \\frac{2\\,\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Double-angle formula}\\hfill \\\\ &amp; =&amp; \\frac{2\\,\\mathrm{tan}\\,\\theta \\left(\\frac{1}{\\mathrm{tan}\\,\\theta }\\right)}{\\left(1-{\\mathrm{tan}}^{2}\\theta \\right)\\left(\\frac{1}{\\mathrm{tan}\\,\\theta }\\right)}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Multiply by a term that results in desired numerator}.\\hfill \\\\ &amp; =&amp; \\frac{2}{\\frac{1}{\\mathrm{tan}\\,\\theta }-\\frac{{\\mathrm{tan}}^{2}\\theta }{\\mathrm{tan}\\,\\theta }}\\hfill &amp; \\\\ &amp; =&amp; \\frac{2}{\\mathrm{cot}\\,\\theta -\\mathrm{tan}\\,\\theta }\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Use reciprocal identity for }\\frac{1}{\\mathrm{tan}\\,\\theta }.\\hfill \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1849183\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1302366\">Here is a case where the more complicated side of the initial equation appeared on the right, but we chose to work the left side. However, if we had chosen the left side to rewrite, we would have been working backwards to arrive at the equivalency. For example, suppose that we wanted to show<\/p>\n\n<div id=\"fs-id2884163\" class=\"unnumbered aligncenter\">[latex]\\frac{2\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }=\\frac{2}{\\mathrm{cot}\\,\\theta -\\mathrm{tan}\\,\\theta }[\/latex]<\/div>\n<p id=\"fs-id1493924\">Let\u2019s work on the right side.<\/p>\n\n<div id=\"fs-id1199365\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\frac{2}{\\mathrm{cot}\\,\\theta -\\mathrm{tan}\\,\\theta }&amp; =&amp; \\frac{2}{\\frac{1}{\\mathrm{tan}\\,\\theta }-\\mathrm{tan}\\,\\theta }\\left(\\frac{\\mathrm{tan}\\,\\theta }{\\mathrm{tan}\\,\\theta }\\right)\\hfill \\\\ &amp; =&amp; \\frac{2\\,\\mathrm{tan}\\,\\theta }{\\frac{1}{\\overline{)\\mathrm{tan}\\,\\theta }}\\left(\\overline{)\\mathrm{tan}\\,\\theta }\\right)-\\mathrm{tan}\\,\\theta \\left(\\mathrm{tan}\\,\\theta \\right)}\\hfill \\\\ &amp; =&amp; \\frac{2\\,\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1328304\">When using the identities to simplify a trigonometric expression or solve a trigonometric equation, there are usually several paths to a desired result. There is no set rule as to what side should be manipulated. However, we should begin with the guidelines set forth earlier.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1299752\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1174055\">\n<p id=\"fs-id1174056\">Verify the identity:[latex]\\,\\mathrm{cos}\\left(2\\theta \\right)\\mathrm{cos}\\,\\theta ={\\mathrm{cos}}^{3}\\theta -\\mathrm{cos}\\,\\theta \\,{\\mathrm{sin}}^{2}\\theta .[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2243554\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2243554\"]\n<p id=\"fs-id2243554\">[latex]\\mathrm{cos}\\left(2\\theta \\right)\\mathrm{cos}\\,\\theta =\\left({\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\right)\\mathrm{cos}\\,\\theta ={\\mathrm{cos}}^{3}\\theta -\\mathrm{cos}\\,\\theta {\\mathrm{sin}}^{2}\\theta [\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1355314\" class=\"bc-section section\">\n<h3>Use Reduction Formulas to Simplify an Expression<\/h3>\n<p id=\"fs-id1357641\">The double-angle formulas can be used to derive the <span class=\"no-emphasis\">reduction formulas<\/span>, which are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine. They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine. These formulas are especially important in higher-level math courses, calculus in particular. Also called the power-reducing formulas, three identities are included and are easily derived from the double-angle formulas.<\/p>\n<p id=\"fs-id2467783\">We can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Let\u2019s begin with[latex]\\,\\mathrm{cos}\\left(2\\theta \\right)=1-2\\,{\\mathrm{sin}}^{2}\\theta .\\,[\/latex]Solve for[latex]\\,{\\mathrm{sin}}^{2}\\theta :[\/latex]<\/p>\n\n<div id=\"fs-id1434749\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(2\\theta \\right)&amp; =&amp; 1-2\\,{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ \\hfill 2\\,{\\mathrm{sin}}^{2}\\theta &amp; =&amp; 1-\\mathrm{cos}\\left(2\\theta \\right)\\hfill \\\\ \\hfill {\\mathrm{sin}}^{2}\\theta &amp; =&amp; \\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2448468\">Next, we use the formula[latex]\\,\\mathrm{cos}\\left(2\\theta \\right)=2\\,{\\mathrm{cos}}^{2}\\theta -1.\\,[\/latex]Solve for[latex]\\,{\\mathrm{cos}}^{2}\\theta :[\/latex]<\/p>\n\n<div id=\"fs-id1134317\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(2\\theta \\right)&amp; =&amp; \\text{ }2\\,{\\mathrm{cos}}^{2}\\theta -1\\hfill \\\\ \\hfill 1+\\mathrm{cos}\\left(2\\theta \\right)&amp; =&amp; 2\\,{\\mathrm{cos}}^{2}\\theta \\hfill \\\\ \\hfill \\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}&amp; =&amp; {\\mathrm{cos}}^{2}\\theta \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1105729\">The last reduction formula is derived by writing tangent in terms of sine and cosine:<\/p>\n\n<div id=\"fs-id2628300\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill {\\mathrm{tan}}^{2}\\theta &amp; =&amp; \\frac{{\\mathrm{sin}}^{2}\\theta }{{\\mathrm{cos}}^{2}\\theta }\\hfill &amp; \\\\ &amp; =&amp; \\frac{\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}}{\\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}}\\hfill &amp; \\phantom{\\rule{2em}{0ex}}\\text{Substitute the reduction formulas}\\text{.}\\hfill \\\\ &amp; =&amp; \\left(\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}\\right)\\left(\\frac{2}{1+\\mathrm{cos}\\left(2\\theta \\right)}\\right)\\hfill &amp; \\\\ &amp; =&amp; \\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}\\hfill &amp; \\end{array}[\/latex]<\/div>\n<div id=\"fs-id2900248\" class=\"textbox key-takeaways\">\n<h3>Reduction Formulas<\/h3>\n<p id=\"fs-id1864092\">The reduction formulas are summarized as follows:<\/p>\n\n<div id=\"Eq_07_03_04\">[latex]{\\mathrm{sin}}^{2}\\theta =\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}[\/latex]<\/div>\n<div id=\"Eq_07_03_05\">[latex]{\\mathrm{cos}}^{2}\\theta =\\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}[\/latex]<\/div>\n<div id=\"Eq_07_03_06\">[latex]{\\mathrm{tan}}^{2}\\theta =\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id2042766\">\n<div id=\"fs-id2211041\">\n<h3>Writing an Equivalent Expression Not Containing Powers Greater Than 1<\/h3>\n<p id=\"fs-id2893591\">Write an equivalent expression for[latex]\\,{\\mathrm{cos}}^{4}x\\,[\/latex]that does not involve any powers of sine or cosine greater than 1.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2164657\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2164657\"]\n<p id=\"fs-id2164657\">We will apply the reduction formula for cosine twice.<\/p>\n\n<div id=\"fs-id1546208\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill {\\mathrm{cos}}^{4}x&amp; =&amp; {\\left({\\mathrm{cos}}^{2}x\\right)}^{2}\\hfill &amp; \\\\ &amp; =&amp; {\\left(\\frac{1+\\mathrm{cos}\\left(2x\\right)}{2}\\right)}^{2}\\hfill &amp; {\\phantom{\\rule{2em}{0ex}}\\text{Substitute reduction formula for cos}}^{2}x.\\hfill \\\\ &amp; =&amp; \\frac{1}{4}\\left(1+2\\mathrm{cos}\\left(2x\\right)+{\\mathrm{cos}}^{2}\\left(2x\\right)\\right)\\hfill &amp; \\\\ &amp; =&amp; \\frac{1}{4}+\\frac{1}{2}\\,\\mathrm{cos}\\left(2x\\right)+\\frac{1}{4}\\left(\\frac{1+\\mathrm{cos}2\\left(2x\\right)}{2}\\right)\\hfill &amp; {\\phantom{\\rule{2em}{0ex}}\\text{Substitute reduction formula for cos}}^{2}x.\\hfill \\\\ &amp; =&amp; \\frac{1}{4}+\\frac{1}{2}\\,\\mathrm{cos}\\left(2x\\right)+\\frac{1}{8}+\\frac{1}{8}\\,\\mathrm{cos}\\left(4x\\right)\\hfill &amp; \\\\ &amp; =&amp; \\frac{3}{8}+\\frac{1}{2}\\,\\mathrm{cos}\\left(2x\\right)+\\frac{1}{8}\\,\\mathrm{cos}\\left(4x\\right)\\hfill &amp; \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1863883\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1995932\">The solution is found by using the reduction formula twice, as noted, and the perfect square formula from algebra.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1833921\">\n<div id=\"fs-id1154810\">\n<h3>Using the Power-Reducing Formulas to Prove an Identity<\/h3>\n<p id=\"fs-id1154815\">Use the power-reducing formulas to prove<\/p>\n\n<div id=\"fs-id1386362\" class=\"unnumbered aligncenter\">[latex]{\\mathrm{sin}}^{3}\\left(2x\\right)=\\left[\\frac{1}{2}\\,\\mathrm{sin}\\left(2x\\right)\\right]\\,\\left[1-\\mathrm{cos}\\left(4x\\right)\\right][\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1512827\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1512827\"]\n<p id=\"fs-id1512827\">We will work on simplifying the left side of the equation:<\/p>\n\n<div class=\"unnumbered\">[latex]\\begin{array}{cccc}\\hfill {\\mathrm{sin}}^{3}\\left(2x\\right)&amp; =&amp; \\left[\\mathrm{sin}\\left(2x\\right)\\right]\\left[{\\mathrm{sin}}^{2}\\left(2x\\right)\\right]\\hfill &amp; \\\\ &amp; =&amp; \\mathrm{sin}\\left(2x\\right)\\left[\\frac{1-\\mathrm{cos}\\left(4x\\right)}{2}\\right]\\hfill &amp; \\text{Substitute the power-reduction formula}.\\hfill \\\\ &amp; =&amp; \\mathrm{sin}\\left(2x\\right)\\left(\\frac{1}{2}\\right)\\left[1-\\mathrm{cos}\\left(4x\\right)\\right]\\hfill &amp; \\\\ &amp; =&amp; \\frac{1}{2}\\left[\\mathrm{sin}\\left(2x\\right)\\right]\\left[1-\\mathrm{cos}\\left(4x\\right)\\right]\\hfill &amp; \\end{array}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id1615455\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id2715681\">Note that in this example, we substituted<\/p>\n\n<div id=\"fs-id1784090\" class=\"unnumbered aligncenter\">[latex]\\frac{1-\\mathrm{cos}\\left(4x\\right)}{2}[\/latex]<\/div>\n<p id=\"fs-id1059525\">for[latex]\\,{\\mathrm{sin}}^{2}\\left(2x\\right).\\,[\/latex]The formula states<\/p>\n\n<div id=\"fs-id1473243\" class=\"unnumbered aligncenter\">[latex]{\\mathrm{sin}}^{2}\\theta =\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}[\/latex]<\/div>\n<p id=\"eip-id1893537\">We let[latex]\\,\\theta =2x,[\/latex]so[latex]\\,2\\theta =4x.[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1753781\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1154772\">\n<p id=\"fs-id1154773\">Use the power-reducing formulas to prove that[latex]\\,10\\,{\\mathrm{cos}}^{4}x=\\frac{15}{4}+5\\,\\mathrm{cos}\\left(2x\\right)+\\frac{5}{4}\\,\\mathrm{cos}\\left(4x\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1952834\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1952834\"]\n<p id=\"fs-id1952834\">[latex]\\begin{array}{cccc}\\hfill 10{\\mathrm{cos}}^{4}x&amp; =&amp; 10{\\left({\\mathrm{cos}}^{2}x\\right)}^{2}\\hfill &amp; \\\\ &amp; =&amp; 10{\\left[\\frac{1+\\mathrm{cos}\\left(2x\\right)}{2}\\right]}^{2}\\hfill &amp; {\\phantom{\\rule{1em}{0ex}}\\text{Substitute reduction formula for cos}}^{2}x.\\hfill \\\\ &amp; =&amp; \\frac{10}{4}\\left[1+2\\mathrm{cos}\\left(2x\\right)+{\\mathrm{cos}}^{2}\\left(2x\\right)\\right]\\hfill &amp; \\\\ &amp; =&amp; \\frac{10}{4}+\\frac{10}{2}\\mathrm{cos}\\left(2x\\right)+\\frac{10}{4}\\left(\\frac{1+\\mathrm{cos}2\\left(2x\\right)}{2}\\right)\\hfill &amp; {\\phantom{\\rule{1em}{0ex}}\\text{Substitute reduction formula for cos}}^{2}x.\\hfill \\\\ &amp; =&amp; \\frac{10}{4}+\\frac{10}{2}\\mathrm{cos}\\left(2x\\right)+\\frac{10}{8}+\\frac{10}{8}\\mathrm{cos}\\left(4x\\right)\\hfill &amp; \\\\ &amp; =&amp; \\frac{30}{8}+5\\mathrm{cos}\\left(2x\\right)+\\frac{10}{8}\\mathrm{cos}\\left(4x\\right)\\hfill &amp; \\\\ &amp; =&amp; \\frac{15}{4}+5\\mathrm{cos}\\left(2x\\right)+\\frac{5}{4}\\mathrm{cos}\\left(4x\\right)\\hfill &amp; \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h3>Using Half-Angle Formulas to Find Exact Values<\/h3>\n<p id=\"fs-id1615430\">The next set of identities is the set of half-angle formulas, which can be derived from the reduction formulas and we can use when we have an angle that is half the size of a special angle. If we replace[latex]\\,\\theta \\,[\/latex]with[latex]\\,\\frac{\\alpha }{2},[\/latex]the half-angle formula for sine is found by simplifying the equation and solving for[latex]\\,\\mathrm{sin}\\left(\\frac{\\alpha }{2}\\right).\\,[\/latex]Note that the half-angle formulas are preceded by a[latex]\\,\u00b1\\,[\/latex]sign. This does not mean that both the positive and negative expressions are valid. Rather, it depends on the quadrant in which[latex]\\,\\frac{\\alpha }{2}\\,[\/latex]terminates.<\/p>\n<p id=\"fs-id2637834\">The half-angle formula for sine is derived as follows:<\/p>\n\n<div id=\"fs-id2637838\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {\\mathrm{sin}}^{2}\\theta &amp; =&amp; \\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}\\hfill \\\\ \\hfill {\\mathrm{sin}}^{2}\\left(\\frac{\\alpha }{2}\\right)&amp; =&amp; \\frac{1-\\left(\\mathrm{cos}2\\cdot \\frac{\\alpha }{2}\\right)}{2}\\hfill \\\\ &amp; =&amp; \\frac{1-\\mathrm{cos}\\,\\alpha }{2}\\hfill \\\\ \\hfill \\mathrm{sin}\\left(\\frac{\\alpha }{2}\\right)&amp; =&amp; \u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2257759\">To derive the half-angle formula for cosine, we have<\/p>\n\n<div id=\"fs-id2257762\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {\\mathrm{cos}}^{2}\\theta &amp; =&amp; \\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}\\hfill \\\\ \\hfill {\\mathrm{cos}}^{2}\\left(\\frac{\\alpha }{2}\\right)&amp; =&amp; \\frac{1+\\mathrm{cos}\\left(2\\cdot \\frac{\\alpha }{2}\\right)}{2}\\hfill \\\\ &amp; =&amp; \\frac{1+\\mathrm{cos}\\,\\alpha }{2}\\hfill \\\\ \\hfill \\mathrm{cos}\\left(\\frac{\\alpha }{2}\\right)&amp; =&amp; \u00b1\\sqrt{\\frac{1+\\mathrm{cos}\\,\\alpha }{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1471943\">For the tangent identity, we have<\/p>\n\n<div id=\"fs-id1471946\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {\\mathrm{tan}}^{2}\\theta &amp; =&amp; \\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}\\hfill \\\\ \\hfill {\\mathrm{tan}}^{2}\\left(\\frac{\\alpha }{2}\\right)&amp; =&amp; \\frac{1-\\mathrm{cos}\\left(2\\cdot \\frac{\\alpha }{2}\\right)}{1+\\mathrm{cos}\\left(2\\cdot \\frac{\\alpha }{2}\\right)}\\hfill \\\\ &amp; =&amp; \\frac{1-\\mathrm{cos}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }\\hfill \\\\ \\hfill \\mathrm{tan}\\left(\\frac{\\alpha }{2}\\right)&amp; =&amp; \u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }}\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id2223366\" class=\"textbox key-takeaways\">\n<h3>Half-Angle Formulas<\/h3>\n<p id=\"fs-id2001532\">The half-angle formulas are as follows:<\/p>\n\n<div id=\"Eq_07_03_07\">[latex]\\mathrm{sin}\\left(\\frac{\\alpha }{2}\\right)=\u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{2}}[\/latex]<\/div>\n<div id=\"Eq_07_03_08\">[latex]\\mathrm{cos}\\left(\\frac{\\alpha }{2}\\right)=\u00b1\\sqrt{\\frac{1+\\mathrm{cos}\\,\\alpha }{2}}[\/latex]<\/div>\n<div id=\"Eq_07_03_09\">[latex]\\begin{array}{l}\\mathrm{tan}\\left(\\frac{\\alpha }{2}\\right)=\u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{\\mathrm{sin}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{1-\\mathrm{cos}\\,\\alpha }{\\mathrm{sin}\\,\\alpha }\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id2827946\">\n<div id=\"fs-id2827948\">\n<h3>Using a Half-Angle Formula to Find the Exact Value of a Sine Function<\/h3>\n<p id=\"fs-id2827954\">Find[latex]\\,\\mathrm{sin}\\left(15\u00b0\\right)\\,[\/latex]using a half-angle formula.