{"id":3228,"date":"2025-08-15T23:37:00","date_gmt":"2025-08-15T23:37:00","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=3228"},"modified":"2025-10-09T19:10:16","modified_gmt":"2025-10-09T19:10:16","slug":"sine-and-cosine-functions-apply-it","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/sine-and-cosine-functions-apply-it\/","title":{"raw":"Sine and Cosine Functions: Apply It","rendered":"Sine and Cosine Functions: Apply It"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find function values for the sine and cosine of the special angles.<\/li>\r\n \t<li>Use reference angles to evaluate trigonometric functions.<\/li>\r\n \t<li>Evaluate sine and cosine values using a calculator.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Satellite Orbit Analysis<\/h2>\r\n<p class=\"whitespace-normal break-words\">A communications satellite orbits Earth in a circular path. Engineers track the satellite's position relative to a ground station at the center of its orbit. The satellite's path can be modeled using a unit circle, where each point [latex](x, y)[\/latex] corresponds to the satellite's position at angle [latex]t[\/latex].<\/p>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\">On a unit circle with radius 1, for any angle [latex]t[\/latex], the coordinates of the point where the terminal side intersects the circle are [latex](\\cos t, \\sin t)[\/latex]. The cosine gives the x-coordinate and the sine gives the y-coordinate.\r\n\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\">\r\n<p class=\"whitespace-normal break-words\"><strong>How To: Finding Sine and Cosine Using Reference Angles<\/strong><\/p>\r\n\r\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Determine which quadrant the angle is in<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find the reference angle (the acute angle to the x-axis)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find sine and cosine of the reference angle<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Apply appropriate signs based on the quadrant (remember: All Students Take Calculus)\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Quadrant I: both positive<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quadrant II: sine positive, cosine negative<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quadrant III: both negative<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Quadrant IV: sine negative, cosine positive<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">\r\n<p class=\"whitespace-normal break-words\">A satellite begins its orbit at the rightmost position (angle 0) relative to a ground tracking station. Engineers need to determine the satellite's position at various points in its orbit.<\/p>\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li class=\"whitespace-normal break-words\">After rotating [latex]\\frac{5\\pi}{6}[\/latex] radians, find the coordinates on a unit circle corresponding to this angle.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">The satellite's actual orbit has a radius of 8,000 kilometers above the tracking station. What are the satellite's horizontal and vertical distances from the station?<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Find the satellite's position when it has rotated [latex]\\frac{5\\pi}{4}[\/latex] radians from its starting point.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"498102\"]Show Solutions[\/reveal-answer]\r\n[hidden-answer a=\"498102\"]\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>\r\n<p class=\"whitespace-normal break-words\">First, determine the quadrant. Since [latex]\\frac{5\\pi}{6}[\/latex] is between [latex]\\frac{\\pi}{2}[\/latex] and [latex]\\pi[\/latex], the angle is in Quadrant II.