{"id":1380,"date":"2023-06-05T14:51:09","date_gmt":"2023-06-05T14:51:09","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-to-graphs-of-the-sine-and-cosine-function\/"},"modified":"2023-06-05T14:51:09","modified_gmt":"2023-06-05T14:51:09","slug":"solutions-to-graphs-of-the-sine-and-cosine-function","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-to-graphs-of-the-sine-and-cosine-function\/","title":{"raw":"Solutions 48: Graphs of the Sine and Cosine Function","rendered":"Solutions 48: Graphs of the Sine and Cosine Function"},"content":{"raw":"\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\n1. The sine and cosine functions have the property that [latex]f(x+P)=f(x)[\/latex] for a certain <em>P<\/em>. This means that the function values repeat for every <em>P<\/em> units on the x-axis.\n\n3. The absolute value of the constant <em>A<\/em> (amplitude) increases the total range and the constant <em>D<\/em> (vertical shift) shifts the graph vertically.\n\n5. At the point where the terminal side of <em>t<\/em> intersects the unit circle, you can determine that the sin <em>t<\/em> equals the <em>y<\/em>-coordinate of the point.\n\n7. amplitude: [latex]\\frac{2}{3}[\/latex]; period: 2\u03c0; midline: [latex]y=0[\/latex]; maximum: [latex]y=23[\/latex] occurs at [latex]x=0[\/latex]; minimum: [latex]y=\u221223[\/latex] occurs at [latex]x=\\pi[\/latex]; for one period, the graph starts at 0 and ends at 2\u03c0\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004034\/CNX_Precalc_Figure_06_01_202.jpg\" alt=\"A graph of (2\/3)cos(x). Graph has amplitude of 2\/3, period of 2pi, and range of [-2\/3, 2\/3].\">\n\n9. amplitude: 4; period: 2\u03c0; midline: [latex]y=0[\/latex]; maximum [latex]y=4[\/latex] occurs at [latex]x=\\frac{\\pi}{2}[\/latex]; minimum: [latex]y=\u22124[\/latex] occurs at [latex]x=\\frac{3\\pi}{2}[\/latex]; one full period occurs from [latex]x=0[\/latex] to [latex]x=2\u03c0[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004035\/CNX_Precalc_Figure_06_01_204.jpg\" alt=\"A graph of 4sin(x). Graph has amplitude of 4, period of 2pi, and range of [-4, 4].\">\n\n11. amplitude: 1; period: \u03c0; midline: y=0; maximum: y=1 occurs at [latex]x=\\pi[\/latex]; minimum: [latex]y=\u22121[\/latex] occurs at [latex]x=\\frac{\\pi}{2}[\/latex]; one full period is graphed from [latex]x=0[\/latex] to [latex]x=\\pi[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004037\/CNX_Precalc_Figure_06_01_206.jpg\" alt=\"A graph of cos(2x). Graph has amplitude of 1, period of pi, and range of [-1,1].\">\n\n13. amplitude: 4; period: 2; midline: [latex]y=0[\/latex]; maximum: [latex]y=4[\/latex] occurs at [latex]x=0[\/latex]; minimum: [latex]y=\u22124[\/latex] occurs at [latex]x=1[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004039\/CNX_Precalc_Figure_06_01_208.jpg\" alt=\"A graph of 4cos(pi*x). Grpah has amplitude of 4, period of 2, and range of [-4, 4].\">\n\n15. amplitude: 3; period: [latex]\\frac{\\pi}{4}[\/latex]; midline: [latex]y=5[\/latex]; maximum: [latex]y=8[\/latex] occurs at [latex]x=0.12[\/latex]; minimum: [latex]y=2[\/latex] occurs at [latex]x=0.516[\/latex]; horizontal shift: \u22124; vertical translation 5; one period occurs from [latex]x=0[\/latex] to [latex]x=\\frac{\\pi}{4}[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004041\/CNX_Precalc_Figure_06_01_210.jpg\" alt=\"A graph of 3sin(8(x+4))+5. Graph has amplitude of 3, range of [2, 8], and period of pi\/4.