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2281042\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2281042\"]\n<p id=\"fs-id2281042\">Since[latex]\\,15\u00b0=\\frac{30\u00b0}{2},[\/latex]we use the half-angle formula for sine:<\/p>\n\n<div id=\"fs-id1507376\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\,\\frac{30\u00b0}{2}&amp; =&amp; \\sqrt{\\frac{1-\\mathrm{cos}30\u00b0}{2}}\\hfill \\\\ &amp; =&amp; \\sqrt{\\frac{1-\\frac{\\sqrt{3}}{2}}{2}}\\hfill \\\\ &amp; =&amp; \\sqrt{\\frac{\\frac{2-\\sqrt{3}}{2}}{2}}\\hfill \\\\ &amp; =&amp; \\sqrt{\\frac{2-\\sqrt{3}}{4}}\\hfill \\\\ &amp; =&amp; \\frac{\\sqrt{2-\\sqrt{3}}}{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"eip-id2209696\">Remember that we can check the answer with a graphing calculator.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<div id=\"fs-id2020943\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id2594644\">Notice that we used only the positive root because[latex]\\,\\mathrm{sin}\\left(15\u00b0\\right)\\,[\/latex]is positive.<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2784673\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1936207\"><strong>Given the tangent of an angle and the quadrant in which the angle lies, find the exact values of trigonometric functions of half of the angle.<\/strong><\/p>\n\n<ol id=\"fs-id1936213\" type=\"1\">\n \t<li>Draw a triangle to represent the given information.<\/li>\n \t<li>Determine the correct half-angle formula.<\/li>\n \t<li>Substitute values into the formula based on the triangle.<\/li>\n \t<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_07_03_08\" class=\"textbox examples\">\n<div id=\"fs-id2023302\">\n<div id=\"fs-id1274798\">\n<h3>Finding Exact Values Using Half-Angle Identities<\/h3>\n<p id=\"fs-id1274803\">Given that[latex]\\,\\mathrm{tan}\\,\\alpha =\\frac{8}{15}[\/latex] and[latex]\\,\\alpha \\,[\/latex]lies in quadrant III, find the exact value of the following:<\/p>\n\n<ol id=\"fs-id3068887\" type=\"a\">\n \t<li>[latex]\\mathrm{sin}\\left(\\frac{\\alpha }{2}\\right)[\/latex]<\/li>\n \t<li>[latex]\\mathrm{cos}\\left(\\frac{\\alpha }{2}\\right)[\/latex]<\/li>\n \t<li>[latex]\\mathrm{tan}\\left(\\frac{\\alpha }{2}\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1631214\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1631214\"]\n<p id=\"fs-id1631214\">Using the given information, we can draw the triangle shown in <a class=\"autogenerated-content\" href=\"#Figure_07_03_003\">(Figure)<\/a>. Using the Pythagorean Theorem, we find the hypotenuse to be 17. Therefore, we can calculate[latex]\\,\\mathrm{sin}\\,\\alpha =-\\frac{8}{17}\\,[\/latex]and[latex]\\,\\mathrm{cos}\\,\\alpha =-\\frac{15}{17}.[\/latex]<\/p>\n\n<div class=\"medium\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144040\/CNX_Precalc_Figure_07_03_003.jpg\" alt=\"Diagram of a triangle in the x,y-plane. The vertices are at the origin, (-15,0), and (-15,-8). The angle at the origin is alpha. The angle formed by the side (-15,-8) to (-15,0) forms a right angle with the x axis. The hypotenuse across from the right angle is length 17.\" width=\"487\" height=\"289\"> <strong>Figure 3.<\/strong>[\/caption]\n\n<\/div>\n<ol id=\"fs-id2917167\" type=\"a\">\n \t<li>Before we start, we must remember that if[latex]\\,\\alpha \\,[\/latex]is in quadrant III, then[latex]\\,180\u00b0&lt;\\alpha &lt;270\u00b0,[\/latex]so[latex]\\,\\frac{180\u00b0}{2}&lt;\\frac{\\alpha }{2}&lt;\\frac{270\u00b0}{2}.\\,[\/latex]This means that the terminal side of[latex]\\,\\frac{\\alpha }{2}\\,[\/latex]is in quadrant II, since[latex]\\,90\u00b0&lt;\\frac{\\alpha }{2}&lt;135\u00b0.[\/latex]\n<p id=\"fs-id1912327\">To find[latex]\\,\\mathrm{sin}\\,\\frac{\\alpha }{2},[\/latex]we begin by writing the half-angle formula for sine. Then we substitute the value of the cosine we found from the triangle in <a class=\"autogenerated-content\" href=\"#Figure_07_03_003\">(Figure)<\/a> and simplify.<\/p>\n\n<div id=\"fs-id3026942\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\,\\frac{\\alpha }{2}&amp; =&amp; \u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{2}}\\hfill \\\\ &amp; =&amp; \u00b1\\sqrt{\\frac{1-\\left(-\\frac{15}{17}\\right)}{2}}\\hfill \\\\ &amp; =&amp; \u00b1\\sqrt{\\frac{\\frac{32}{17}}{2}}\\hfill \\\\ &amp; =&amp; \u00b1\\sqrt{\\frac{32}{17}\\cdot \\frac{1}{2}}\\hfill \\\\ &amp; =&amp; \u00b1\\sqrt{\\frac{16}{17}}\\hfill \\\\ &amp; =&amp; \u00b1\\frac{4}{\\sqrt{17}}\\hfill \\\\ &amp; =&amp; \\frac{4\\sqrt{17}}{17}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2335322\">We choose the positive value of[latex]\\,\\mathrm{sin}\\,\\frac{\\alpha }{2}\\,[\/latex]because the angle terminates in quadrant II and sine is positive in quadrant II.<\/p>\n<\/li>\n \t<li>To find[latex]\\,\\mathrm{cos}\\,\\frac{\\alpha }{2},[\/latex]we will write the half-angle formula for cosine, substitute the value of the cosine we found from the triangle in <a class=\"autogenerated-content\" href=\"#Figure_07_03_003\">(Figure)<\/a>, and simplify.\n<div id=\"fs-id1882356\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\,\\frac{\\alpha }{2}&amp; =&amp; \u00b1\\sqrt{\\frac{1+\\mathrm{cos}\\,\\alpha }{2}}\\hfill \\\\ &amp; =&amp; \u00b1\\sqrt{\\frac{1+\\left(-\\frac{15}{17}\\right)}{2}}\\hfill \\\\ &amp; =&amp; \u00b1\\sqrt{\\frac{\\frac{2}{17}}{2}}\\hfill \\\\ &amp; =&amp; \u00b1\\sqrt{\\frac{2}{17}\\cdot \\frac{1}{2}}\\hfill \\\\ &amp; =&amp; \u00b1\\sqrt{\\frac{1}{17}}\\hfill \\\\ &amp; =&amp; -\\frac{\\sqrt{17}}{17}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1823306\">We choose the negative value of[latex]\\,\\mathrm{cos}\\,\\frac{\\alpha }{2}\\,[\/latex]because the angle is in quadrant II because cosine is negative in quadrant II.<\/p>\n<\/li>\n \t<li>To find[latex]\\,\\mathrm{tan}\\,\\frac{\\alpha }{2},[\/latex]we write the half-angle formula for tangent. Again, we substitute the value of the cosine we found from the triangle in <a class=\"autogenerated-content\" href=\"#Figure_07_03_003\">(Figure)<\/a> and simplify.\n<div id=\"fs-id2233932\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{tan}\\,\\frac{\\alpha }{2}&amp; =&amp; \u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }}\\hfill \\\\ &amp; =&amp; \u00b1\\sqrt{\\frac{1-\\left(-\\frac{15}{17}\\right)}{1+\\left(-\\frac{15}{17}\\right)}}\\hfill \\\\ &amp; =&amp; \u00b1\\sqrt{\\frac{\\frac{32}{17}}{\\frac{2}{17}}}\\hfill \\\\ &amp; =&amp; \u00b1\\sqrt{\\frac{32}{2}}\\hfill \\\\ &amp; =&amp; -\\sqrt{16}\\hfill \\\\ &amp; =&amp; -4\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2738260\">We choose the negative value of[latex]\\,\\mathrm{tan}\\,\\frac{\\alpha }{2}\\,[\/latex]because[latex]\\,\\frac{\\alpha }{2}\\,[\/latex]lies in quadrant II, and tangent is negative in quadrant II.[\/hidden-answer]<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2235070\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id2235079\">\n<p id=\"fs-id2235080\">Given that[latex]\\,\\mathrm{sin}\\,\\alpha =-\\frac{4}{5}\\,[\/latex]and[latex]\\,\\alpha \\,[\/latex]lies in quadrant IV, find the exact value of[latex]\\,\\mathrm{cos}\\,\\left(\\frac{\\alpha }{2}\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1762529\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1762529\"]\n<p id=\"fs-id1762529\">[latex]-\\frac{2}{\\sqrt{5}}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2034648\" class=\"key-equations\">\n<div id=\"fs-id2923004\">\n<div class=\"textbox examples\">\n<div id=\"fs-id2923004\">\n<h3>Finding the Measurement of a Half Angle<\/h3>\n<p id=\"fs-id2923009\">Now, we will return to the problem posed at the beginning of the section. A bicycle ramp is constructed for high-level competition with an angle of[latex]\\,\\theta \\,[\/latex]formed by the ramp and the ground. Another ramp is to be constructed half as steep for novice competition. If[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{5}{3}\\,[\/latex]for higher-level competition, what is the measurement of the angle for novice competition?<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1697991\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1697991\"]\n<p id=\"fs-id1697991\">Since the angle for novice competition measures half the steepness of the angle for the high level competition, and[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{5}{3}\\,[\/latex]for high competition, we can find[latex]\\,\\mathrm{cos}\\,\\theta \\,[\/latex]from the right triangle and the Pythagorean theorem so that we can use the half-angle identities. See <a class=\"autogenerated-content\" href=\"#Figure_07_03_004\">(Figure)<\/a>.<\/p>\n\n<div id=\"fs-id2119600\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {3}^{2}+{5}^{2}&amp; =&amp; 34\\hfill \\\\ \\hfill c&amp; =&amp; \\sqrt{34}\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"small\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144046\/CNX_Precalc_Figure_07_03_004.jpg\" alt=\"Image of a right triangle with sides 3, 5, and rad34. Rad 34 is the hypotenuse, and 3 is the base. The angle formed by the hypotenuse and base is theta. The angle between the side of length 3 and side of length 5 is a right angle.\" width=\"487\" height=\"210\"> <strong>Figure 4.<\/strong>[\/caption]\n\n<\/div>\n<p id=\"fs-id2256411\">We see that[latex]\\,\\mathrm{cos}\\,\\theta =\\frac{3}{\\sqrt{34}}=\\frac{3\\sqrt{34}}{34}.\\,[\/latex]We can use the half-angle formula for tangent:[latex]\\,\\mathrm{tan}\\,\\frac{\\theta }{2}=\\sqrt{\\frac{1-\\mathrm{cos}\\,\\theta }{1+\\mathrm{cos}\\,\\theta }}.\\,[\/latex]Since[latex]\\,\\mathrm{tan}\\,\\theta \\,[\/latex]is in the first quadrant, so is[latex]\\,\\mathrm{tan}\\,\\frac{\\theta }{2}.\\,[\/latex]<\/p>\n\n<div id=\"fs-id1539352\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{tan}\\,\\frac{\\theta }{2}&amp; =&amp; \\sqrt{\\frac{1-\\frac{3\\sqrt{34}}{34}}{1+\\frac{3\\sqrt{34}}{34}}}\\hfill \\\\ &amp; =&amp; \\sqrt{\\frac{\\frac{34-3\\sqrt{34}}{34}}{\\frac{34+3\\sqrt{34}}{34}}}\\hfill \\\\ &amp; =&amp; \\sqrt{\\frac{34-3\\sqrt{34}}{34+3\\sqrt{34}}}\\hfill \\\\ &amp; \\approx &amp; 0.57\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2132071\">We can take the inverse tangent to find the angle:[latex]\\,{\\mathrm{tan}}^{-1}\\left(0.57\\right)\\approx 29.7\u00b0.\\,[\/latex]So the angle of the ramp for novice competition is[latex]\\,\\approx 29.7\u00b0.[\/latex][\/hidden-answer]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id2454717\">Access these online resources for additional instruction and practice with double-angle, half-angle, and reduction formulas.<\/p>\n\n<ul id=\"fs-id2034632\">\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/doubleangiden\">Double-Angle Identities<\/a><\/li>\n \t<li><a href=\"http:\/\/openstaxcollege.org\/l\/halfangleident\">Half-Angle Identities<\/a><\/li>\n<\/ul>\n<h3>Key Equations<\/h3>\n<table id=\"fs-id2034654\" summary=\"..\"><colgroup> <col> <col><\/colgroup>\n<tbody>\n<tr>\n<td>Double-angle formulas<\/td>\n<td>[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\left(2\\theta \\right)&amp; =&amp; 2\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta \\hfill \\\\ \\hfill \\mathrm{cos}\\left(2\\theta \\right)&amp; =&amp; {\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ &amp; =&amp; 1-2{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ &amp; =&amp; 2{\\mathrm{cos}}^{2}\\theta -1\\hfill \\\\ \\hfill \\mathrm{tan}\\left(2\\theta \\right)&amp; =&amp; \\frac{2\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reduction formulas<\/td>\n<td>[latex]\\begin{array}{ccc}\\hfill {\\mathrm{sin}}^{2}\\theta &amp; =&amp; \\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}\\hfill \\\\ \\hfill {\\mathrm{cos}}^{2}\\theta &amp; =&amp; \\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}\\hfill \\\\ \\hfill {\\mathrm{tan}}^{2}\\theta &amp; =&amp; \\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Half-angle formulas<\/td>\n<td>[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\,\\frac{\\alpha }{2}&amp; =&amp; \u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{2}}\\hfill \\\\ \\hfill \\mathrm{cos}\\,\\frac{\\alpha }{2}&amp; =&amp; \u00b1\\sqrt{\\frac{1+\\mathrm{cos}\\,\\alpha }{2}}\\hfill \\\\ \\hfill \\mathrm{tan}\\,\\frac{\\alpha }{2}&amp; =&amp; \u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }}\\hfill \\\\ &amp; =&amp; \\frac{\\mathrm{sin}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }\\hfill \\\\ &amp; =&amp; \\frac{1-\\mathrm{cos}\\,\\alpha }{\\mathrm{sin}\\,\\alpha }\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1792287\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1792293\">\n \t<li>Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. See <a class=\"autogenerated-content\" href=\"#Example_07_03_01\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_07_03_02\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_07_03_03\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_07_03_04\">(Figure)<\/a>.<\/li>\n \t<li>Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. See <a class=\"autogenerated-content\" href=\"#Example_07_03_05\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_07_03_06\">(Figure)<\/a>.<\/li>\n \t<li>Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not. See <a class=\"autogenerated-content\" href=\"#Example_07_03_07\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_07_03_08\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_07_05_09\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1596266\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1596270\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1596275\">\n<div id=\"fs-id1596276\">\n<p id=\"fs-id1596277\">Explain how to determine the reduction identities from the double-angle identity[latex]\\,\\mathrm{cos}\\left(2x\\right)={\\mathrm{cos}}^{2}x-{\\mathrm{sin}}^{2}x.[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2031070\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2031070\"]\n<p id=\"fs-id2031070\">Use the Pythagorean identities and isolate the squared term.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2031074\">\n<div id=\"fs-id2031075\">\n<p id=\"fs-id2031076\">Explain how to determine the double-angle formula for[latex]\\,\\mathrm{tan}\\left(2x\\right)\\,[\/latex]using the double-angle formulas for[latex]\\,\\mathrm{cos}\\left(2x\\right)\\,[\/latex]and[latex]\\,\\mathrm{sin}\\left(2x\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1703961\">\n<p id=\"fs-id1703962\">We can determine the half-angle formula for[latex]\\,\\mathrm{tan}\\left(\\frac{x}{2}\\right)=\\frac{\\sqrt{1-\\mathrm{cos}\\,x}}{\\sqrt{1+\\mathrm{cos}\\,x}}\\,[\/latex]by dividing the formula for[latex]\\,\\mathrm{sin}\\left(\\frac{x}{2}\\right)\\,[\/latex]by[latex]\\,\\mathrm{cos}\\left(\\frac{x}{2}\\right).\\,[\/latex]Explain how to determine two formulas for[latex]\\,\\mathrm{tan}\\left(\\frac{x}{2}\\right)\\,[\/latex] that do not involve any square roots.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2165532\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2165532\"]\n<p id=\"fs-id2165532\">[latex]\\,\\frac{1-\\mathrm{cos}\\,x}{\\mathrm{sin}\\,x},\\frac{\\mathrm{sin}\\,x}{1+\\mathrm{cos}\\,x},[\/latex]multiplying the top and bottom by[latex]\\,\\sqrt{1-\\mathrm{cos}\\,x}\\,[\/latex]and[latex]\\,\\sqrt{1+\\mathrm{cos}\\,x},[\/latex]respectively.<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2467445\">\n<div id=\"fs-id2467446\">\n<p id=\"fs-id2467447\">For the half-angle formula given in the previous exercise for[latex]\\,\\mathrm{tan}\\left(\\frac{x}{2}\\right),[\/latex]explain why dividing by 0 is not a concern. (Hint: examine the values of[latex]\\,\\mathrm{cos}\\,x\\,[\/latex]necessary for the denominator to be 0.)