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Find the reference angle: [latex]\\begin{aligned} \\text{Reference angle} &amp;= \\pi - \\frac{5\\pi}{6} \\ &amp;= \\frac{6\\pi}{6} - \\frac{5\\pi}{6} \\ &amp;= \\frac{\\pi}{6} \\end{aligned}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For the reference angle [latex]\\frac{\\pi}{6}[\/latex] (30\u00b0), we know from our special angles: [latex]\\cos\\left(\\frac{\\pi}{6}\\right) = \\frac{\\sqrt{3}}{2} \\text{ and } \\sin\\left(\\frac{\\pi}{6}\\right) = \\frac{1}{2}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">In Quadrant II, cosine is negative and sine is positive: [latex]\\cos\\left(\\frac{5\\pi}{6}\\right) = -\\frac{\\sqrt{3}}{2} \\text{ and } \\sin\\left(\\frac{5\\pi}{6}\\right) = \\frac{1}{2}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">The unit circle coordinates are [latex]\\left(-\\frac{\\sqrt{3}}{2}, \\frac{1}{2}\\right)[\/latex].<\/p>\r\n<\/li>\r\n \t<li>\r\n<p class=\"whitespace-normal break-words\">For a circular orbit with radius 8,000 kilometers, multiply the unit circle coordinates by 8,000:<\/p>\r\n<p class=\"whitespace-normal break-words\">Horizontal distance from station: [latex]8{,}000 \\times \\left(-\\frac{\\sqrt{3}}{2}\\right) \\approx 8{,}000 \\times (-0.866) = -6{,}928[\/latex] km<\/p>\r\n<p class=\"whitespace-normal break-words\">Vertical distance from station: [latex]8{,}000 \\times \\frac{1}{2} = 4{,}000[\/latex] km<\/p>\r\n<p class=\"whitespace-normal break-words\">The satellite is approximately 6,928 km to the west of the tracking station and 4,000 km north of the station.<\/p>\r\n<\/li>\r\n \t<li>\r\n<p class=\"whitespace-normal break-words\">First, determine the quadrant. Since [latex]\\frac{5\\pi}{4}[\/latex] is between [latex]\\pi[\/latex] and [latex]\\frac{3\\pi}{2}[\/latex], the angle is in Quadrant III.<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">Find the reference angle: [latex]\\begin{aligned} \\text{Reference angle} &amp;= \\frac{5\\pi}{4} - \\pi \\ &amp;= \\frac{5\\pi}{4} - \\frac{4\\pi}{4} \\ &amp;= \\frac{\\pi}{4} \\end{aligned}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">For the reference angle [latex]\\frac{\\pi}{4}[\/latex] (45\u00b0): [latex]\\cos\\left(\\frac{\\pi}{4}\\right) = \\frac{\\sqrt{2}}{2} \\text{ and } \\sin\\left(\\frac{\\pi}{4}\\right) = \\frac{\\sqrt{2}}{2}[\/latex]<\/p>\r\n<p class=\"whitespace-pre-wrap break-words\">In Quadrant III, both cosine and sine are negative: [latex]\\cos\\left(\\frac{5\\pi}{4}\\right) = -\\frac{\\sqrt{2}}{2} \\text{ and } \\sin\\left(\\frac{5\\pi}{4}\\right) = -\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">The unit circle coordinates are [latex]\\left(-\\frac{\\sqrt{2}}{2}, -\\frac{\\sqrt{2}}{2}\\right)[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">On the 8,000 km orbit:<\/p>\r\n\r\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\r\n \t<li class=\"whitespace-normal break-words\">Horizontal: [latex]8{,}000 \\times \\left(-\\frac{\\sqrt{2}}{2}\\right) \\approx -5{,}657[\/latex] km (west of station)<\/li>\r\n \t<li class=\"whitespace-normal break-words\">Vertical: [latex]8{,}000 \\times \\left(-\\frac{\\sqrt{2}}{2}\\right) \\approx -5{,}657[\/latex] km (south of station)<\/li>\r\n<\/ul>\r\n<p class=\"whitespace-normal break-words\">The satellite is approximately 5,657 km southwest of the tracking station.<\/p>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox proTip\" aria-label=\"Pro Tip\">To find coordinates on any circle with radius [latex]r[\/latex], multiply the unit circle coordinates by [latex]r[\/latex]: [latex](x, y) = (r\\cos t, r\\sin t)[\/latex].\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<p class=\"whitespace-normal break-words\">A satellite is at an angle of [latex]\\frac{2\\pi}{3}[\/latex] radians from its starting position.<\/p>\r\n\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li class=\"whitespace-normal break-words\">Find the unit circle coordinates for this angle.<\/li>\r\n \t<li class=\"whitespace-normal break-words\">If the satellite orbits at 6,500 km from the tracking station, what are its horizontal and vertical distances from the station?<\/li>\r\n<\/ol>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find function values for the sine and cosine of the special angles.<\/li>\n<li>Use reference angles to evaluate trigonometric functions.