\">\n\n17. amplitude: 5; period: [latex]\\frac{2\\pi}{5}; midline: [latex]y=\u22122[\/latex]; maximum: [latex]y=3[\/latex] occurs at [latex]x=0.08[\/latex]; minimum: [latex]y=\u22127[\/latex] occurs at [latex]x=0.71[\/latex]; phase shift:\u22124; vertical translation:\u22122; one full period can be graphed on [latex]x=0[\/latex] to [latex]x=\\frac{2\\pi}{5}[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004042\/CNX_Precalc_Figure_06_01_212.jpg\" alt=\"A graph of 5sin(5x+20)-2. Graph has an amplitude of 5, period of 2pi\/5, and range of [-7,3].\">\n\n19. amplitude: 1; period: 2\u03c0; midline: y=1; maximum:[latex]y=2[\/latex] occurs at [latex]x=2.09[\/latex]; maximum:[latex]y=2[\/latex] occurs at[latex]t=2.09[\/latex]; minimum:[latex]y=0[\/latex] occurs at [latex]t=5.24[\/latex]; phase shift: [latex]\u2212\\frac{\\pi}{3}[\/latex]; vertical translation: 1; one full period is from [latex]t=0[\/latex] to [latex]t=2\u03c0[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004044\/CNX_Precalc_Figure_06_01_214.jpg\" alt=\"A graph of -cos(t+pi\/3)+1. Graph has amplitude of 1, period of 2pi, and range of [0,2]. Phase shifted pi\/3 to the left.\">\n\n21. amplitude: 1; period: 4\u03c0; midline: [latex]y=0[\/latex]; maximum: [latex]y=1[\/latex] occurs at [latex]t=11.52[\/latex]; minimum: [latex]y=\u22121[\/latex] occurs at [latex]t=5.24[\/latex]; phase shift: \u2212[latex]\\frac{10\\pi}{3}[\/latex]; vertical shift: 0\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004047\/CNX_Precalc_Figure_06_01_216.jpg\" alt=\"A graph of -sin((1\/2)*t + 5pi\/3). Graph has amplitude of 1, range of [-1,1], period of 4pi, and a phase shift of -10pi\/3.\">\n\n23. amplitude: 2; midline: [latex]y=\u22123[\/latex]; period: 4; equation: [latex]f(x)=2\\sin\\left(\\frac{\\pi}{2}x\\right)\u22123[\/latex]\n\n25. amplitude: 2; period: 5; midline: [latex]y=3[\/latex]; equation: [latex]f(x)=\u22122\\cos\\left(\\frac{2\\pi}{5}x\\right)+3[\/latex]\n\n27. amplitude: 4; period: 2; midline: [latex]y=0[\/latex]; equation: [latex]f(x)=\u22124\\cos\\left(\\pi\\left(x\u2212\\frac{\\pi}{2}\\right)\\right)[\/latex]\n\n29. amplitude: 2; period: 2; midline [latex]y=1[\/latex]; equation: [latex]f(x)=2\\cos\\left(\\frac{\\pi}{x}\\right)+1[\/latex]\n\n31.&nbsp;[latex]\\frac{\\pi}{6},\\frac{5\\pi}{6}[\/latex]\n\n33.&nbsp;[latex]\\frac{\\pi}{4},\\frac{3\\pi}{4}[\/latex]\n\n35.&nbsp;[latex]\\frac{3\\pi}{2}[\/latex]\n\n37.&nbsp;[latex]\\frac{\\pi}{2},\\frac{3\\pi}{2}[\/latex]\n\n39.&nbsp;[latex]\\frac{\\pi}{2},\\frac{3\\pi}{2}[\/latex]\n\n41.&nbsp;[latex]\\frac{\\pi}{6},\\frac{11\\pi}{6}[\/latex]\n\n43. The graph appears linear. The linear functions dominate the shape of the graph for large values of x.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004049\/CNX_Precalc_Figure_06_01_227.jpg\" alt=\"A sinusoidal graph that increases like the function y=x, shown from 0 to 100.\">\n\n45. The graph is symmetric with respect to the <em>y<\/em>-axis and there is no amplitude because the function is not periodic.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004052\/CNX_Precalc_Figure_06_01_229.jpg\" alt=\"A sinusoidal graph that has increasing peaks and decreasing lows as the absolute value of x increases.\">\n\n47.\na. Amplitude: 12.5; period: 10; midline: [latex]y=13.5[\/latex];\nb. [latex]h(t)=12.5\\sin\\left(\\frac{\\pi}{5}\\left(t\u22122.5\\right)\\right)+13.5;[\/latex]\nc. 26 ft\n","rendered":"<h2>Solutions to Odd-Numbered Exercises<\/h2>\n<p>1. The sine and cosine functions have the property that [latex]f(x+P)=f(x)[\/latex] for a certain <em>P<\/em>. This means that the function values repeat for every <em>P<\/em> units on the x-axis.