<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2084633\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id2084638\">For the following exercises, find the exact values of a)[latex]\\,\\mathrm{sin}\\left(2x\\right),[\/latex] b)[latex]\\,\\mathrm{cos}\\left(2x\\right),[\/latex] and c)[latex]\\,\\mathrm{tan}\\left(2x\\right)\\,[\/latex]without solving for[latex]\\,x.[\/latex]<\/p>\n\n<div id=\"fs-id2141772\">\n<div id=\"fs-id2141773\">\n<p id=\"fs-id2141774\">If[latex]\\,\\mathrm{sin}\\,x=\\frac{1}{8},[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant I.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1910225\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1910225\"]a)[latex]\\,\\frac{3\\sqrt{7}}{32}\\,[\/latex]b)[latex]\\,\\frac{31}{32}\\,[\/latex]c)[latex]\\,\\frac{3\\sqrt{7}}{31}[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id2160373\">\n<div id=\"fs-id2160374\">\n<p id=\"fs-id2160375\">If[latex]\\,\\mathrm{cos}\\,x=\\frac{2}{3},[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant I.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2708435\">\n<div id=\"fs-id2708436\">\n<p id=\"fs-id2708437\">If[latex]\\,\\mathrm{cos}\\,x=-\\frac{1}{2},[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant III.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2701891\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2701891\"]a)[latex]\\,\\frac{\\sqrt{3}}{2}\\,[\/latex]b)[latex]\\,-\\frac{1}{2}\\,[\/latex]c)[latex]\\,-\\sqrt{3}\\,[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id2130388\">\n<div id=\"fs-id2130389\">\n<p id=\"fs-id2130390\">If[latex]\\,\\mathrm{tan}\\,x=-8,[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant IV.<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2126716\">For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.<\/p>\n\n<div id=\"fs-id2126720\">\n<div id=\"fs-id2126721\">\n<p id=\"fs-id2126722\">[latex]\\mathrm{cos}\\left(2\\theta \\right)=\\frac{3}{5}\\,[\/latex]and[latex]\\,90\u00b0\\le \\theta \\le 180\u00b0[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2056415\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2056415\"]\n<p id=\"fs-id2056415\">[latex]\\mathrm{cos}\\,\\theta =-\\frac{2\\sqrt{5}}{5},\\mathrm{sin}\\,\\theta =\\frac{\\sqrt{5}}{5},\\mathrm{tan}\\,\\theta =-\\frac{1}{2},\\mathrm{csc}\\,\\theta =\\sqrt{5},\\mathrm{sec}\\,\\theta =-\\frac{\\sqrt{5}}{2},\\mathrm{cot}\\,\\theta =-2[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1431936\">\n<div id=\"fs-id1431937\">\n<p id=\"fs-id1431938\">[latex]\\mathrm{cos}\\left(2\\theta \\right)=\\frac{1}{\\sqrt{2}}\\,[\/latex]and[latex]\\,180\u00b0\\le \\theta \\le 270\u00b0[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2633017\">For the following exercises, simplify to one trigonometric expression.<\/p>\n\n<div id=\"fs-id2633020\">\n<div id=\"fs-id2633021\">\n<p id=\"fs-id2633022\">[latex]2\\,\\mathrm{sin}\\left(\\frac{\\pi }{4}\\right)\\,2\\,\\mathrm{cos}\\left(\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1575788\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1575788\"]\n<p id=\"fs-id1575788\">[latex]2\\,\\mathrm{sin}\\left(\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2915511\">\n<div id=\"fs-id2915512\">\n<p id=\"fs-id2915513\">[latex]4\\,\\mathrm{sin}\\left(\\frac{\\pi }{8}\\right)\\,\\mathrm{cos}\\left(\\frac{\\pi }{8}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2079018\">For the following exercises, find the exact value using half-angle formulas.<\/p>\n\n<div id=\"fs-id2079021\">\n<div id=\"fs-id2079022\">\n<p id=\"fs-id2079023\">[latex]\\,\\mathrm{sin}\\left(\\frac{\\pi }{8}\\right)\\,[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1267577\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1267577\"]\n<p id=\"fs-id1267577\">[latex]\\frac{\\sqrt{2-\\sqrt{2}}}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1256220\">\n<div id=\"fs-id1256221\">\n<p id=\"fs-id1256222\">[latex]\\mathrm{cos}\\left(-\\frac{11\\pi }{12}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2077632\">\n<div id=\"fs-id2077633\">\n<p id=\"fs-id2077634\">[latex]\\mathrm{sin}\\left(\\frac{11\\pi }{12}\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2786199\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2786199\"]\n<p id=\"fs-id2786199\">[latex]\\frac{\\sqrt{2-\\sqrt{3}}}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1613024\">\n<div id=\"fs-id1613025\">\n<p id=\"fs-id1613026\">[latex]\\mathrm{cos}\\left(\\frac{7\\pi }{8}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1274464\">\n<div id=\"fs-id1274465\">\n<p id=\"fs-id1274466\">[latex]\\mathrm{tan}\\left(\\frac{5\\pi }{12}\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1541638\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1541638\"]\n<p id=\"fs-id1541638\">[latex]2+\\sqrt{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1562395\">\n<div id=\"fs-id1562396\">\n<p id=\"fs-id1562397\">[latex]\\mathrm{tan}\\left(-\\frac{3\\pi }{12}\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1525779\">\n<div id=\"fs-id1525780\">\n<p id=\"fs-id1525781\">[latex]\\mathrm{tan}\\left(-\\frac{3\\pi }{8}\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2564655\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2564655\"]\n<p id=\"fs-id2564655\">[latex]-1-\\sqrt{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id1971088\">For the following exercises, find the exact values of a)[latex]\\,\\mathrm{sin}\\left(\\frac{x}{2}\\right),[\/latex] b)[latex]\\,\\mathrm{cos}\\left(\\frac{x}{2}\\right),[\/latex] and c)[latex]\\,\\mathrm{tan}\\left(\\frac{x}{2}\\right)[\/latex]without solving for[latex]\\,x,\\,[\/latex]when[latex]\\,0\u00b0\\le x\\le 360\u00b0.[\/latex]<\/p>\n\n<div id=\"fs-id2030106\">\n<div id=\"fs-id2030108\">\n<p id=\"fs-id2030109\">If[latex]\\,\\mathrm{tan}\\,x=-\\frac{4}{3},[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant IV.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2823463\">\n<div id=\"fs-id2823464\">\n<p id=\"fs-id2823465\">If[latex]\\,\\mathrm{sin}\\,x=-\\frac{12}{13},[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant III.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2156083\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2156083\"]a)[latex]\\,\\frac{3\\sqrt{13}}{13}\\,[\/latex]b)[latex]\\,-\\frac{2\\sqrt{13}}{13}\\,[\/latex]c)[latex]\\,-\\frac{3}{2}\\,[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<div id=\"fs-id2633748\">\n<div id=\"fs-id2633749\">\n<p id=\"fs-id2633750\">If[latex]\\,\\mathrm{csc}\\,x=7,[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant II.<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1743158\">\n<div id=\"fs-id1743159\">\n<p id=\"fs-id1743160\">If[latex]\\,\\mathrm{sec}\\,x=-4,[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant II.<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2234965\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2234965\"]a)[latex]\\,\\frac{\\sqrt{10}}{4}\\,[\/latex]b)[latex]\\,\\frac{\\sqrt{6}}{4}\\,[\/latex]c)[latex]\\,\\frac{\\sqrt{15}}{3}\\,[\/latex][\/hidden-answer]<\/div>\n<\/div>\n<p id=\"fs-id1617236\">For the following exercises, use <a class=\"autogenerated-content\" href=\"#Figure_07_03_201\">(Figure)<\/a> to find the requested half and double angles.<\/p>\n\n<div id=\"Figure_07_03_201\" class=\"small wp-caption aligncenter\">\n\n[caption id=\"\" align=\"aligncenter\" width=\"325\" class=\"small\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144053\/CNX_Precalc_Figure_07_03_201.jpg\" alt=\"Image of a right triangle. The base is length 12, and the height is length 5. The angle between the base and the height is 90 degrees, the angle between the base and the hypotenuse is theta, and the angle between the height and the hypotenuse is alpha degrees.\" width=\"325\" height=\"152\"> <strong>Figure 5.<\/strong>[\/caption]\n\n<\/div>\n<div id=\"fs-id1926566\">\n<div id=\"fs-id1926567\">\n<p id=\"fs-id1926568\">Find[latex]\\,\\mathrm{sin}\\left(2\\theta \\right),\\mathrm{cos}\\left(2\\theta \\right),[\/latex]and[latex]\\,\\mathrm{tan}\\left(2\\theta \\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2148715\">\n<div id=\"fs-id2148716\">\n<p id=\"fs-id2148717\">Find[latex]\\,\\mathrm{sin}\\left(2\\alpha \\right),\\mathrm{cos}\\left(2\\alpha \\right),[\/latex]and[latex]\\,\\mathrm{tan}\\left(2\\alpha \\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1475703\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1475703\"]\n<p id=\"fs-id1475703\">[latex]\\frac{120}{169},\u2013\\frac{119}{169},\u2013\\frac{120}{119}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1929612\">\n<div id=\"fs-id1929613\">\n<p id=\"fs-id1929614\">Find[latex]\\,\\mathrm{sin}\\left(\\frac{\\theta }{2}\\right),\\mathrm{cos}\\left(\\frac{\\theta }{2}\\right),[\/latex]and[latex]\\,\\mathrm{tan}\\left(\\frac{\\theta }{2}\\right).[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2802108\">\n<div id=\"fs-id2802109\">\n<p id=\"fs-id2802110\">Find[latex]\\,\\mathrm{sin}\\left(\\frac{\\alpha }{2}\\right),\\mathrm{cos}\\left(\\frac{\\alpha }{2}\\right),[\/latex]and[latex]\\,\\mathrm{tan}\\left(\\frac{\\alpha }{2}\\right).[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1740867\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1740867\"]\n<p id=\"fs-id1740867\">[latex]\\frac{2\\sqrt{13}}{13},\\frac{3\\sqrt{13}}{13},\\frac{2}{3}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2780640\">For the following exercises, simplify each expression. Do not evaluate.<\/p>\n\n<div id=\"fs-id2780643\">\n<div id=\"fs-id2780644\">\n<p id=\"fs-id2780645\">[latex]{\\mathrm{cos}}^{2}\\left(28\u00b0\\right)-{\\mathrm{sin}}^{2}\\left(28\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1754046\">\n<div id=\"fs-id2334589\">\n<p id=\"fs-id2334590\">[latex]2{\\mathrm{cos}}^{2}\\left(37\u00b0\\right)-1[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1615373\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1615373\"]\n<p id=\"fs-id1615373\">[latex]\\mathrm{cos}\\left(74\u00b0\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1868300\">\n<div id=\"fs-id1868301\">\n<p id=\"fs-id1868302\">[latex]1-2\\,{\\mathrm{sin}}^{2}\\left(17\u00b0\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1363902\">\n<div id=\"fs-id2067709\">\n<p id=\"fs-id2067710\">[latex]{\\mathrm{cos}}^{2}\\left(9x\\right)-{\\mathrm{sin}}^{2}\\left(9x\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1615567\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1615567\"]\n<p id=\"fs-id1615567\">[latex]\\mathrm{cos}\\left(18x\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1806937\">\n<div id=\"fs-id1806938\">\n<p id=\"fs-id1806939\">[latex]4\\,\\mathrm{sin}\\left(8x\\right)\\,\\mathrm{cos}\\left(8x\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2164318\">\n<div id=\"fs-id2164319\">\n<p id=\"fs-id2164320\">[latex]6\\,\\mathrm{sin}\\left(5x\\right)\\,\\mathrm{cos}\\left(5x\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2906542\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2906542\"]\n<p id=\"fs-id2906542\">[latex]3\\mathrm{sin}\\left(10x\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2586821\">For the following exercises, prove the given identity.<\/p>\n\n<div id=\"fs-id2586824\">\n<div id=\"fs-id2586825\">\n<p id=\"fs-id2586826\">[latex]{\\left(\\mathrm{sin}\\,t-\\mathrm{cos}\\,t\\right)}^{2}=1-\\mathrm{sin}\\left(2t\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2617871\">\n<div id=\"fs-id2617872\">\n<p id=\"fs-id2617873\">[latex]\\mathrm{sin}\\left(2x\\right)=-2\\,\\mathrm{sin}\\left(-x\\right)\\,\\mathrm{cos}\\left(-x\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2038543\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2038543\"]\n<p id=\"fs-id2038543\">[latex]-2\\,\\mathrm{sin}\\left(-x\\right)\\mathrm{cos}\\left(-x\\right)=-2\\left(-\\mathrm{sin}\\left(x\\right)\\mathrm{cos}\\left(x\\right)\\right)=\\mathrm{sin}\\left(2x\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2116113\">\n<div id=\"fs-id2116114\">\n<p id=\"fs-id2116115\">[latex]\\mathrm{cot}\\,x-\\mathrm{tan}\\,x=2\\,\\mathrm{cot}\\left(2x\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1919766\">\n<div id=\"fs-id1919767\">\n<p id=\"fs-id1919768\">[latex]\\frac{\\mathrm{sin}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}{\\mathrm{tan}}^{2}\\theta ={\\mathrm{tan}}^{3}\\,\\theta [\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2259858\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2259858\"]\n<p id=\"fs-id2259858\">[latex]\\begin{array}{ccc}\\hfill \\frac{\\mathrm{sin}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}{\\mathrm{tan}}^{2}\\theta &amp; =&amp; \\frac{2\\mathrm{sin}\\left(\\theta \\right)\\mathrm{cos}\\left(\\theta \\right)}{1+{\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta }{\\mathrm{tan}}^{2}\\theta =\\hfill \\\\ \\hfill \\frac{2\\mathrm{sin}\\left(\\theta \\right)\\mathrm{cos}\\left(\\theta \\right)}{2{\\mathrm{cos}}^{2}\\theta }{\\mathrm{tan}}^{2}\\theta &amp; =&amp; \\frac{\\mathrm{sin}\\left(\\theta \\right)}{\\mathrm{cos}\\,\\theta }{\\mathrm{tan}}^{2}\\theta =\\hfill \\\\ \\hfill \\mathrm{cot}\\left(\\theta \\right){\\mathrm{tan}}^{2}\\theta &amp; =&amp; {\\mathrm{tan}}^{3}\\,\\theta \\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<p id=\"fs-id2307988\">For the following exercises, rewrite the expression with an exponent no higher than 1.<\/p>\n\n<div id=\"fs-id2307992\">\n<div id=\"fs-id2307993\">\n<p id=\"fs-id2307994\">[latex]{\\mathrm{cos}}^{2}\\left(5x\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1217358\">\n<div id=\"fs-id1217359\">\n<p id=\"fs-id1217360\">[latex]{\\mathrm{cos}}^{2}\\left(6x\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2296538\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2296538\"]\n<p id=\"fs-id2296538\">[latex]\\frac{1+\\mathrm{cos}\\left(12x\\right)}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2148526\">\n<div id=\"fs-id2148527\">\n<p id=\"fs-id2148528\">[latex]{\\mathrm{sin}}^{4}\\left(8x\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1669604\">\n<div id=\"fs-id1669605\">\n<p id=\"fs-id1669606\">[latex]{\\mathrm{sin}}^{4}\\left(3x\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2430151\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2430151\"]\n<p id=\"fs-id2430151\">[latex]\\frac{3+\\mathrm{cos}\\left(12x\\right)-4\\mathrm{cos}\\left(6x\\right)}{8}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2595632\">\n<div id=\"fs-id2595633\">\n<p id=\"fs-id2595634\">[latex]{\\mathrm{cos}}^{2}x{\\,\\mathrm{sin}}^{4}x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2261679\">\n<div id=\"fs-id2261680\">\n<p id=\"fs-id2261681\">[latex]{\\mathrm{cos}}^{4}x{\\,\\mathrm{sin}}^{2}x[\/latex]<\/p>\n\n<\/div>\n<div id=\"fs-id505042\" class=\"solution textbox shaded\">[reveal-answer q=\"fs-id505042\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id505042\"]\n<p id=\"fs-id505044\">[latex]\\frac{2+\\mathrm{cos}\\left(2x\\right)-2\\mathrm{cos}\\left(4x\\right)-\\mathrm{cos}\\left(6x\\right)}{32}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1341258\">\n<div id=\"fs-id1341259\">\n<p id=\"fs-id1341260\">[latex]{\\mathrm{tan}}^{2}x{\\,\\mathrm{sin}}^{2}x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2957879\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1154359\">For the following exercises, reduce the equations to powers of one, and then check the answer graphically.