<\/li>\n<li>Evaluate sine and cosine values using a calculator.<\/li>\n<\/ul>\n<\/section>\n<h2 class=\"text-xl font-bold text-text-100 mt-1 -mb-0.5\">Satellite Orbit Analysis<\/h2>\n<p class=\"whitespace-normal break-words\">A communications satellite orbits Earth in a circular path. Engineers track the satellite&#8217;s position relative to a ground station at the center of its orbit. The satellite&#8217;s path can be modeled using a unit circle, where each point [latex](x, y)[\/latex] corresponds to the satellite&#8217;s position at angle [latex]t[\/latex].<\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\">On a unit circle with radius 1, for any angle [latex]t[\/latex], the coordinates of the point where the terminal side intersects the circle are [latex](\\cos t, \\sin t)[\/latex]. The cosine gives the x-coordinate and the sine gives the y-coordinate.<\/p>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\">\n<p class=\"whitespace-normal break-words\"><strong>How To: Finding Sine and Cosine Using Reference Angles<\/strong><\/p>\n<ol class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Determine which quadrant the angle is in<\/li>\n<li class=\"whitespace-normal break-words\">Find the reference angle (the acute angle to the x-axis)<\/li>\n<li class=\"whitespace-normal break-words\">Find sine and cosine of the reference angle<\/li>\n<li class=\"whitespace-normal break-words\">Apply appropriate signs based on the quadrant (remember: All Students Take Calculus)\n<ul class=\"[&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Quadrant I: both positive<\/li>\n<li class=\"whitespace-normal break-words\">Quadrant II: sine positive, cosine negative<\/li>\n<li class=\"whitespace-normal break-words\">Quadrant III: both negative<\/li>\n<li class=\"whitespace-normal break-words\">Quadrant IV: sine negative, cosine positive<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p class=\"whitespace-normal break-words\">A satellite begins its orbit at the rightmost position (angle 0) relative to a ground tracking station. Engineers need to determine the satellite&#8217;s position at various points in its orbit.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li class=\"whitespace-normal break-words\">After rotating [latex]\\frac{5\\pi}{6}[\/latex] radians, find the coordinates on a unit circle corresponding to this angle.<\/li>\n<li class=\"whitespace-normal break-words\">The satellite&#8217;s actual orbit has a radius of 8,000 kilometers above the tracking station. What are the satellite&#8217;s horizontal and vertical distances from the station?<\/li>\n<li class=\"whitespace-normal break-words\">Find the satellite&#8217;s position when it has rotated [latex]\\frac{5\\pi}{4}[\/latex] radians from its starting point.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q498102\">Show Solutions<\/button><\/p>\n<div id=\"q498102\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li>\n<p class=\"whitespace-normal break-words\">First, determine the quadrant. Since [latex]\\frac{5\\pi}{6}[\/latex] is between [latex]\\frac{\\pi}{2}[\/latex] and [latex]\\pi[\/latex], the angle is in Quadrant II.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Find the reference angle: [latex]\\begin{aligned} \\text{Reference angle} &= \\pi - \\frac{5\\pi}{6} \\ &= \\frac{6\\pi}{6} - \\frac{5\\pi}{6} \\ &= \\frac{\\pi}{6} \\end{aligned}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">For the reference angle [latex]\\frac{\\pi}{6}[\/latex] (30\u00b0), we know from our special angles: [latex]\\cos\\left(\\frac{\\pi}{6}\\right) = \\frac{\\sqrt{3}}{2} \\text{ and } \\sin\\left(\\frac{\\pi}{6}\\right) = \\frac{1}{2}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">In Quadrant II, cosine is negative and sine is positive: [latex]\\cos\\left(\\frac{5\\pi}{6}\\right) = -\\frac{\\sqrt{3}}{2} \\text{ and } \\sin\\left(\\frac{5\\pi}{6}\\right) = \\frac{1}{2}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">The unit circle coordinates are [latex]\\left(-\\frac{\\sqrt{3}}{2}, \\frac{1}{2}\\right)[\/latex].