<\/p>\n<p>3. The absolute value of the constant <em>A<\/em> (amplitude) increases the total range and the constant <em>D<\/em> (vertical shift) shifts the graph vertically.<\/p>\n<p>5. At the point where the terminal side of <em>t<\/em> intersects the unit circle, you can determine that the sin <em>t<\/em> equals the <em>y<\/em>-coordinate of the point.<\/p>\n<p>7. amplitude: [latex]\\frac{2}{3}[\/latex]; period: 2\u03c0; midline: [latex]y=0[\/latex]; maximum: [latex]y=23[\/latex] occurs at [latex]x=0[\/latex]; minimum: [latex]y=\u221223[\/latex] occurs at [latex]x=\\pi[\/latex]; for one period, the graph starts at 0 and ends at 2\u03c0<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004034\/CNX_Precalc_Figure_06_01_202.jpg\" alt=\"A graph of (2\/3)cos(x). Graph has amplitude of 2\/3, period of 2pi, and range of [-2\/3, 2\/3].\" \/><\/p>\n<p>9. amplitude: 4; period: 2\u03c0; midline: [latex]y=0[\/latex]; maximum [latex]y=4[\/latex] occurs at [latex]x=\\frac{\\pi}{2}[\/latex]; minimum: [latex]y=\u22124[\/latex] occurs at [latex]x=\\frac{3\\pi}{2}[\/latex]; one full period occurs from [latex]x=0[\/latex] to [latex]x=2\u03c0[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004035\/CNX_Precalc_Figure_06_01_204.jpg\" alt=\"A graph of 4sin(x). Graph has amplitude of 4, period of 2pi, and range of [-4, 4].\" \/><\/p>\n<p>11. amplitude: 1; period: \u03c0; midline: y=0; maximum: y=1 occurs at [latex]x=\\pi[\/latex]; minimum: [latex]y=\u22121[\/latex] occurs at [latex]x=\\frac{\\pi}{2}[\/latex]; one full period is graphed from [latex]x=0[\/latex] to [latex]x=\\pi[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004037\/CNX_Precalc_Figure_06_01_206.jpg\" alt=\"A graph of cos(2x). Graph has amplitude of 1, period of pi, and range of [-1,1].\" \/><\/p>\n<p>13. amplitude: 4; period: 2; midline: [latex]y=0[\/latex]; maximum: [latex]y=4[\/latex] occurs at [latex]x=0[\/latex]; minimum: [latex]y=\u22124[\/latex] occurs at [latex]x=1[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004039\/CNX_Precalc_Figure_06_01_208.jpg\" alt=\"A graph of 4cos(pi*x). Grpah has amplitude of 4, period of 2, and range of [-4, 4].\" \/><\/p>\n<p>15. amplitude: 3; period: [latex]\\frac{\\pi}{4}[\/latex]; midline: [latex]y=5[\/latex]; maximum: [latex]y=8[\/latex] occurs at [latex]x=0.12[\/latex]; minimum: [latex]y=2[\/latex] occurs at [latex]x=0.516[\/latex]; horizontal shift: \u22124; vertical translation 5; one period occurs from [latex]x=0[\/latex] to [latex]x=\\frac{\\pi}{4}[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004041\/CNX_Precalc_Figure_06_01_210.jpg\" alt=\"A graph of 3sin(8(x+4))+5. Graph has amplitude of 3, range of [2, 8], and period of pi\/4.\" \/><\/p>\n<p>17. amplitude: 5; period: [latex]\\frac{2\\pi}{5}; midline: [latex]y=\u22122[\/latex]; maximum: [latex]y=3[\/latex] occurs at [latex]x=0.08[\/latex]; minimum: [latex]y=\u22127[\/latex] occurs at [latex]x=0.71[\/latex]; phase shift:\u22124; vertical translation:\u22122; one full period can be graphed on [latex]x=0[\/latex] to [latex]x=\\frac{2\\pi}{5}[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004042\/CNX_Precalc_Figure_06_01_212.jpg\" alt=\"A graph of 5sin(5x+20)-2. Graph has an amplitude of 5, period of 2pi\/5, and range of [-7,3].\" \/><\/p>\n<p>19. amplitude: 1; period: 2\u03c0; midline: y=1; maximum:[latex]y=2[\/latex] occurs at [latex]x=2.09[\/latex]; maximum:[latex]y=2[\/latex] occurs at[latex]t=2.09[\/latex]; minimum:[latex]y=0[\/latex] occurs at [latex]t=5.24[\/latex]; phase shift: [latex]\u2212\\frac{\\pi}{3}[\/latex]; vertical translation: 1; one full period is from [latex]t=0[\/latex] to [latex]t=2\u03c0[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004044\/CNX_Precalc_Figure_06_01_214.jpg\" alt=\"A graph of -cos(t+pi\/3)+1. Graph has amplitude of 1, period of 2pi, and range of [0,2]. Phase shifted pi\/3 to the left.