<\/p>\n\n<div id=\"fs-id1154363\">\n<div id=\"fs-id1154364\">\n<p id=\"fs-id1154365\">[latex]{\\mathrm{tan}}^{4}x[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1154392\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1154392\"]\n<p id=\"fs-id1154392\">[latex]\\frac{3+\\mathrm{cos}\\left(4x\\right)-4\\mathrm{cos}\\left(2x\\right)}{3+\\mathrm{cos}\\left(4x\\right)+4\\mathrm{cos}\\left(2x\\right)}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id3460275\">\n<div id=\"fs-id3460276\">\n<p id=\"fs-id3460277\">[latex]{\\mathrm{sin}}^{2}\\left(2x\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2477648\">\n<div id=\"fs-id2477650\">\n<p id=\"fs-id2477651\">[latex]{\\mathrm{sin}}^{2}x{\\,\\mathrm{cos}}^{2}x[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2896955\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2896955\"]\n<p id=\"fs-id2896955\">[latex]\\frac{1-\\mathrm{cos}\\left(4x\\right)}{8}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2281010\">\n<div id=\"fs-id2281011\">\n<p id=\"fs-id2281012\">[latex]{\\mathrm{tan}}^{2}x\\,\\mathrm{sin}\\,x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id1737544\">\n<div id=\"fs-id1737545\">\n<p id=\"fs-id1737546\">[latex]{\\mathrm{tan}}^{4}x{\\,\\mathrm{cos}}^{2}x[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1899789\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1899789\"]\n<p id=\"fs-id1899789\">[latex]\\frac{3+\\mathrm{cos}\\left(4x\\right)-4\\mathrm{cos}\\left(2x\\right)}{4\\left(\\mathrm{cos}\\left(2x\\right)+1\\right)}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1617272\">\n<div id=\"fs-id1617273\">\n<p id=\"fs-id1617274\">[latex]{\\mathrm{cos}}^{2}x\\,\\mathrm{sin}\\left(2x\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2738099\">\n<div id=\"fs-id2738100\">\n<p id=\"fs-id2738101\">[latex]{\\mathrm{cos}}^{2}\\left(2x\\right)\\mathrm{sin}\\,x[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2186001\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2186001\"]\n<p id=\"fs-id2186001\">[latex]\\frac{\\left(1+\\mathrm{cos}\\left(4x\\right)\\right)\\mathrm{sin}\\,x}{2}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2164535\">\n<div id=\"fs-id2164536\">\n<p id=\"fs-id2164537\">[latex]{\\mathrm{tan}}^{2}\\left(\\frac{x}{2}\\right)\\,\\mathrm{sin}\\,x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<p id=\"fs-id2675176\">For the following exercises, algebraically find an equivalent function, only in terms of[latex]\\,\\mathrm{sin}\\,x\\,[\/latex]and\/or[latex]\\,\\mathrm{cos}\\,x,[\/latex]and then check the answer by graphing both functions.<\/p>\n\n<div id=\"fs-id1926621\">\n<div id=\"fs-id1926622\">\n<p id=\"fs-id1926623\">[latex]\\mathrm{sin}\\left(4x\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1643964\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1643964\"]\n<p id=\"fs-id1643964\">[latex]4\\mathrm{sin}\\,x\\mathrm{cos}\\,x\\left({\\mathrm{cos}}^{2}x-{\\mathrm{sin}}^{2}x\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2796947\">\n<div id=\"fs-id2796948\">\n<p id=\"fs-id2796949\">[latex]\\mathrm{cos}\\left(4x\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2796973\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1875788\">For the following exercises, prove the identities.<\/p>\n\n<div id=\"fs-id1875791\">\n<div id=\"fs-id1875792\">\n<p id=\"fs-id1875794\">[latex]\\mathrm{sin}\\left(2x\\right)=\\frac{2\\,\\mathrm{tan}\\,x}{1+{\\mathrm{tan}}^{2}x}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2749007\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2749007\"]\n<p id=\"fs-id2749007\">[latex]\\begin{array}{l}\\frac{2\\mathrm{tan}\\,x}{1+{\\mathrm{tan}}^{2}x}=\\frac{\\frac{2\\mathrm{sin}\\,x}{\\mathrm{cos}\\,x}}{1+\\frac{{\\mathrm{sin}}^{2}x}{{\\mathrm{cos}}^{2}x}}=\\frac{\\frac{2\\mathrm{sin}\\,x}{\\mathrm{cos}\\,x}}{\\frac{{\\mathrm{cos}}^{2}x+{\\mathrm{sin}}^{2}x}{{\\mathrm{cos}}^{2}x}}=\\\\ \\frac{2\\mathrm{sin}\\,x}{\\mathrm{cos}\\,x}.\\frac{{\\mathrm{cos}}^{2}x}{1}=2\\mathrm{sin}\\,x\\mathrm{cos}\\,x=\\mathrm{sin}\\left(2x\\right)\\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id1086887\">\n<div id=\"fs-id1086888\">\n<p id=\"fs-id1086889\">[latex]\\mathrm{cos}\\left(2\\alpha \\right)=\\frac{1-{\\mathrm{tan}}^{2}\\alpha }{1+{\\mathrm{tan}}^{2}\\alpha }[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2609211\">\n<div id=\"fs-id2609212\">\n<p id=\"fs-id2609214\">[latex]\\mathrm{tan}\\left(2x\\right)=\\frac{2\\,\\mathrm{sin}\\,x\\,\\mathrm{cos}\\,x}{2{\\mathrm{cos}}^{2}x-1}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1790130\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1790130\"]\n<p id=\"fs-id1790130\">[latex]\\frac{2\\mathrm{sin}\\,x\\mathrm{cos}\\,x}{2{\\mathrm{cos}}^{2}x-1}=\\frac{\\mathrm{sin}\\left(2x\\right)}{\\mathrm{cos}\\left(2x\\right)}=\\mathrm{tan}\\left(2x\\right)[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2106580\">\n<div id=\"fs-id2106582\">\n<p id=\"fs-id2106583\">[latex]{\\left({\\mathrm{sin}}^{2}x-1\\right)}^{2}=\\mathrm{cos}\\left(2x\\right)+{\\mathrm{sin}}^{4}x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id3032804\">\n<div id=\"fs-id1882179\">\n<p id=\"fs-id1882180\">[latex]\\mathrm{sin}\\left(3x\\right)=3\\,\\mathrm{sin}\\,x\\,{\\mathrm{cos}}^{2}x-{\\mathrm{sin}}^{3}x[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2569931\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2569931\"]\n<p id=\"fs-id2569931\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\left(x+2x\\right)&amp; =&amp; \\mathrm{sin}\\,x\\mathrm{cos}\\left(2x\\right)+\\mathrm{sin}\\left(2x\\right)\\mathrm{cos}\\,x\\hfill \\\\ &amp; =&amp; \\mathrm{sin}\\,x\\left({\\mathrm{cos}}^{2}x-{\\mathrm{sin}}^{2}x\\right)+2\\mathrm{sin}\\,x\\mathrm{cos}\\,x\\mathrm{cos}\\,x\\hfill \\\\ &amp; =&amp; \\mathrm{sin}\\,x{\\mathrm{cos}}^{2}x-{\\mathrm{sin}}^{3}x+2\\mathrm{sin}\\,x{\\mathrm{cos}}^{2}x\\hfill \\\\ &amp; =&amp; 3\\mathrm{sin}\\,x{\\mathrm{cos}}^{2}x-{\\mathrm{sin}}^{3}x\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1673819\">[latex]\\mathrm{cos}\\left(3x\\right)={\\mathrm{cos}}^{3}x-3{\\mathrm{sin}}^{2}x\\,\\mathrm{cos}\\,x[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2049232\">\n<div id=\"fs-id2049234\">\n<p id=\"fs-id2049235\">[latex]\\frac{1+\\mathrm{cos}\\left(2t\\right)}{\\mathrm{sin}\\left(2t\\right)-\\mathrm{cos}\\,t}=\\frac{2\\,\\mathrm{cos}\\,t}{2\\,\\mathrm{sin}\\,t-1}[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id1506519\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id1506519\"]\n<p id=\"fs-id1506519\">[latex]\\begin{array}{ccc}\\hfill \\frac{1+\\mathrm{cos}\\left(2t\\right)}{\\mathrm{sin}\\left(2t\\right)-\\mathrm{cos}t}&amp; =&amp; \\frac{1+2{\\mathrm{cos}}^{2}t-1}{2\\mathrm{sin}t\\mathrm{cos}t-\\mathrm{cos}t}\\hfill \\\\ &amp; =&amp; \\frac{2{\\mathrm{cos}}^{2}t}{\\mathrm{cos}t\\left(2\\mathrm{sin}t-1\\right)}\\hfill \\\\ &amp; =&amp; \\frac{2\\mathrm{cos}t}{2\\mathrm{sin}t-1}\\hfill \\end{array}[\/latex]<\/p>\n[\/hidden-answer]\n\n<\/div>\n<\/div>\n<div id=\"fs-id2042946\">\n<div id=\"fs-id2042947\">\n<p id=\"fs-id2042948\">[latex]\\mathrm{sin}\\left(16x\\right)=16\\,\\mathrm{sin}\\,x\\,\\mathrm{cos}\\,x\\,\\mathrm{cos}\\left(2x\\right)\\mathrm{cos}\\left(4x\\right)\\mathrm{cos}\\left(8x\\right)[\/latex]<\/p>\n\n<\/div>\n<\/div>\n<div id=\"fs-id2234996\">\n<div id=\"fs-id2234998\">\n<p id=\"fs-id2234999\">[latex]\\mathrm{cos}\\left(16x\\right)=\\left({\\mathrm{cos}}^{2}\\left(4x\\right)-{\\mathrm{sin}}^{2}\\left(4x\\right)-\\mathrm{sin}\\left(8x\\right)\\right)\\left({\\mathrm{cos}}^{2}\\left(4x\\right)-{\\mathrm{sin}}^{2}\\left(4x\\right)+\\mathrm{sin}\\left(8x\\right)\\right)[\/latex]<\/p>\n\n<\/div>\n<div class=\"textbox shaded\">[reveal-answer q=\"fs-id2141257\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"fs-id2141257\"]\n<p id=\"fs-id2141257\">[latex]\\begin{array}{ccc}\\hfill \\left({\\mathrm{cos}}^{2}\\left(4x\\right)-{\\mathrm{sin}}^{2}\\left(4x\\right)-\\mathrm{sin}\\left(8x\\right)\\right)\\left({\\mathrm{cos}}^{2}\\left(4x\\right)-{\\mathrm{sin}}^{2}\\left(4x\\right)+\\mathrm{sin}\\left(8x\\right)\\right)&amp; =&amp; \\\\ &amp; =&amp; \\left(\\mathrm{cos}\\left(8x\\right)-\\mathrm{sin}\\left(8x\\right)\\right)\\left(\\mathrm{cos}\\left(8x\\right)+\\mathrm{sin}\\left(8x\\right)\\right)\\hfill \\\\ &amp; =&amp; {\\mathrm{cos}}^{2}\\left(8x\\right)-{\\mathrm{sin}}^{2}\\left(8x\\right)\\hfill \\\\ &amp; =&amp; \\mathrm{cos}\\left(16x\\right)\\hfill \\end{array}[\/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-id2253485\">\n \t<dt>double-angle formulas<\/dt>\n \t<dd id=\"fs-id2253488\">identities derived from the sum formulas for sine, cosine, and tangent in which the angles are equal<\/dd>\n<\/dl>\n<dl id=\"fs-id2253492\">\n \t<dt>half-angle formulas<\/dt>\n \t<dd id=\"fs-id2253495\">identities derived from the reduction formulas and used to determine half-angle values of trigonometric functions<\/dd>\n<\/dl>\n<dl id=\"fs-id2253499\">\n \t<dt>reduction formulas<\/dt>\n \t<dd id=\"fs-id2253503\">identities derived from the double-angle formulas and used to reduce the power of a trigonometric function<\/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>Use double-angle formulas to find exact values.<\/li>\n<li>Use double-angle formulas to verify identities.<\/li>\n<li>Use reduction formulas to simplify an expression.<\/li>\n<li>Use half-angle formulas to find exact values.<\/li>\n<\/ul>\n<\/div>\n<div class=\"wp-caption aligncenter\">\n<figure style=\"width: 979px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144031\/CNX_Precalc_Figure_07_03_001.jpg\" alt=\"Picture of two bicycle ramps, one with a steep slope and one with a gentle slope.\" width=\"979\" height=\"287\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 1. <\/strong>Bicycle ramps for advanced riders have a steeper incline than those designed for novices.<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1368511\">Bicycle ramps made for competition (see <a class=\"autogenerated-content\" href=\"#Figure_07_03_001\">(Figure)<\/a>) must vary in height depending on the skill level of the competitors. For advanced competitors, the angle formed by the ramp and the ground should be[latex]\\,\\theta \\,[\/latex]such that[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{5}{3}.\\,[\/latex]The angle is divided in half for novices. What is the steepness of the ramp for novices? In this section, we will investigate three additional categories of identities that we can use to answer questions such as this one.<\/p>\n<div id=\"fs-id1944514\" class=\"bc-section section\">\n<h3>Using Double-Angle Formulas to Find Exact Values<\/h3>\n<p>In the previous section, we used addition and subtraction formulas for trigonometric functions. Now, we take another look at those same formulas. The <span class=\"no-emphasis\">double-angle formulas<\/span> are a special case of the sum formulas, where[latex]\\,\\alpha =\\beta .\\,[\/latex]Deriving the double-angle formula for sine begins with the sum formula,<\/p>\n<div id=\"fs-id1388755\" class=\"unnumbered aligncenter\">[latex]\\mathrm{sin}\\left(\\alpha +\\beta \\right)=\\mathrm{sin}\\,\\alpha \\,\\mathrm{cos}\\,\\beta +\\mathrm{cos}\\,\\alpha \\,\\mathrm{sin}\\,\\beta[\/latex]<\/div>\n<p id=\"fs-id1017955\">If we let[latex]\\,\\alpha =\\beta =\\theta ,[\/latex]then we have<\/p>\n<div id=\"fs-id1958300\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\left(\\theta +\\theta \\right)& =& \\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta +\\mathrm{cos}\\,\\theta \\,\\mathrm{sin}\\,\\theta \\hfill \\\\ \\hfill \\mathrm{sin}\\left(2\\theta \\right)& =& 2\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1691770\">Deriving the double-angle for cosine gives us three options. First, starting from the sum formula,[latex]\\,\\mathrm{cos}\\left(\\alpha +\\beta \\right)=\\mathrm{cos}\\,\\alpha \\,\\mathrm{cos}\\,\\beta -\\mathrm{sin}\\,\\alpha \\,\\mathrm{sin}\\,\\beta ,[\/latex]and letting[latex]\\,\\alpha =\\beta =\\theta ,[\/latex]we have<\/p>\n<div id=\"fs-id2073447\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(\\theta +\\theta \\right)& =& \\mathrm{cos}\\,\\theta \\,\\mathrm{cos}\\,\\theta -\\mathrm{sin}\\,\\theta \\,\\mathrm{sin}\\,\\theta \\hfill \\\\ \\hfill \\mathrm{cos}\\left(2\\theta \\right)& =& {\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1679855\">Using the Pythagorean properties, we can expand this double-angle formula for cosine and get two more variations. The first variation is:<\/p>\n<div id=\"fs-id1082227\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(2\\theta \\right)& =& {\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ & =& \\left(1-{\\mathrm{sin}}^{2}\\theta \\right)-{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ & =& 1-2{\\mathrm{sin}}^{2}\\theta \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1874226\">The second variation is:<\/p>\n<div id=\"fs-id1154694\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(2\\theta \\right)& =& {\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ & =& {\\mathrm{cos}}^{2}\\theta -\\left(1-{\\mathrm{cos}}^{2}\\theta \\right)\\hfill \\\\ & =& 2\\,{\\mathrm{cos}}^{2}\\theta -1\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1541852\">Similarly, to derive the double-angle formula for tangent, replacing[latex]\\,\\alpha =\\beta =\\theta \\,[\/latex]in the sum formula gives<\/p>\n<div id=\"fs-id1688220\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{tan}\\left(\\alpha +\\beta \\right)& =& \\frac{\\mathrm{tan}\\,\\alpha +\\mathrm{tan}\\,\\beta }{1-\\mathrm{tan}\\,\\alpha \\,\\mathrm{tan}\\,\\beta }\\hfill \\\\ \\hfill \\mathrm{tan}\\left(\\theta +\\theta \\right)& =& \\frac{\\mathrm{tan}\\,\\theta +\\mathrm{tan}\\,\\theta }{1-\\mathrm{tan}\\,\\theta \\,\\mathrm{tan}\\,\\theta }\\hfill \\\\ \\hfill \\mathrm{tan}\\left(2\\theta \\right)& =& \\frac{2\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id1992296\" class=\"textbox key-takeaways\">\n<h3>Double-Angle Formulas<\/h3>\n<p id=\"fs-id1681037\">The double-angle formulas are summarized as follows:<\/p>\n<div id=\"Eq_07_03_01\">[latex]\\begin{array}{ccc}\\hfill \\phantom{\\rule{.45em}{0ex}}\\mathrm{sin}\\left(2\\theta \\right)& =& 2\\,\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta \\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<div id=\"Eq_07_03_02\">[latex]\\begin{array}{ccc}\\hfill \\phantom{\\rule{1.5em}{0ex}}\\mathrm{cos}\\left(2\\theta \\right)& =& {\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ & =& 1-2\\,{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ & =& 2\\,{\\mathrm{cos}}^{2}\\theta -1\\hfill \\end{array}[\/latex]<\/div>\n<div><\/div>\n<div id=\"Eq_07_03_03\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{tan}\\left(2\\theta \\right)& =& \\frac{2\\,\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div id=\"fs-id1578259\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1342627\"><strong>Given the tangent of an angle and the quadrant in which it is located, use the double-angle formulas to find the exact value.<br \/>\n<\/strong><\/p>\n<ol id=\"fs-id2248451\" type=\"1\">\n<li>Draw a triangle to reflect the given information.<\/li>\n<li>Determine the correct double-angle formula.<\/li>\n<li>Substitute values into the formula based on the triangle.