<\/p>\n<\/li>\n<li>\n<p class=\"whitespace-normal break-words\">For a circular orbit with radius 8,000 kilometers, multiply the unit circle coordinates by 8,000:<\/p>\n<p class=\"whitespace-normal break-words\">Horizontal distance from station: [latex]8{,}000 \\times \\left(-\\frac{\\sqrt{3}}{2}\\right) \\approx 8{,}000 \\times (-0.866) = -6{,}928[\/latex] km<\/p>\n<p class=\"whitespace-normal break-words\">Vertical distance from station: [latex]8{,}000 \\times \\frac{1}{2} = 4{,}000[\/latex] km<\/p>\n<p class=\"whitespace-normal break-words\">The satellite is approximately 6,928 km to the west of the tracking station and 4,000 km north of the station.<\/p>\n<\/li>\n<li>\n<p class=\"whitespace-normal break-words\">First, determine the quadrant. Since [latex]\\frac{5\\pi}{4}[\/latex] is between [latex]\\pi[\/latex] and [latex]\\frac{3\\pi}{2}[\/latex], the angle is in Quadrant III.<\/p>\n<p class=\"whitespace-pre-wrap break-words\">Find the reference angle: [latex]\\begin{aligned} \\text{Reference angle} &= \\frac{5\\pi}{4} - \\pi \\ &= \\frac{5\\pi}{4} - \\frac{4\\pi}{4} \\ &= \\frac{\\pi}{4} \\end{aligned}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">For the reference angle [latex]\\frac{\\pi}{4}[\/latex] (45\u00b0): [latex]\\cos\\left(\\frac{\\pi}{4}\\right) = \\frac{\\sqrt{2}}{2} \\text{ and } \\sin\\left(\\frac{\\pi}{4}\\right) = \\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<p class=\"whitespace-pre-wrap break-words\">In Quadrant III, both cosine and sine are negative: [latex]\\cos\\left(\\frac{5\\pi}{4}\\right) = -\\frac{\\sqrt{2}}{2} \\text{ and } \\sin\\left(\\frac{5\\pi}{4}\\right) = -\\frac{\\sqrt{2}}{2}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">The unit circle coordinates are [latex]\\left(-\\frac{\\sqrt{2}}{2}, -\\frac{\\sqrt{2}}{2}\\right)[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">On the 8,000 km orbit:<\/p>\n<ul class=\"&#091;&amp;:not(:last-child)_ul&#093;:pb-1 &#091;&amp;:not(:last-child)_ol&#093;:pb-1 list-disc space-y-1.5 pl-7\">\n<li class=\"whitespace-normal break-words\">Horizontal: [latex]8{,}000 \\times \\left(-\\frac{\\sqrt{2}}{2}\\right) \\approx -5{,}657[\/latex] km (west of station)<\/li>\n<li class=\"whitespace-normal break-words\">Vertical: [latex]8{,}000 \\times \\left(-\\frac{\\sqrt{2}}{2}\\right) \\approx -5{,}657[\/latex] km (south of station)<\/li>\n<\/ul>\n<p class=\"whitespace-normal break-words\">The satellite is approximately 5,657 km southwest of the tracking station.<\/p>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox proTip\" aria-label=\"Pro Tip\">To find coordinates on any circle with radius [latex]r[\/latex], multiply the unit circle coordinates by [latex]r[\/latex]: [latex](x, y) = (r\\cos t, r\\sin t)[\/latex].<\/p>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<p class=\"whitespace-normal break-words\">A satellite is at an angle of [latex]\\frac{2\\pi}{3}[\/latex] radians from its starting position.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li class=\"whitespace-normal break-words\">Find the unit circle coordinates for this angle.<\/li>\n<li class=\"whitespace-normal break-words\">If the satellite orbits at 6,500 km from the tracking station, what are its horizontal and vertical distances from the station?<\/li>\n<\/ol>\n<\/section>\n","protected":false},"author":67,"menu_order":23,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":178,"module-header":"apply_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3228"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3228\/revisions"}],"predecessor-version":[{"id":4590,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3228\/revisions\/4590"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/178"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/3228\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=3228"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=3228"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=3228"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=3228"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}