\" \/><\/p>\n<p>21. amplitude: 1; period: 4\u03c0; midline: [latex]y=0[\/latex]; maximum: [latex]y=1[\/latex] occurs at [latex]t=11.52[\/latex]; minimum: [latex]y=\u22121[\/latex] occurs at [latex]t=5.24[\/latex]; phase shift: \u2212[latex]\\frac{10\\pi}{3}[\/latex]; vertical shift: 0<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004047\/CNX_Precalc_Figure_06_01_216.jpg\" alt=\"A graph of -sin((1\/2)*t + 5pi\/3). Graph has amplitude of 1, range of [-1,1], period of 4pi, and a phase shift of -10pi\/3.\" \/><\/p>\n<p>23. amplitude: 2; midline: [latex]y=\u22123[\/latex]; period: 4; equation: [latex]f(x)=2\\sin\\left(\\frac{\\pi}{2}x\\right)\u22123[\/latex]<\/p>\n<p>25. amplitude: 2; period: 5; midline: [latex]y=3[\/latex]; equation: [latex]f(x)=\u22122\\cos\\left(\\frac{2\\pi}{5}x\\right)+3[\/latex]<\/p>\n<p>27. amplitude: 4; period: 2; midline: [latex]y=0[\/latex]; equation: [latex]f(x)=\u22124\\cos\\left(\\pi\\left(x\u2212\\frac{\\pi}{2}\\right)\\right)[\/latex]<\/p>\n<p>29. amplitude: 2; period: 2; midline [latex]y=1[\/latex]; equation: [latex]f(x)=2\\cos\\left(\\frac{\\pi}{x}\\right)+1[\/latex]<\/p>\n<p>31.&nbsp;[latex]\\frac{\\pi}{6},\\frac{5\\pi}{6}[\/latex]<\/p>\n<p>33.&nbsp;[latex]\\frac{\\pi}{4},\\frac{3\\pi}{4}[\/latex]<\/p>\n<p>35.&nbsp;[latex]\\frac{3\\pi}{2}[\/latex]<\/p>\n<p>37.&nbsp;[latex]\\frac{\\pi}{2},\\frac{3\\pi}{2}[\/latex]<\/p>\n<p>39.&nbsp;[latex]\\frac{\\pi}{2},\\frac{3\\pi}{2}[\/latex]<\/p>\n<p>41.&nbsp;[latex]\\frac{\\pi}{6},\\frac{11\\pi}{6}[\/latex]<\/p>\n<p>43. The graph appears linear. The linear functions dominate the shape of the graph for large values of x.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004049\/CNX_Precalc_Figure_06_01_227.jpg\" alt=\"A sinusoidal graph that increases like the function y=x, shown from 0 to 100.\" \/><\/p>\n<p>45. The graph is symmetric with respect to the <em>y<\/em>-axis and there is no amplitude because the function is not periodic.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004052\/CNX_Precalc_Figure_06_01_229.jpg\" alt=\"A sinusoidal graph that has increasing peaks and decreasing lows as the absolute value of x increases.\" \/><\/p>\n<p>47.<br \/>\na. Amplitude: 12.5; period: 10; midline: [latex]y=13.5[\/latex];<br \/>\nb. [latex]h(t)=12.5\\sin\\left(\\frac{\\pi}{5}\\left(t\u22122.5\\right)\\right)+13.5;[\/latex]<br \/>\nc. 26 ft<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1380\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":503070,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1380","chapter","type-chapter","status-publish","hentry"],"part":1375,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1380","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/users\/503070"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1380\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/parts\/1375"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1380\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/media?parent=1380"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapter-type?post=1380"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/contributor?post=1380"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/license?post=1380"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}