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1764242\">\n<div id=\"fs-id1256954\">\n<h3>Using a Double-Angle Formula to Find the Exact Value Involving Tangent<\/h3>\n<p id=\"fs-id1491583\">Given that[latex]\\,\\mathrm{tan}\\,\\theta =-\\frac{3}{4}\\,[\/latex]and[latex]\\,\\theta \\,[\/latex]is in quadrant II, find the following:<\/p>\n<ol id=\"fs-id2182373\" type=\"a\">\n<li>[latex]\\mathrm{sin}\\left(2\\theta \\right)[\/latex]<\/li>\n<li>[latex]\\mathrm{cos}\\left(2\\theta \\right)[\/latex]<\/li>\n<li>[latex]\\mathrm{tan}\\left(2\\theta \\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1418852\">If we draw a triangle to reflect the information given, we can find the values needed to solve the problems on the image. We are given[latex]\\,\\mathrm{tan}\\,\\theta =-\\frac{3}{4},[\/latex]such that[latex]\\,\\theta \\,[\/latex]is in quadrant II. The tangent of an angle is equal to the opposite side over the adjacent side, and because[latex]\\,\\theta \\,[\/latex]is in the second quadrant, the adjacent side is on the <em>x<\/em>-axis and is negative. Use the <span class=\"no-emphasis\">Pythagorean Theorem<\/span> to find the length of the hypotenuse:<\/p>\n<div id=\"fs-id2303132\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {\\left(-4\\right)}^{2}+{\\left(3\\right)}^{2}& =& {c}^{2}\\hfill \\\\ \\hfill 16+9& =& {c}^{2}\\hfill \\\\ \\hfill 25& =& {c}^{2}\\hfill \\\\ \\hfill c& =& 5\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1208430\">Now we can draw a triangle similar to the one shown in <a class=\"autogenerated-content\" href=\"#Figure_07_03_02\">(Figure)<\/a>.<\/p>\n<div id=\"Figure_07_03_02\" 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\/19144034\/CNX_Precalc_Figure_07_03_002.jpg\" alt=\"Diagram of a triangle in the x,y-plane. The vertices are at the origin, (-4,0), and (-4,3). The angle at the origin is theta. The angle formed by the side (-4,3) to (-4,0) forms a right angle with the x axis. The hypotenuse across from the right angle is length 5.\" width=\"487\" height=\"251\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 2.<\/strong><\/figcaption><\/figure>\n<\/div>\n<ol id=\"fs-id1754093\" type=\"a\">\n<li>Let\u2019s begin by writing the double-angle formula for sine.\n<div id=\"fs-id1298188\" class=\"unnumbered aligncenter\">[latex]\\mathrm{sin}\\left(2\\theta \\right)=2\\,\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta[\/latex]<\/div>\n<p id=\"fs-id948689\">We see that we to need to find[latex]\\,\\mathrm{sin}\\,\\theta \\,[\/latex]and[latex]\\,\\mathrm{cos}\\,\\theta .\\,[\/latex]Based on <a class=\"autogenerated-content\" href=\"#Figure_07_03_02\">(Figure)<\/a>, we see that the hypotenuse equals 5, so[latex]\\,\\mathrm{sin}\\,\\theta =\\frac{3}{5},[\/latex]and[latex]\\,\\mathrm{cos}\\,\\theta =-\\frac{4}{5}.\\,[\/latex]Substitute these values into the equation, and simplify.<\/p>\n<p id=\"fs-id1494112\">Thus,<\/p>\n<div id=\"fs-id1316292\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\left(2\\theta \\right)& =& 2\\left(\\frac{3}{5}\\right)\\left(-\\frac{4}{5}\\right)\\hfill \\\\ & =& -\\frac{24}{25}\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>Write the double-angle formula for cosine.\n<div id=\"fs-id2347943\" class=\"unnumbered aligncenter\">[latex]\\mathrm{cos}\\left(2\\theta \\right)={\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta[\/latex]<\/div>\n<p id=\"fs-id1793867\">Again, substitute the values of the sine and cosine into the equation, and simplify.<\/p>\n<div id=\"fs-id1530574\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(2\\theta \\right)& =& {\\left(-\\frac{4}{5}\\right)}^{2}-{\\left(\\frac{3}{5}\\right)}^{2}\\hfill \\\\ & =& \\frac{16}{25}-\\frac{9}{25}\\hfill \\\\ & =& \\frac{7}{25}\\hfill \\end{array}[\/latex]<\/div>\n<\/li>\n<li>Write the double-angle formula for tangent.\n<div id=\"fs-id2078890\" class=\"unnumbered aligncenter\">[latex]\\mathrm{tan}\\left(2\\theta \\right)=\\frac{2\\,\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }[\/latex]<\/div>\n<p id=\"fs-id2130326\">In this formula, we need the tangent, which we were given as[latex]\\,\\mathrm{tan}\\,\\theta =-\\frac{3}{4}.\\,[\/latex]Substitute this value into the equation, and simplify.<\/p>\n<div id=\"fs-id2121569\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{tan}\\left(2\\theta \\right)& =& \\frac{2\\left(-\\frac{3}{4}\\right)}{1-{\\left(-\\frac{3}{4}\\right)}^{2}}\\hfill \\\\ & =& \\frac{-\\frac{3}{2}}{1-\\frac{9}{16}}\\hfill \\\\ & =& -\\frac{3}{2}\\left(\\frac{16}{7}\\right)\\hfill \\\\ & =& -\\frac{24}{7}\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1864098\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id2336091\">\n<p id=\"fs-id1506209\">Given[latex]\\,\\mathrm{sin}\\,\\alpha =\\frac{5}{8},[\/latex]with[latex]\\,\\theta \\,[\/latex]in quadrant I, find[latex]\\,\\mathrm{cos}\\left(2\\alpha \\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1338288\">[latex]\\mathrm{cos}\\left(2\\alpha \\right)=\\frac{7}{32}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1400362\">\n<div id=\"fs-id2327940\">\n<h3>Using the Double-Angle Formula for Cosine without Exact Values<\/h3>\n<p id=\"fs-id2303223\">Use the double-angle formula for cosine to write[latex]\\,\\mathrm{cos}\\left(6x\\right)\\,[\/latex]in terms of[latex]\\,\\mathrm{cos}\\left(3x\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<div id=\"fs-id2038748\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(6x\\right)& =& \\mathrm{cos}\\left(3x+3x\\right)\\hfill \\\\ & =& \\mathrm{cos}\\,3x\\,\\mathrm{cos}\\,3x-\\mathrm{sin}\\,3x\\,\\mathrm{sin}\\,3x\\hfill \\\\ & =& {\\mathrm{cos}}^{2}3x-{\\mathrm{sin}}^{2}3x\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1167807\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1514609\">This example illustrates that we can use the double-angle formula without having exact values. It emphasizes that the pattern is what we need to remember and that identities are true for all values in the domain of the trigonometric function.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2249050\" class=\"bc-section section\">\n<h3>Using Double-Angle Formulas to Verify Identities<\/h3>\n<p id=\"fs-id1313018\">Establishing identities using the double-angle formulas is performed using the same steps we used to derive the sum and difference formulas. Choose the more complicated side of the equation and rewrite it until it matches the other side.<\/p>\n<div class=\"textbox examples\">\n<div id=\"fs-id728598\">\n<div id=\"fs-id2712986\">\n<h3>Using the Double-Angle Formulas to Verify an Identity<\/h3>\n<p id=\"fs-id2067948\">Verify the following identity using double-angle formulas:<\/p>\n<div id=\"fs-id1169806\" class=\"unnumbered aligncenter\">[latex]1+\\mathrm{sin}\\left(2\\theta \\right)={\\left(\\mathrm{sin}\\,\\theta +\\mathrm{cos}\\,\\theta \\right)}^{2}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2208344\">We will work on the right side of the equal sign and rewrite the expression until it matches the left side.<\/p>\n<div id=\"fs-id1478975\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {\\left(\\mathrm{sin}\\,\\theta +\\mathrm{cos}\\,\\theta \\right)}^{2}& =& {\\mathrm{sin}}^{2}\\theta +2\\,\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta +{\\mathrm{cos}}^{2}\\theta \\hfill \\\\ & =& \\left({\\mathrm{sin}}^{2}\\theta +{\\mathrm{cos}}^{2}\\theta \\right)+2\\,\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta \\hfill \\\\ & =& 1+2\\,\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta \\hfill \\\\ & =& 1+\\mathrm{sin}\\left(2\\theta \\right)\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1364527\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1351871\">This process is not complicated, as long as we recall the perfect square formula from algebra:<\/p>\n<div id=\"fs-id1205403\" class=\"unnumbered aligncenter\">[latex]{\\left(a\u00b1b\\right)}^{2}={a}^{2}\u00b12ab+{b}^{2}[\/latex]<\/div>\n<p id=\"fs-id2431590\">where[latex]\\,a=\\mathrm{sin}\\,\\theta \\,[\/latex]and[latex]\\,b=\\mathrm{cos}\\,\\theta .\\,[\/latex]Part of being successful in mathematics is the ability to recognize patterns. While the terms or symbols may change, the algebra remains consistent.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1897349\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id2301016\">\n<p id=\"fs-id1530074\">Verify the identity:[latex]\\,{\\mathrm{cos}}^{4}\\theta -{\\mathrm{sin}}^{4}\\theta =\\mathrm{cos}\\left(2\\theta \\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1313281\">[latex]{\\mathrm{cos}}^{4}\\theta -{\\mathrm{sin}}^{4}\\theta =\\left({\\mathrm{cos}}^{2}\\theta +{\\mathrm{sin}}^{2}\\theta \\right)\\left({\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\right)=\\mathrm{cos}\\left(2\\theta \\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1524820\">\n<div id=\"fs-id1316219\">\n<h3>Verifying a Double-Angle Identity for Tangent<\/h3>\n<p id=\"fs-id1319888\">Verify the identity:<\/p>\n<div id=\"fs-id1486278\" class=\"unnumbered aligncenter\">[latex]\\mathrm{tan}\\left(2\\theta \\right)=\\frac{2}{\\mathrm{cot}\\,\\theta -\\mathrm{tan}\\,\\theta }[\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1333908\">In this case, we will work with the left side of the equation and simplify or rewrite until it equals the right side of the equation.<\/p>\n<div id=\"fs-id1199236\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill \\mathrm{tan}\\left(2\\theta \\right)& =& \\frac{2\\,\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Double-angle formula}\\hfill \\\\ & =& \\frac{2\\,\\mathrm{tan}\\,\\theta \\left(\\frac{1}{\\mathrm{tan}\\,\\theta }\\right)}{\\left(1-{\\mathrm{tan}}^{2}\\theta \\right)\\left(\\frac{1}{\\mathrm{tan}\\,\\theta }\\right)}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Multiply by a term that results in desired numerator}.\\hfill \\\\ & =& \\frac{2}{\\frac{1}{\\mathrm{tan}\\,\\theta }-\\frac{{\\mathrm{tan}}^{2}\\theta }{\\mathrm{tan}\\,\\theta }}\\hfill & \\\\ & =& \\frac{2}{\\mathrm{cot}\\,\\theta -\\mathrm{tan}\\,\\theta }\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Use reciprocal identity for }\\frac{1}{\\mathrm{tan}\\,\\theta }.\\hfill \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1849183\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1302366\">Here is a case where the more complicated side of the initial equation appeared on the right, but we chose to work the left side. However, if we had chosen the left side to rewrite, we would have been working backwards to arrive at the equivalency. For example, suppose that we wanted to show<\/p>\n<div id=\"fs-id2884163\" class=\"unnumbered aligncenter\">[latex]\\frac{2\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }=\\frac{2}{\\mathrm{cot}\\,\\theta -\\mathrm{tan}\\,\\theta }[\/latex]<\/div>\n<p id=\"fs-id1493924\">Let\u2019s work on the right side.<\/p>\n<div id=\"fs-id1199365\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\frac{2}{\\mathrm{cot}\\,\\theta -\\mathrm{tan}\\,\\theta }& =& \\frac{2}{\\frac{1}{\\mathrm{tan}\\,\\theta }-\\mathrm{tan}\\,\\theta }\\left(\\frac{\\mathrm{tan}\\,\\theta }{\\mathrm{tan}\\,\\theta }\\right)\\hfill \\\\ & =& \\frac{2\\,\\mathrm{tan}\\,\\theta }{\\frac{1}{\\overline{)\\mathrm{tan}\\,\\theta }}\\left(\\overline{)\\mathrm{tan}\\,\\theta }\\right)-\\mathrm{tan}\\,\\theta \\left(\\mathrm{tan}\\,\\theta \\right)}\\hfill \\\\ & =& \\frac{2\\,\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1328304\">When using the identities to simplify a trigonometric expression or solve a trigonometric equation, there are usually several paths to a desired result. There is no set rule as to what side should be manipulated. However, we should begin with the guidelines set forth earlier.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1299752\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1174055\">\n<p id=\"fs-id1174056\">Verify the identity:[latex]\\,\\mathrm{cos}\\left(2\\theta \\right)\\mathrm{cos}\\,\\theta ={\\mathrm{cos}}^{3}\\theta -\\mathrm{cos}\\,\\theta \\,{\\mathrm{sin}}^{2}\\theta .[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2243554\">[latex]\\mathrm{cos}\\left(2\\theta \\right)\\mathrm{cos}\\,\\theta =\\left({\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\right)\\mathrm{cos}\\,\\theta ={\\mathrm{cos}}^{3}\\theta -\\mathrm{cos}\\,\\theta {\\mathrm{sin}}^{2}\\theta[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1355314\" class=\"bc-section section\">\n<h3>Use Reduction Formulas to Simplify an Expression<\/h3>\n<p id=\"fs-id1357641\">The double-angle formulas can be used to derive the <span class=\"no-emphasis\">reduction formulas<\/span>, which are formulas we can use to reduce the power of a given expression involving even powers of sine or cosine. They allow us to rewrite the even powers of sine or cosine in terms of the first power of cosine. These formulas are especially important in higher-level math courses, calculus in particular. Also called the power-reducing formulas, three identities are included and are easily derived from the double-angle formulas.<\/p>\n<p id=\"fs-id2467783\">We can use two of the three double-angle formulas for cosine to derive the reduction formulas for sine and cosine. Let\u2019s begin with[latex]\\,\\mathrm{cos}\\left(2\\theta \\right)=1-2\\,{\\mathrm{sin}}^{2}\\theta .\\,[\/latex]Solve for[latex]\\,{\\mathrm{sin}}^{2}\\theta :[\/latex]<\/p>\n<div id=\"fs-id1434749\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(2\\theta \\right)& =& 1-2\\,{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ \\hfill 2\\,{\\mathrm{sin}}^{2}\\theta & =& 1-\\mathrm{cos}\\left(2\\theta \\right)\\hfill \\\\ \\hfill {\\mathrm{sin}}^{2}\\theta & =& \\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2448468\">Next, we use the formula[latex]\\,\\mathrm{cos}\\left(2\\theta \\right)=2\\,{\\mathrm{cos}}^{2}\\theta -1.\\,[\/latex]Solve for[latex]\\,{\\mathrm{cos}}^{2}\\theta :[\/latex]<\/p>\n<div id=\"fs-id1134317\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\left(2\\theta \\right)& =& \\text{ }2\\,{\\mathrm{cos}}^{2}\\theta -1\\hfill \\\\ \\hfill 1+\\mathrm{cos}\\left(2\\theta \\right)& =& 2\\,{\\mathrm{cos}}^{2}\\theta \\hfill \\\\ \\hfill \\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}& =& {\\mathrm{cos}}^{2}\\theta \\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1105729\">The last reduction formula is derived by writing tangent in terms of sine and cosine:<\/p>\n<div id=\"fs-id2628300\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill {\\mathrm{tan}}^{2}\\theta & =& \\frac{{\\mathrm{sin}}^{2}\\theta }{{\\mathrm{cos}}^{2}\\theta }\\hfill & \\\\ & =& \\frac{\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}}{\\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}}\\hfill & \\phantom{\\rule{2em}{0ex}}\\text{Substitute the reduction formulas}\\text{.}\\hfill \\\\ & =& \\left(\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}\\right)\\left(\\frac{2}{1+\\mathrm{cos}\\left(2\\theta \\right)}\\right)\\hfill & \\\\ & =& \\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}\\hfill & \\end{array}[\/latex]<\/div>\n<div id=\"fs-id2900248\" class=\"textbox key-takeaways\">\n<h3>Reduction Formulas<\/h3>\n<p id=\"fs-id1864092\">The reduction formulas are summarized as follows:<\/p>\n<div id=\"Eq_07_03_04\">[latex]{\\mathrm{sin}}^{2}\\theta =\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}[\/latex]<\/div>\n<div id=\"Eq_07_03_05\">[latex]{\\mathrm{cos}}^{2}\\theta =\\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}[\/latex]<\/div>\n<div id=\"Eq_07_03_06\">[latex]{\\mathrm{tan}}^{2}\\theta =\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id2042766\">\n<div id=\"fs-id2211041\">\n<h3>Writing an Equivalent Expression Not Containing Powers Greater Than 1<\/h3>\n<p id=\"fs-id2893591\">Write an equivalent expression for[latex]\\,{\\mathrm{cos}}^{4}x\\,[\/latex]that does not involve any powers of sine or cosine greater than 1.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2164657\">We will apply the reduction formula for cosine twice.<\/p>\n<div id=\"fs-id1546208\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{cccc}\\hfill {\\mathrm{cos}}^{4}x& =& {\\left({\\mathrm{cos}}^{2}x\\right)}^{2}\\hfill & \\\\ & =& {\\left(\\frac{1+\\mathrm{cos}\\left(2x\\right)}{2}\\right)}^{2}\\hfill & {\\phantom{\\rule{2em}{0ex}}\\text{Substitute reduction formula for cos}}^{2}x.\\hfill \\\\ & =& \\frac{1}{4}\\left(1+2\\mathrm{cos}\\left(2x\\right)+{\\mathrm{cos}}^{2}\\left(2x\\right)\\right)\\hfill & \\\\ & =& \\frac{1}{4}+\\frac{1}{2}\\,\\mathrm{cos}\\left(2x\\right)+\\frac{1}{4}\\left(\\frac{1+\\mathrm{cos}2\\left(2x\\right)}{2}\\right)\\hfill & {\\phantom{\\rule{2em}{0ex}}\\text{Substitute reduction formula for cos}}^{2}x.\\hfill \\\\ & =& \\frac{1}{4}+\\frac{1}{2}\\,\\mathrm{cos}\\left(2x\\right)+\\frac{1}{8}+\\frac{1}{8}\\,\\mathrm{cos}\\left(4x\\right)\\hfill & \\\\ & =& \\frac{3}{8}+\\frac{1}{2}\\,\\mathrm{cos}\\left(2x\\right)+\\frac{1}{8}\\,\\mathrm{cos}\\left(4x\\right)\\hfill & \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1863883\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id1995932\">The solution is found by using the reduction formula twice, as noted, and the perfect square formula from algebra.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id1833921\">\n<div id=\"fs-id1154810\">\n<h3>Using the Power-Reducing Formulas to Prove an Identity<\/h3>\n<p id=\"fs-id1154815\">Use the power-reducing formulas to prove<\/p>\n<div id=\"fs-id1386362\" class=\"unnumbered aligncenter\">[latex]{\\mathrm{sin}}^{3}\\left(2x\\right)=\\left[\\frac{1}{2}\\,\\mathrm{sin}\\left(2x\\right)\\right]\\,\\left[1-\\mathrm{cos}\\left(4x\\right)\\right][\/latex]<\/div>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1512827\">We will work on simplifying the left side of the equation:<\/p>\n<div class=\"unnumbered\">[latex]\\begin{array}{cccc}\\hfill {\\mathrm{sin}}^{3}\\left(2x\\right)& =& \\left[\\mathrm{sin}\\left(2x\\right)\\right]\\left[{\\mathrm{sin}}^{2}\\left(2x\\right)\\right]\\hfill & \\\\ & =& \\mathrm{sin}\\left(2x\\right)\\left[\\frac{1-\\mathrm{cos}\\left(4x\\right)}{2}\\right]\\hfill & \\text{Substitute the power-reduction formula}.\\hfill \\\\ & =& \\mathrm{sin}\\left(2x\\right)\\left(\\frac{1}{2}\\right)\\left[1-\\mathrm{cos}\\left(4x\\right)\\right]\\hfill & \\\\ & =& \\frac{1}{2}\\left[\\mathrm{sin}\\left(2x\\right)\\right]\\left[1-\\mathrm{cos}\\left(4x\\right)\\right]\\hfill & \\end{array}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1615455\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id2715681\">Note that in this example, we substituted<\/p>\n<div id=\"fs-id1784090\" class=\"unnumbered aligncenter\">[latex]\\frac{1-\\mathrm{cos}\\left(4x\\right)}{2}[\/latex]<\/div>\n<p id=\"fs-id1059525\">for[latex]\\,{\\mathrm{sin}}^{2}\\left(2x\\right).\\,[\/latex]The formula states<\/p>\n<div id=\"fs-id1473243\" class=\"unnumbered aligncenter\">[latex]{\\mathrm{sin}}^{2}\\theta =\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}[\/latex]<\/div>\n<p id=\"eip-id1893537\">We let[latex]\\,\\theta =2x,[\/latex]so[latex]\\,2\\theta =4x.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1753781\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id1154772\">\n<p id=\"fs-id1154773\">Use the power-reducing formulas to prove that[latex]\\,10\\,{\\mathrm{cos}}^{4}x=\\frac{15}{4}+5\\,\\mathrm{cos}\\left(2x\\right)+\\frac{5}{4}\\,\\mathrm{cos}\\left(4x\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1952834\">[latex]\\begin{array}{cccc}\\hfill 10{\\mathrm{cos}}^{4}x& =& 10{\\left({\\mathrm{cos}}^{2}x\\right)}^{2}\\hfill & \\\\ & =& 10{\\left[\\frac{1+\\mathrm{cos}\\left(2x\\right)}{2}\\right]}^{2}\\hfill & {\\phantom{\\rule{1em}{0ex}}\\text{Substitute reduction formula for cos}}^{2}x.\\hfill \\\\ & =& \\frac{10}{4}\\left[1+2\\mathrm{cos}\\left(2x\\right)+{\\mathrm{cos}}^{2}\\left(2x\\right)\\right]\\hfill & \\\\ & =& \\frac{10}{4}+\\frac{10}{2}\\mathrm{cos}\\left(2x\\right)+\\frac{10}{4}\\left(\\frac{1+\\mathrm{cos}2\\left(2x\\right)}{2}\\right)\\hfill & {\\phantom{\\rule{1em}{0ex}}\\text{Substitute reduction formula for cos}}^{2}x.\\hfill \\\\ & =& \\frac{10}{4}+\\frac{10}{2}\\mathrm{cos}\\left(2x\\right)+\\frac{10}{8}+\\frac{10}{8}\\mathrm{cos}\\left(4x\\right)\\hfill & \\\\ & =& \\frac{30}{8}+5\\mathrm{cos}\\left(2x\\right)+\\frac{10}{8}\\mathrm{cos}\\left(4x\\right)\\hfill & \\\\ & =& \\frac{15}{4}+5\\mathrm{cos}\\left(2x\\right)+\\frac{5}{4}\\mathrm{cos}\\left(4x\\right)\\hfill & \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\">\n<h3>Using Half-Angle Formulas to Find Exact Values<\/h3>\n<p id=\"fs-id1615430\">The next set of identities is the set of half-angle formulas, which can be derived from the reduction formulas and we can use when we have an angle that is half the size of a special angle. If we replace[latex]\\,\\theta \\,[\/latex]with[latex]\\,\\frac{\\alpha }{2},[\/latex]the half-angle formula for sine is found by simplifying the equation and solving for[latex]\\,\\mathrm{sin}\\left(\\frac{\\alpha }{2}\\right).\\,[\/latex]Note that the half-angle formulas are preceded by a[latex]\\,\u00b1\\,[\/latex]sign. This does not mean that both the positive and negative expressions are valid. Rather, it depends on the quadrant in which[latex]\\,\\frac{\\alpha }{2}\\,[\/latex]terminates.<\/p>\n<p id=\"fs-id2637834\">The half-angle formula for sine is derived as follows:<\/p>\n<div id=\"fs-id2637838\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {\\mathrm{sin}}^{2}\\theta & =& \\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}\\hfill \\\\ \\hfill {\\mathrm{sin}}^{2}\\left(\\frac{\\alpha }{2}\\right)& =& \\frac{1-\\left(\\mathrm{cos}2\\cdot \\frac{\\alpha }{2}\\right)}{2}\\hfill \\\\ & =& \\frac{1-\\mathrm{cos}\\,\\alpha }{2}\\hfill \\\\ \\hfill \\mathrm{sin}\\left(\\frac{\\alpha }{2}\\right)& =& \u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2257759\">To derive the half-angle formula for cosine, we have<\/p>\n<div id=\"fs-id2257762\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {\\mathrm{cos}}^{2}\\theta & =& \\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}\\hfill \\\\ \\hfill {\\mathrm{cos}}^{2}\\left(\\frac{\\alpha }{2}\\right)& =& \\frac{1+\\mathrm{cos}\\left(2\\cdot \\frac{\\alpha }{2}\\right)}{2}\\hfill \\\\ & =& \\frac{1+\\mathrm{cos}\\,\\alpha }{2}\\hfill \\\\ \\hfill \\mathrm{cos}\\left(\\frac{\\alpha }{2}\\right)& =& \u00b1\\sqrt{\\frac{1+\\mathrm{cos}\\,\\alpha }{2}}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1471943\">For the tangent identity, we have<\/p>\n<div id=\"fs-id1471946\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {\\mathrm{tan}}^{2}\\theta & =& \\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}\\hfill \\\\ \\hfill {\\mathrm{tan}}^{2}\\left(\\frac{\\alpha }{2}\\right)& =& \\frac{1-\\mathrm{cos}\\left(2\\cdot \\frac{\\alpha }{2}\\right)}{1+\\mathrm{cos}\\left(2\\cdot \\frac{\\alpha }{2}\\right)}\\hfill \\\\ & =& \\frac{1-\\mathrm{cos}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }\\hfill \\\\ \\hfill \\mathrm{tan}\\left(\\frac{\\alpha }{2}\\right)& =& \u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }}\\hfill \\end{array}[\/latex]<\/div>\n<div id=\"fs-id2223366\" class=\"textbox key-takeaways\">\n<h3>Half-Angle Formulas<\/h3>\n<p id=\"fs-id2001532\">The half-angle formulas are as follows:<\/p>\n<div id=\"Eq_07_03_07\">[latex]\\mathrm{sin}\\left(\\frac{\\alpha }{2}\\right)=\u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{2}}[\/latex]<\/div>\n<div id=\"Eq_07_03_08\">[latex]\\mathrm{cos}\\left(\\frac{\\alpha }{2}\\right)=\u00b1\\sqrt{\\frac{1+\\mathrm{cos}\\,\\alpha }{2}}[\/latex]<\/div>\n<div id=\"Eq_07_03_09\">[latex]\\begin{array}{l}\\mathrm{tan}\\left(\\frac{\\alpha }{2}\\right)=\u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }}\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{\\mathrm{sin}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }\\hfill \\\\ \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,=\\frac{1-\\mathrm{cos}\\,\\alpha }{\\mathrm{sin}\\,\\alpha }\\hfill \\end{array}[\/latex]<\/div>\n<\/div>\n<div class=\"textbox examples\">\n<div id=\"fs-id2827946\">\n<div id=\"fs-id2827948\">\n<h3>Using a Half-Angle Formula to Find the Exact Value of a Sine Function<\/h3>\n<p id=\"fs-id2827954\">Find[latex]\\,\\mathrm{sin}\\left(15\u00b0\\right)\\,[\/latex]using a half-angle formula.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2281042\">Since[latex]\\,15\u00b0=\\frac{30\u00b0}{2},[\/latex]we use the half-angle formula for sine:<\/p>\n<div id=\"fs-id1507376\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\,\\frac{30\u00b0}{2}& =& \\sqrt{\\frac{1-\\mathrm{cos}30\u00b0}{2}}\\hfill \\\\ & =& \\sqrt{\\frac{1-\\frac{\\sqrt{3}}{2}}{2}}\\hfill \\\\ & =& \\sqrt{\\frac{\\frac{2-\\sqrt{3}}{2}}{2}}\\hfill \\\\ & =& \\sqrt{\\frac{2-\\sqrt{3}}{4}}\\hfill \\\\ & =& \\frac{\\sqrt{2-\\sqrt{3}}}{2}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"eip-id2209696\">Remember that we can check the answer with a graphing calculator.<\/p>\n<\/details>\n<\/div>\n<div id=\"fs-id2020943\">\n<h4>Analysis<\/h4>\n<p id=\"fs-id2594644\">Notice that we used only the positive root because[latex]\\,\\mathrm{sin}\\left(15\u00b0\\right)\\,[\/latex]is positive.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2784673\" class=\"precalculus howto textbox tryit\">\n<h3>How To<\/h3>\n<p id=\"fs-id1936207\"><strong>Given the tangent of an angle and the quadrant in which the angle lies, find the exact values of trigonometric functions of half of the angle.<\/strong><\/p>\n<ol id=\"fs-id1936213\" type=\"1\">\n<li>Draw a triangle to represent the given information.<\/li>\n<li>Determine the correct half-angle formula.<\/li>\n<li>Substitute values into the formula based on the triangle.<\/li>\n<li>Simplify.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_07_03_08\" class=\"textbox examples\">\n<div id=\"fs-id2023302\">\n<div id=\"fs-id1274798\">\n<h3>Finding Exact Values Using Half-Angle Identities<\/h3>\n<p id=\"fs-id1274803\">Given that[latex]\\,\\mathrm{tan}\\,\\alpha =\\frac{8}{15}[\/latex] and[latex]\\,\\alpha \\,[\/latex]lies in quadrant III, find the exact value of the following:<\/p>\n<ol id=\"fs-id3068887\" type=\"a\">\n<li>[latex]\\mathrm{sin}\\left(\\frac{\\alpha }{2}\\right)[\/latex]<\/li>\n<li>[latex]\\mathrm{cos}\\left(\\frac{\\alpha }{2}\\right)[\/latex]<\/li>\n<li>[latex]\\mathrm{tan}\\left(\\frac{\\alpha }{2}\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1631214\">Using the given information, we can draw the triangle shown in <a class=\"autogenerated-content\" href=\"#Figure_07_03_003\">(Figure)<\/a>. Using the Pythagorean Theorem, we find the hypotenuse to be 17. Therefore, we can calculate[latex]\\,\\mathrm{sin}\\,\\alpha =-\\frac{8}{17}\\,[\/latex]and[latex]\\,\\mathrm{cos}\\,\\alpha =-\\frac{15}{17}.[\/latex]<\/p>\n<div class=\"medium\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144040\/CNX_Precalc_Figure_07_03_003.jpg\" alt=\"Diagram of a triangle in the x,y-plane. The vertices are at the origin, (-15,0), and (-15,-8). The angle at the origin is alpha. The angle formed by the side (-15,-8) to (-15,0) forms a right angle with the x axis. The hypotenuse across from the right angle is length 17.\" width=\"487\" height=\"289\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 3.<\/strong><\/figcaption><\/figure>\n<\/div>\n<ol id=\"fs-id2917167\" type=\"a\">\n<li>Before we start, we must remember that if[latex]\\,\\alpha \\,[\/latex]is in quadrant III, then[latex]\\,180\u00b0<\\alpha <270\u00b0,[\/latex]so[latex]\\,\\frac{180\u00b0}{2}<\\frac{\\alpha }{2}<\\frac{270\u00b0}{2}.\\,[\/latex]This means that the terminal side of[latex]\\,\\frac{\\alpha }{2}\\,[\/latex]is in quadrant II, since[latex]\\,90\u00b0<\\frac{\\alpha }{2}<135\u00b0.[\/latex]\n\n\n<p id=\"fs-id1912327\">To find[latex]\\,\\mathrm{sin}\\,\\frac{\\alpha }{2},[\/latex]we begin by writing the half-angle formula for sine. Then we substitute the value of the cosine we found from the triangle in <a class=\"autogenerated-content\" href=\"#Figure_07_03_003\">(Figure)<\/a> and simplify.<\/p>\n<div id=\"fs-id3026942\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\,\\frac{\\alpha }{2}& =& \u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{2}}\\hfill \\\\ & =& \u00b1\\sqrt{\\frac{1-\\left(-\\frac{15}{17}\\right)}{2}}\\hfill \\\\ & =& \u00b1\\sqrt{\\frac{\\frac{32}{17}}{2}}\\hfill \\\\ & =& \u00b1\\sqrt{\\frac{32}{17}\\cdot \\frac{1}{2}}\\hfill \\\\ & =& \u00b1\\sqrt{\\frac{16}{17}}\\hfill \\\\ & =& \u00b1\\frac{4}{\\sqrt{17}}\\hfill \\\\ & =& \\frac{4\\sqrt{17}}{17}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2335322\">We choose the positive value of[latex]\\,\\mathrm{sin}\\,\\frac{\\alpha }{2}\\,[\/latex]because the angle terminates in quadrant II and sine is positive in quadrant II.<\/p>\n<\/li>\n<li>To find[latex]\\,\\mathrm{cos}\\,\\frac{\\alpha }{2},[\/latex]we will write the half-angle formula for cosine, substitute the value of the cosine we found from the triangle in <a class=\"autogenerated-content\" href=\"#Figure_07_03_003\">(Figure)<\/a>, and simplify.\n<div id=\"fs-id1882356\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{cos}\\,\\frac{\\alpha }{2}& =& \u00b1\\sqrt{\\frac{1+\\mathrm{cos}\\,\\alpha }{2}}\\hfill \\\\ & =& \u00b1\\sqrt{\\frac{1+\\left(-\\frac{15}{17}\\right)}{2}}\\hfill \\\\ & =& \u00b1\\sqrt{\\frac{\\frac{2}{17}}{2}}\\hfill \\\\ & =& \u00b1\\sqrt{\\frac{2}{17}\\cdot \\frac{1}{2}}\\hfill \\\\ & =& \u00b1\\sqrt{\\frac{1}{17}}\\hfill \\\\ & =& -\\frac{\\sqrt{17}}{17}\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id1823306\">We choose the negative value of[latex]\\,\\mathrm{cos}\\,\\frac{\\alpha }{2}\\,[\/latex]because the angle is in quadrant II because cosine is negative in quadrant II.<\/p>\n<\/li>\n<li>To find[latex]\\,\\mathrm{tan}\\,\\frac{\\alpha }{2},[\/latex]we write the half-angle formula for tangent. Again, we substitute the value of the cosine we found from the triangle in <a class=\"autogenerated-content\" href=\"#Figure_07_03_003\">(Figure)<\/a> and simplify.\n<div id=\"fs-id2233932\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{tan}\\,\\frac{\\alpha }{2}& =& \u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }}\\hfill \\\\ & =& \u00b1\\sqrt{\\frac{1-\\left(-\\frac{15}{17}\\right)}{1+\\left(-\\frac{15}{17}\\right)}}\\hfill \\\\ & =& \u00b1\\sqrt{\\frac{\\frac{32}{17}}{\\frac{2}{17}}}\\hfill \\\\ & =& \u00b1\\sqrt{\\frac{32}{2}}\\hfill \\\\ & =& -\\sqrt{16}\\hfill \\\\ & =& -4\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2738260\">We choose the negative value of[latex]\\,\\mathrm{tan}\\,\\frac{\\alpha }{2}\\,[\/latex]because[latex]\\,\\frac{\\alpha }{2}\\,[\/latex]lies in quadrant II, and tangent is negative in quadrant II.<\/details>\n<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2235070\" class=\"textbox tryit\">\n<h3>Try It<\/h3>\n<div>\n<div id=\"fs-id2235079\">\n<p id=\"fs-id2235080\">Given that[latex]\\,\\mathrm{sin}\\,\\alpha =-\\frac{4}{5}\\,[\/latex]and[latex]\\,\\alpha \\,[\/latex]lies in quadrant IV, find the exact value of[latex]\\,\\mathrm{cos}\\,\\left(\\frac{\\alpha }{2}\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1762529\">[latex]-\\frac{2}{\\sqrt{5}}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2034648\" class=\"key-equations\">\n<div id=\"fs-id2923004\">\n<div class=\"textbox examples\">\n<div id=\"fs-id2923004\">\n<h3>Finding the Measurement of a Half Angle<\/h3>\n<p id=\"fs-id2923009\">Now, we will return to the problem posed at the beginning of the section. A bicycle ramp is constructed for high-level competition with an angle of[latex]\\,\\theta \\,[\/latex]formed by the ramp and the ground. Another ramp is to be constructed half as steep for novice competition. If[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{5}{3}\\,[\/latex]for higher-level competition, what is the measurement of the angle for novice competition?<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1697991\">Since the angle for novice competition measures half the steepness of the angle for the high level competition, and[latex]\\,\\mathrm{tan}\\,\\theta =\\frac{5}{3}\\,[\/latex]for high competition, we can find[latex]\\,\\mathrm{cos}\\,\\theta \\,[\/latex]from the right triangle and the Pythagorean theorem so that we can use the half-angle identities. See <a class=\"autogenerated-content\" href=\"#Figure_07_03_004\">(Figure)<\/a>.<\/p>\n<div id=\"fs-id2119600\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill {3}^{2}+{5}^{2}& =& 34\\hfill \\\\ \\hfill c& =& \\sqrt{34}\\hfill \\end{array}[\/latex]<\/div>\n<div class=\"small\">\n<figure style=\"width: 487px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3252\/2018\/07\/19144046\/CNX_Precalc_Figure_07_03_004.jpg\" alt=\"Image of a right triangle with sides 3, 5, and rad34. Rad 34 is the hypotenuse, and 3 is the base. The angle formed by the hypotenuse and base is theta. The angle between the side of length 3 and side of length 5 is a right angle.\" width=\"487\" height=\"210\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 4.<\/strong><\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id2256411\">We see that[latex]\\,\\mathrm{cos}\\,\\theta =\\frac{3}{\\sqrt{34}}=\\frac{3\\sqrt{34}}{34}.\\,[\/latex]We can use the half-angle formula for tangent:[latex]\\,\\mathrm{tan}\\,\\frac{\\theta }{2}=\\sqrt{\\frac{1-\\mathrm{cos}\\,\\theta }{1+\\mathrm{cos}\\,\\theta }}.\\,[\/latex]Since[latex]\\,\\mathrm{tan}\\,\\theta \\,[\/latex]is in the first quadrant, so is[latex]\\,\\mathrm{tan}\\,\\frac{\\theta }{2}.\\,[\/latex]<\/p>\n<div id=\"fs-id1539352\" class=\"unnumbered aligncenter\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{tan}\\,\\frac{\\theta }{2}& =& \\sqrt{\\frac{1-\\frac{3\\sqrt{34}}{34}}{1+\\frac{3\\sqrt{34}}{34}}}\\hfill \\\\ & =& \\sqrt{\\frac{\\frac{34-3\\sqrt{34}}{34}}{\\frac{34+3\\sqrt{34}}{34}}}\\hfill \\\\ & =& \\sqrt{\\frac{34-3\\sqrt{34}}{34+3\\sqrt{34}}}\\hfill \\\\ & \\approx & 0.57\\hfill \\end{array}[\/latex]<\/div>\n<p id=\"fs-id2132071\">We can take the inverse tangent to find the angle:[latex]\\,{\\mathrm{tan}}^{-1}\\left(0.57\\right)\\approx 29.7\u00b0.\\,[\/latex]So the angle of the ramp for novice competition is[latex]\\,\\approx 29.7\u00b0.[\/latex]<\/details>\n<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id2454717\">Access these online resources for additional instruction and practice with double-angle, half-angle, and reduction formulas.<\/p>\n<ul id=\"fs-id2034632\">\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/doubleangiden\">Double-Angle Identities<\/a><\/li>\n<li><a href=\"http:\/\/openstaxcollege.org\/l\/halfangleident\">Half-Angle Identities<\/a><\/li>\n<\/ul>\n<h3>Key Equations<\/h3>\n<table id=\"fs-id2034654\" summary=\"..\">\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<tbody>\n<tr>\n<td>Double-angle formulas<\/td>\n<td>[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\left(2\\theta \\right)& =& 2\\mathrm{sin}\\,\\theta \\,\\mathrm{cos}\\,\\theta \\hfill \\\\ \\hfill \\mathrm{cos}\\left(2\\theta \\right)& =& {\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ & =& 1-2{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ & =& 2{\\mathrm{cos}}^{2}\\theta -1\\hfill \\\\ \\hfill \\mathrm{tan}\\left(2\\theta \\right)& =& \\frac{2\\mathrm{tan}\\,\\theta }{1-{\\mathrm{tan}}^{2}\\theta }\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Reduction formulas<\/td>\n<td>[latex]\\begin{array}{ccc}\\hfill {\\mathrm{sin}}^{2}\\theta & =& \\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}\\hfill \\\\ \\hfill {\\mathrm{cos}}^{2}\\theta & =& \\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}\\hfill \\\\ \\hfill {\\mathrm{tan}}^{2}\\theta & =& \\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Half-angle formulas<\/td>\n<td>[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\,\\frac{\\alpha }{2}& =& \u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{2}}\\hfill \\\\ \\hfill \\mathrm{cos}\\,\\frac{\\alpha }{2}& =& \u00b1\\sqrt{\\frac{1+\\mathrm{cos}\\,\\alpha }{2}}\\hfill \\\\ \\hfill \\mathrm{tan}\\,\\frac{\\alpha }{2}& =& \u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }}\\hfill \\\\ & =& \\frac{\\mathrm{sin}\\,\\alpha }{1+\\mathrm{cos}\\,\\alpha }\\hfill \\\\ & =& \\frac{1-\\mathrm{cos}\\,\\alpha }{\\mathrm{sin}\\,\\alpha }\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1792287\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1792293\">\n<li>Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. See <a class=\"autogenerated-content\" href=\"#Example_07_03_01\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_07_03_02\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_07_03_03\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_07_03_04\">(Figure)<\/a>.<\/li>\n<li>Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. See <a class=\"autogenerated-content\" href=\"#Example_07_03_05\">(Figure)<\/a> and <a class=\"autogenerated-content\" href=\"#Example_07_03_06\">(Figure)<\/a>.<\/li>\n<li>Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not. See <a class=\"autogenerated-content\" href=\"#Example_07_03_07\">(Figure)<\/a>, <a class=\"autogenerated-content\" href=\"#Example_07_03_08\">(Figure)<\/a>, and <a class=\"autogenerated-content\" href=\"#Example_07_05_09\">(Figure)<\/a>.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1596266\" class=\"textbox exercises\">\n<h3>Section Exercises<\/h3>\n<div id=\"fs-id1596270\" class=\"bc-section section\">\n<h4>Verbal<\/h4>\n<div id=\"fs-id1596275\">\n<div id=\"fs-id1596276\">\n<p id=\"fs-id1596277\">Explain how to determine the reduction identities from the double-angle identity[latex]\\,\\mathrm{cos}\\left(2x\\right)={\\mathrm{cos}}^{2}x-{\\mathrm{sin}}^{2}x.[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2031070\">Use the Pythagorean identities and isolate the squared term.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2031074\">\n<div id=\"fs-id2031075\">\n<p id=\"fs-id2031076\">Explain how to determine the double-angle formula for[latex]\\,\\mathrm{tan}\\left(2x\\right)\\,[\/latex]using the double-angle formulas for[latex]\\,\\mathrm{cos}\\left(2x\\right)\\,[\/latex]and[latex]\\,\\mathrm{sin}\\left(2x\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div>\n<div id=\"fs-id1703961\">\n<p id=\"fs-id1703962\">We can determine the half-angle formula for[latex]\\,\\mathrm{tan}\\left(\\frac{x}{2}\\right)=\\frac{\\sqrt{1-\\mathrm{cos}\\,x}}{\\sqrt{1+\\mathrm{cos}\\,x}}\\,[\/latex]by dividing the formula for[latex]\\,\\mathrm{sin}\\left(\\frac{x}{2}\\right)\\,[\/latex]by[latex]\\,\\mathrm{cos}\\left(\\frac{x}{2}\\right).\\,[\/latex]Explain how to determine two formulas for[latex]\\,\\mathrm{tan}\\left(\\frac{x}{2}\\right)\\,[\/latex] that do not involve any square roots.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2165532\">[latex]\\,\\frac{1-\\mathrm{cos}\\,x}{\\mathrm{sin}\\,x},\\frac{\\mathrm{sin}\\,x}{1+\\mathrm{cos}\\,x},[\/latex]multiplying the top and bottom by[latex]\\,\\sqrt{1-\\mathrm{cos}\\,x}\\,[\/latex]and[latex]\\,\\sqrt{1+\\mathrm{cos}\\,x},[\/latex]respectively.<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2467445\">\n<div id=\"fs-id2467446\">\n<p id=\"fs-id2467447\">For the half-angle formula given in the previous exercise for[latex]\\,\\mathrm{tan}\\left(\\frac{x}{2}\\right),[\/latex]explain why dividing by 0 is not a concern. (Hint: examine the values of[latex]\\,\\mathrm{cos}\\,x\\,[\/latex]necessary for the denominator to be 0.)<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2084633\" class=\"bc-section section\">\n<h4>Algebraic<\/h4>\n<p id=\"fs-id2084638\">For the following exercises, find the exact values of a)[latex]\\,\\mathrm{sin}\\left(2x\\right),[\/latex] b)[latex]\\,\\mathrm{cos}\\left(2x\\right),[\/latex] and c)[latex]\\,\\mathrm{tan}\\left(2x\\right)\\,[\/latex]without solving for[latex]\\,x.[\/latex]<\/p>\n<div id=\"fs-id2141772\">\n<div id=\"fs-id2141773\">\n<p id=\"fs-id2141774\">If[latex]\\,\\mathrm{sin}\\,x=\\frac{1}{8},[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant I.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>a)[latex]\\,\\frac{3\\sqrt{7}}{32}\\,[\/latex]b)[latex]\\,\\frac{31}{32}\\,[\/latex]c)[latex]\\,\\frac{3\\sqrt{7}}{31}[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2160373\">\n<div id=\"fs-id2160374\">\n<p id=\"fs-id2160375\">If[latex]\\,\\mathrm{cos}\\,x=\\frac{2}{3},[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant I.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2708435\">\n<div id=\"fs-id2708436\">\n<p id=\"fs-id2708437\">If[latex]\\,\\mathrm{cos}\\,x=-\\frac{1}{2},[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant III.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>a)[latex]\\,\\frac{\\sqrt{3}}{2}\\,[\/latex]b)[latex]\\,-\\frac{1}{2}\\,[\/latex]c)[latex]\\,-\\sqrt{3}\\,[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2130388\">\n<div id=\"fs-id2130389\">\n<p id=\"fs-id2130390\">If[latex]\\,\\mathrm{tan}\\,x=-8,[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant IV.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2126716\">For the following exercises, find the values of the six trigonometric functions if the conditions provided hold.<\/p>\n<div id=\"fs-id2126720\">\n<div id=\"fs-id2126721\">\n<p id=\"fs-id2126722\">[latex]\\mathrm{cos}\\left(2\\theta \\right)=\\frac{3}{5}\\,[\/latex]and[latex]\\,90\u00b0\\le \\theta \\le 180\u00b0[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2056415\">[latex]\\mathrm{cos}\\,\\theta =-\\frac{2\\sqrt{5}}{5},\\mathrm{sin}\\,\\theta =\\frac{\\sqrt{5}}{5},\\mathrm{tan}\\,\\theta =-\\frac{1}{2},\\mathrm{csc}\\,\\theta =\\sqrt{5},\\mathrm{sec}\\,\\theta =-\\frac{\\sqrt{5}}{2},\\mathrm{cot}\\,\\theta =-2[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1431936\">\n<div id=\"fs-id1431937\">\n<p id=\"fs-id1431938\">[latex]\\mathrm{cos}\\left(2\\theta \\right)=\\frac{1}{\\sqrt{2}}\\,[\/latex]and[latex]\\,180\u00b0\\le \\theta \\le 270\u00b0[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2633017\">For the following exercises, simplify to one trigonometric expression.<\/p>\n<div id=\"fs-id2633020\">\n<div id=\"fs-id2633021\">\n<p id=\"fs-id2633022\">[latex]2\\,\\mathrm{sin}\\left(\\frac{\\pi }{4}\\right)\\,2\\,\\mathrm{cos}\\left(\\frac{\\pi }{4}\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1575788\">[latex]2\\,\\mathrm{sin}\\left(\\frac{\\pi }{2}\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2915511\">\n<div id=\"fs-id2915512\">\n<p id=\"fs-id2915513\">[latex]4\\,\\mathrm{sin}\\left(\\frac{\\pi }{8}\\right)\\,\\mathrm{cos}\\left(\\frac{\\pi }{8}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2079018\">For the following exercises, find the exact value using half-angle formulas.<\/p>\n<div id=\"fs-id2079021\">\n<div id=\"fs-id2079022\">\n<p id=\"fs-id2079023\">[latex]\\,\\mathrm{sin}\\left(\\frac{\\pi }{8}\\right)\\,[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1267577\">[latex]\\frac{\\sqrt{2-\\sqrt{2}}}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1256220\">\n<div id=\"fs-id1256221\">\n<p id=\"fs-id1256222\">[latex]\\mathrm{cos}\\left(-\\frac{11\\pi }{12}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2077632\">\n<div id=\"fs-id2077633\">\n<p id=\"fs-id2077634\">[latex]\\mathrm{sin}\\left(\\frac{11\\pi }{12}\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2786199\">[latex]\\frac{\\sqrt{2-\\sqrt{3}}}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1613024\">\n<div id=\"fs-id1613025\">\n<p id=\"fs-id1613026\">[latex]\\mathrm{cos}\\left(\\frac{7\\pi }{8}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1274464\">\n<div id=\"fs-id1274465\">\n<p id=\"fs-id1274466\">[latex]\\mathrm{tan}\\left(\\frac{5\\pi }{12}\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1541638\">[latex]2+\\sqrt{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1562395\">\n<div id=\"fs-id1562396\">\n<p id=\"fs-id1562397\">[latex]\\mathrm{tan}\\left(-\\frac{3\\pi }{12}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1525779\">\n<div id=\"fs-id1525780\">\n<p id=\"fs-id1525781\">[latex]\\mathrm{tan}\\left(-\\frac{3\\pi }{8}\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2564655\">[latex]-1-\\sqrt{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1971088\">For the following exercises, find the exact values of a)[latex]\\,\\mathrm{sin}\\left(\\frac{x}{2}\\right),[\/latex] b)[latex]\\,\\mathrm{cos}\\left(\\frac{x}{2}\\right),[\/latex] and c)[latex]\\,\\mathrm{tan}\\left(\\frac{x}{2}\\right)[\/latex]without solving for[latex]\\,x,\\,[\/latex]when[latex]\\,0\u00b0\\le x\\le 360\u00b0.[\/latex]<\/p>\n<div id=\"fs-id2030106\">\n<div id=\"fs-id2030108\">\n<p id=\"fs-id2030109\">If[latex]\\,\\mathrm{tan}\\,x=-\\frac{4}{3},[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant IV.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2823463\">\n<div id=\"fs-id2823464\">\n<p id=\"fs-id2823465\">If[latex]\\,\\mathrm{sin}\\,x=-\\frac{12}{13},[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant III.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>a)[latex]\\,\\frac{3\\sqrt{13}}{13}\\,[\/latex]b)[latex]\\,-\\frac{2\\sqrt{13}}{13}\\,[\/latex]c)[latex]\\,-\\frac{3}{2}\\,[\/latex]<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2633748\">\n<div id=\"fs-id2633749\">\n<p id=\"fs-id2633750\">If[latex]\\,\\mathrm{csc}\\,x=7,[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant II.<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1743158\">\n<div id=\"fs-id1743159\">\n<p id=\"fs-id1743160\">If[latex]\\,\\mathrm{sec}\\,x=-4,[\/latex]and[latex]\\,x\\,[\/latex]is in quadrant II.<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p>a)[latex]\\,\\frac{\\sqrt{10}}{4}\\,[\/latex]b)[latex]\\,\\frac{\\sqrt{6}}{4}\\,[\/latex]c)[latex]\\,\\frac{\\sqrt{15}}{3}\\,[\/latex]<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id1617236\">For the following exercises, use <a class=\"autogenerated-content\" href=\"#Figure_07_03_201\">(Figure)<\/a> to find the requested half and double angles.<\/p>\n<div id=\"Figure_07_03_201\" class=\"small wp-caption aligncenter\">\n<figure style=\"width: 325px\" 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\/19144053\/CNX_Precalc_Figure_07_03_201.jpg\" alt=\"Image of a right triangle. The base is length 12, and the height is length 5. The angle between the base and the height is 90 degrees, the angle between the base and the hypotenuse is theta, and the angle between the height and the hypotenuse is alpha degrees.\" width=\"325\" height=\"152\" \/><figcaption class=\"wp-caption-text\"><strong>Figure 5.<\/strong><\/figcaption><\/figure>\n<\/div>\n<div id=\"fs-id1926566\">\n<div id=\"fs-id1926567\">\n<p id=\"fs-id1926568\">Find[latex]\\,\\mathrm{sin}\\left(2\\theta \\right),\\mathrm{cos}\\left(2\\theta \\right),[\/latex]and[latex]\\,\\mathrm{tan}\\left(2\\theta \\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2148715\">\n<div id=\"fs-id2148716\">\n<p id=\"fs-id2148717\">Find[latex]\\,\\mathrm{sin}\\left(2\\alpha \\right),\\mathrm{cos}\\left(2\\alpha \\right),[\/latex]and[latex]\\,\\mathrm{tan}\\left(2\\alpha \\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1475703\">[latex]\\frac{120}{169},\u2013\\frac{119}{169},\u2013\\frac{120}{119}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1929612\">\n<div id=\"fs-id1929613\">\n<p id=\"fs-id1929614\">Find[latex]\\,\\mathrm{sin}\\left(\\frac{\\theta }{2}\\right),\\mathrm{cos}\\left(\\frac{\\theta }{2}\\right),[\/latex]and[latex]\\,\\mathrm{tan}\\left(\\frac{\\theta }{2}\\right).[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2802108\">\n<div id=\"fs-id2802109\">\n<p id=\"fs-id2802110\">Find[latex]\\,\\mathrm{sin}\\left(\\frac{\\alpha }{2}\\right),\\mathrm{cos}\\left(\\frac{\\alpha }{2}\\right),[\/latex]and[latex]\\,\\mathrm{tan}\\left(\\frac{\\alpha }{2}\\right).[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1740867\">[latex]\\frac{2\\sqrt{13}}{13},\\frac{3\\sqrt{13}}{13},\\frac{2}{3}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2780640\">For the following exercises, simplify each expression. Do not evaluate.<\/p>\n<div id=\"fs-id2780643\">\n<div id=\"fs-id2780644\">\n<p id=\"fs-id2780645\">[latex]{\\mathrm{cos}}^{2}\\left(28\u00b0\\right)-{\\mathrm{sin}}^{2}\\left(28\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1754046\">\n<div id=\"fs-id2334589\">\n<p id=\"fs-id2334590\">[latex]2{\\mathrm{cos}}^{2}\\left(37\u00b0\\right)-1[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1615373\">[latex]\\mathrm{cos}\\left(74\u00b0\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1868300\">\n<div id=\"fs-id1868301\">\n<p id=\"fs-id1868302\">[latex]1-2\\,{\\mathrm{sin}}^{2}\\left(17\u00b0\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1363902\">\n<div id=\"fs-id2067709\">\n<p id=\"fs-id2067710\">[latex]{\\mathrm{cos}}^{2}\\left(9x\\right)-{\\mathrm{sin}}^{2}\\left(9x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1615567\">[latex]\\mathrm{cos}\\left(18x\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1806937\">\n<div id=\"fs-id1806938\">\n<p id=\"fs-id1806939\">[latex]4\\,\\mathrm{sin}\\left(8x\\right)\\,\\mathrm{cos}\\left(8x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2164318\">\n<div id=\"fs-id2164319\">\n<p id=\"fs-id2164320\">[latex]6\\,\\mathrm{sin}\\left(5x\\right)\\,\\mathrm{cos}\\left(5x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2906542\">[latex]3\\mathrm{sin}\\left(10x\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2586821\">For the following exercises, prove the given identity.<\/p>\n<div id=\"fs-id2586824\">\n<div id=\"fs-id2586825\">\n<p id=\"fs-id2586826\">[latex]{\\left(\\mathrm{sin}\\,t-\\mathrm{cos}\\,t\\right)}^{2}=1-\\mathrm{sin}\\left(2t\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2617871\">\n<div id=\"fs-id2617872\">\n<p id=\"fs-id2617873\">[latex]\\mathrm{sin}\\left(2x\\right)=-2\\,\\mathrm{sin}\\left(-x\\right)\\,\\mathrm{cos}\\left(-x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2038543\">[latex]-2\\,\\mathrm{sin}\\left(-x\\right)\\mathrm{cos}\\left(-x\\right)=-2\\left(-\\mathrm{sin}\\left(x\\right)\\mathrm{cos}\\left(x\\right)\\right)=\\mathrm{sin}\\left(2x\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2116113\">\n<div id=\"fs-id2116114\">\n<p id=\"fs-id2116115\">[latex]\\mathrm{cot}\\,x-\\mathrm{tan}\\,x=2\\,\\mathrm{cot}\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1919766\">\n<div id=\"fs-id1919767\">\n<p id=\"fs-id1919768\">[latex]\\frac{\\mathrm{sin}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}{\\mathrm{tan}}^{2}\\theta ={\\mathrm{tan}}^{3}\\,\\theta[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2259858\">[latex]\\begin{array}{ccc}\\hfill \\frac{\\mathrm{sin}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}{\\mathrm{tan}}^{2}\\theta & =& \\frac{2\\mathrm{sin}\\left(\\theta \\right)\\mathrm{cos}\\left(\\theta \\right)}{1+{\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta }{\\mathrm{tan}}^{2}\\theta =\\hfill \\\\ \\hfill \\frac{2\\mathrm{sin}\\left(\\theta \\right)\\mathrm{cos}\\left(\\theta \\right)}{2{\\mathrm{cos}}^{2}\\theta }{\\mathrm{tan}}^{2}\\theta & =& \\frac{\\mathrm{sin}\\left(\\theta \\right)}{\\mathrm{cos}\\,\\theta }{\\mathrm{tan}}^{2}\\theta =\\hfill \\\\ \\hfill \\mathrm{cot}\\left(\\theta \\right){\\mathrm{tan}}^{2}\\theta & =& {\\mathrm{tan}}^{3}\\,\\theta \\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<p id=\"fs-id2307988\">For the following exercises, rewrite the expression with an exponent no higher than 1.<\/p>\n<div id=\"fs-id2307992\">\n<div id=\"fs-id2307993\">\n<p id=\"fs-id2307994\">[latex]{\\mathrm{cos}}^{2}\\left(5x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1217358\">\n<div id=\"fs-id1217359\">\n<p id=\"fs-id1217360\">[latex]{\\mathrm{cos}}^{2}\\left(6x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2296538\">[latex]\\frac{1+\\mathrm{cos}\\left(12x\\right)}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2148526\">\n<div id=\"fs-id2148527\">\n<p id=\"fs-id2148528\">[latex]{\\mathrm{sin}}^{4}\\left(8x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1669604\">\n<div id=\"fs-id1669605\">\n<p id=\"fs-id1669606\">[latex]{\\mathrm{sin}}^{4}\\left(3x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2430151\">[latex]\\frac{3+\\mathrm{cos}\\left(12x\\right)-4\\mathrm{cos}\\left(6x\\right)}{8}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2595632\">\n<div id=\"fs-id2595633\">\n<p id=\"fs-id2595634\">[latex]{\\mathrm{cos}}^{2}x{\\,\\mathrm{sin}}^{4}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2261679\">\n<div id=\"fs-id2261680\">\n<p id=\"fs-id2261681\">[latex]{\\mathrm{cos}}^{4}x{\\,\\mathrm{sin}}^{2}x[\/latex]<\/p>\n<\/div>\n<div id=\"fs-id505042\" class=\"solution textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id505044\">[latex]\\frac{2+\\mathrm{cos}\\left(2x\\right)-2\\mathrm{cos}\\left(4x\\right)-\\mathrm{cos}\\left(6x\\right)}{32}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1341258\">\n<div id=\"fs-id1341259\">\n<p id=\"fs-id1341260\">[latex]{\\mathrm{tan}}^{2}x{\\,\\mathrm{sin}}^{2}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2957879\" class=\"bc-section section\">\n<h4>Technology<\/h4>\n<p id=\"fs-id1154359\">For the following exercises, reduce the equations to powers of one, and then check the answer graphically.<\/p>\n<div id=\"fs-id1154363\">\n<div id=\"fs-id1154364\">\n<p id=\"fs-id1154365\">[latex]{\\mathrm{tan}}^{4}x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1154392\">[latex]\\frac{3+\\mathrm{cos}\\left(4x\\right)-4\\mathrm{cos}\\left(2x\\right)}{3+\\mathrm{cos}\\left(4x\\right)+4\\mathrm{cos}\\left(2x\\right)}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id3460275\">\n<div id=\"fs-id3460276\">\n<p id=\"fs-id3460277\">[latex]{\\mathrm{sin}}^{2}\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2477648\">\n<div id=\"fs-id2477650\">\n<p id=\"fs-id2477651\">[latex]{\\mathrm{sin}}^{2}x{\\,\\mathrm{cos}}^{2}x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2896955\">[latex]\\frac{1-\\mathrm{cos}\\left(4x\\right)}{8}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2281010\">\n<div id=\"fs-id2281011\">\n<p id=\"fs-id2281012\">[latex]{\\mathrm{tan}}^{2}x\\,\\mathrm{sin}\\,x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id1737544\">\n<div id=\"fs-id1737545\">\n<p id=\"fs-id1737546\">[latex]{\\mathrm{tan}}^{4}x{\\,\\mathrm{cos}}^{2}x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1899789\">[latex]\\frac{3+\\mathrm{cos}\\left(4x\\right)-4\\mathrm{cos}\\left(2x\\right)}{4\\left(\\mathrm{cos}\\left(2x\\right)+1\\right)}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1617272\">\n<div id=\"fs-id1617273\">\n<p id=\"fs-id1617274\">[latex]{\\mathrm{cos}}^{2}x\\,\\mathrm{sin}\\left(2x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2738099\">\n<div id=\"fs-id2738100\">\n<p id=\"fs-id2738101\">[latex]{\\mathrm{cos}}^{2}\\left(2x\\right)\\mathrm{sin}\\,x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2186001\">[latex]\\frac{\\left(1+\\mathrm{cos}\\left(4x\\right)\\right)\\mathrm{sin}\\,x}{2}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2164535\">\n<div id=\"fs-id2164536\">\n<p id=\"fs-id2164537\">[latex]{\\mathrm{tan}}^{2}\\left(\\frac{x}{2}\\right)\\,\\mathrm{sin}\\,x[\/latex]<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id2675176\">For the following exercises, algebraically find an equivalent function, only in terms of[latex]\\,\\mathrm{sin}\\,x\\,[\/latex]and\/or[latex]\\,\\mathrm{cos}\\,x,[\/latex]and then check the answer by graphing both functions.<\/p>\n<div id=\"fs-id1926621\">\n<div id=\"fs-id1926622\">\n<p id=\"fs-id1926623\">[latex]\\mathrm{sin}\\left(4x\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1643964\">[latex]4\\mathrm{sin}\\,x\\mathrm{cos}\\,x\\left({\\mathrm{cos}}^{2}x-{\\mathrm{sin}}^{2}x\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2796947\">\n<div id=\"fs-id2796948\">\n<p id=\"fs-id2796949\">[latex]\\mathrm{cos}\\left(4x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2796973\" class=\"bc-section section\">\n<h4>Extensions<\/h4>\n<p id=\"fs-id1875788\">For the following exercises, prove the identities.<\/p>\n<div id=\"fs-id1875791\">\n<div id=\"fs-id1875792\">\n<p id=\"fs-id1875794\">[latex]\\mathrm{sin}\\left(2x\\right)=\\frac{2\\,\\mathrm{tan}\\,x}{1+{\\mathrm{tan}}^{2}x}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2749007\">[latex]\\begin{array}{l}\\frac{2\\mathrm{tan}\\,x}{1+{\\mathrm{tan}}^{2}x}=\\frac{\\frac{2\\mathrm{sin}\\,x}{\\mathrm{cos}\\,x}}{1+\\frac{{\\mathrm{sin}}^{2}x}{{\\mathrm{cos}}^{2}x}}=\\frac{\\frac{2\\mathrm{sin}\\,x}{\\mathrm{cos}\\,x}}{\\frac{{\\mathrm{cos}}^{2}x+{\\mathrm{sin}}^{2}x}{{\\mathrm{cos}}^{2}x}}=\\\\ \\frac{2\\mathrm{sin}\\,x}{\\mathrm{cos}\\,x}.\\frac{{\\mathrm{cos}}^{2}x}{1}=2\\mathrm{sin}\\,x\\mathrm{cos}\\,x=\\mathrm{sin}\\left(2x\\right)\\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id1086887\">\n<div id=\"fs-id1086888\">\n<p id=\"fs-id1086889\">[latex]\\mathrm{cos}\\left(2\\alpha \\right)=\\frac{1-{\\mathrm{tan}}^{2}\\alpha }{1+{\\mathrm{tan}}^{2}\\alpha }[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2609211\">\n<div id=\"fs-id2609212\">\n<p id=\"fs-id2609214\">[latex]\\mathrm{tan}\\left(2x\\right)=\\frac{2\\,\\mathrm{sin}\\,x\\,\\mathrm{cos}\\,x}{2{\\mathrm{cos}}^{2}x-1}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1790130\">[latex]\\frac{2\\mathrm{sin}\\,x\\mathrm{cos}\\,x}{2{\\mathrm{cos}}^{2}x-1}=\\frac{\\mathrm{sin}\\left(2x\\right)}{\\mathrm{cos}\\left(2x\\right)}=\\mathrm{tan}\\left(2x\\right)[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2106580\">\n<div id=\"fs-id2106582\">\n<p id=\"fs-id2106583\">[latex]{\\left({\\mathrm{sin}}^{2}x-1\\right)}^{2}=\\mathrm{cos}\\left(2x\\right)+{\\mathrm{sin}}^{4}x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id3032804\">\n<div id=\"fs-id1882179\">\n<p id=\"fs-id1882180\">[latex]\\mathrm{sin}\\left(3x\\right)=3\\,\\mathrm{sin}\\,x\\,{\\mathrm{cos}}^{2}x-{\\mathrm{sin}}^{3}x[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2569931\">[latex]\\begin{array}{ccc}\\hfill \\mathrm{sin}\\left(x+2x\\right)& =& \\mathrm{sin}\\,x\\mathrm{cos}\\left(2x\\right)+\\mathrm{sin}\\left(2x\\right)\\mathrm{cos}\\,x\\hfill \\\\ & =& \\mathrm{sin}\\,x\\left({\\mathrm{cos}}^{2}x-{\\mathrm{sin}}^{2}x\\right)+2\\mathrm{sin}\\,x\\mathrm{cos}\\,x\\mathrm{cos}\\,x\\hfill \\\\ & =& \\mathrm{sin}\\,x{\\mathrm{cos}}^{2}x-{\\mathrm{sin}}^{3}x+2\\mathrm{sin}\\,x{\\mathrm{cos}}^{2}x\\hfill \\\\ & =& 3\\mathrm{sin}\\,x{\\mathrm{cos}}^{2}x-{\\mathrm{sin}}^{3}x\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div>\n<div>\n<p id=\"fs-id1673819\">[latex]\\mathrm{cos}\\left(3x\\right)={\\mathrm{cos}}^{3}x-3{\\mathrm{sin}}^{2}x\\,\\mathrm{cos}\\,x[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2049232\">\n<div id=\"fs-id2049234\">\n<p id=\"fs-id2049235\">[latex]\\frac{1+\\mathrm{cos}\\left(2t\\right)}{\\mathrm{sin}\\left(2t\\right)-\\mathrm{cos}\\,t}=\\frac{2\\,\\mathrm{cos}\\,t}{2\\,\\mathrm{sin}\\,t-1}[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id1506519\">[latex]\\begin{array}{ccc}\\hfill \\frac{1+\\mathrm{cos}\\left(2t\\right)}{\\mathrm{sin}\\left(2t\\right)-\\mathrm{cos}t}& =& \\frac{1+2{\\mathrm{cos}}^{2}t-1}{2\\mathrm{sin}t\\mathrm{cos}t-\\mathrm{cos}t}\\hfill \\\\ & =& \\frac{2{\\mathrm{cos}}^{2}t}{\\mathrm{cos}t\\left(2\\mathrm{sin}t-1\\right)}\\hfill \\\\ & =& \\frac{2\\mathrm{cos}t}{2\\mathrm{sin}t-1}\\hfill \\end{array}[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div id=\"fs-id2042946\">\n<div id=\"fs-id2042947\">\n<p id=\"fs-id2042948\">[latex]\\mathrm{sin}\\left(16x\\right)=16\\,\\mathrm{sin}\\,x\\,\\mathrm{cos}\\,x\\,\\mathrm{cos}\\left(2x\\right)\\mathrm{cos}\\left(4x\\right)\\mathrm{cos}\\left(8x\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<div id=\"fs-id2234996\">\n<div id=\"fs-id2234998\">\n<p id=\"fs-id2234999\">[latex]\\mathrm{cos}\\left(16x\\right)=\\left({\\mathrm{cos}}^{2}\\left(4x\\right)-{\\mathrm{sin}}^{2}\\left(4x\\right)-\\mathrm{sin}\\left(8x\\right)\\right)\\left({\\mathrm{cos}}^{2}\\left(4x\\right)-{\\mathrm{sin}}^{2}\\left(4x\\right)+\\mathrm{sin}\\left(8x\\right)\\right)[\/latex]<\/p>\n<\/div>\n<div class=\"textbox shaded\">\n<details>\n<summary>Show Solution<\/summary>\n<p id=\"fs-id2141257\">[latex]\\begin{array}{ccc}\\hfill \\left({\\mathrm{cos}}^{2}\\left(4x\\right)-{\\mathrm{sin}}^{2}\\left(4x\\right)-\\mathrm{sin}\\left(8x\\right)\\right)\\left({\\mathrm{cos}}^{2}\\left(4x\\right)-{\\mathrm{sin}}^{2}\\left(4x\\right)+\\mathrm{sin}\\left(8x\\right)\\right)& =& \\\\ & =& \\left(\\mathrm{cos}\\left(8x\\right)-\\mathrm{sin}\\left(8x\\right)\\right)\\left(\\mathrm{cos}\\left(8x\\right)+\\mathrm{sin}\\left(8x\\right)\\right)\\hfill \\\\ & =& {\\mathrm{cos}}^{2}\\left(8x\\right)-{\\mathrm{sin}}^{2}\\left(8x\\right)\\hfill \\\\ & =& \\mathrm{cos}\\left(16x\\right)\\hfill \\end{array}[\/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-id2253485\">\n<dt>double-angle formulas<\/dt>\n<dd id=\"fs-id2253488\">identities derived from the sum formulas for sine, cosine, and tangent in which the angles are equal<\/dd>\n<\/dl>\n<dl id=\"fs-id2253492\">\n<dt>half-angle formulas<\/dt>\n<dd id=\"fs-id2253495\">identities derived from the reduction formulas and used to determine half-angle values of trigonometric functions<\/dd>\n<\/dl>\n<dl id=\"fs-id2253499\">\n<dt>reduction formulas<\/dt>\n<dd id=\"fs-id2253503\">identities derived from the double-angle formulas and used to reduce the power of a trigonometric function<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":291,"menu_order":4,"template":"","meta":{"pb_show_title":null,"pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-141","chapter","type-chapter","status-publish","hentry"],"part":134,"_links":{"self":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/141","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\/141\/revisions"}],"predecessor-version":[{"id":142,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/141\/revisions\/142"}],"part":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/parts\/134"}],"metadata":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapters\/141\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/media?parent=141"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/pressbooks\/v2\/chapter-type?post=141"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/contributor?post=141"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/integrations.pressbooks.network\/testinternalcloneforcomparison\/wp-json\/wp\/v2\/license?post=141"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}