{"id":1910,"date":"2025-07-30T21:20:20","date_gmt":"2025-07-30T21:20:20","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1910"},"modified":"2025-10-13T19:51:18","modified_gmt":"2025-10-13T19:51:18","slug":"graphs-of-the-sine-and-cosine-function-apply-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-the-sine-and-cosine-function-apply-it-1\/","title":{"raw":"Graphs of the Sine and Cosine Function: Apply It 1","rendered":"Graphs of the Sine and Cosine Function: Apply It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Determine amplitude, period, phase shift, and vertical shift of a sine or cosine graph from its equation.<\/li>\r\n \t<li>Graph transformations of y=cos x and y=sin x .<\/li>\r\n \t<li>Determine a function formula that would have a given sinusoidal graph.<\/li>\r\n \t<li>Determine functions that model circular and periodic motion.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Using Transformations of Sine and Cosine Functions<\/h2>\r\nWe can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, <strong>circular motion<\/strong> can be modeled using either the sine or <strong>cosine function<\/strong>.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">A point rotates around a circle of radius 3 centered at the origin. Sketch a graph of the <em>y<\/em>-coordinate of the point as a function of the angle of rotation.\r\n\r\n[reveal-answer q=\"140255\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"140255\"]\r\n\r\nRecall that, for a point on a circle of radius r, the y-coordinate of the point is [latex]y=r\\sin(x)[\/latex], so in this case, we get the equation [latex]y(x)=3\\sin(x)[\/latex]. The constant 3 causes a vertical stretch of the y-values of the function by a factor of 3, which we can see in the graph.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003959\/CNX_Precalc_Figure_06_01_023.jpg\" alt=\"A graph of 3sin(x). Graph has period of 2pi, amplitude of 3, and range of [-3,3].\" width=\"487\" height=\"319\" \/>\r\n<h4>Analysis of the Solution<\/h4>\r\nNotice that the period of the function is still 2\u03c0; as we travel around the circle, we return to the point (3,0) for [latex]x=2\\pi,4\\pi,6\\pi,\\dots[\/latex] Because the outputs of the graph will now oscillate between \u20133 and 3, the amplitude of the sine wave is 3.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"bcc-box bcc-success\">\r\n\r\nWhat is the amplitude of the function [latex]f(x)=7\\cos(x)[\/latex]? Sketch a graph of this function.\r\n\r\n[reveal-answer q=\"317443\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"317443\"]\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004029\/CNX_Precalc_Figure_06_01_024.jpg\" alt=\"A graph of 7cos(x). Graph has amplitude of 7, period of 2pi, and range of [-7,7].\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled <em>P<\/em>. Sketch a graph of the height above the ground of the point <em>P<\/em> as the circle is rotated; then find a function that gives the height in terms of the angle of rotation.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004001\/CNX_Precalc_Figure_06_01_025.jpg\" alt=\"An illustration of a circle lifted 4 feet off the ground. Circle has radius of 3 ft. There is a point P labeled on the circle's circumference.\" width=\"487\" height=\"300\" \/>\r\n\r\n[reveal-answer q=\"367979\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"367979\"]\r\n\r\nSketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004004\/CNX_Precalc_Figure_06_01_026.jpg\" alt=\"A graph of -3cox(x)+4. Graph has midline at y=4, amplitude of 3, and period of 2pi.\" width=\"487\" height=\"521\" \/>\r\n\r\nAlthough we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other. Let\u2019s use a cosine function because it starts at the highest or lowest value, while a <strong>sine function<\/strong> starts at the middle value. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection.\r\n\r\nSecond, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example.\r\n\r\nFinally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these transformations together, we find that\r\n<p style=\"text-align: center;\">[latex]y=\u22123\\cos(x)+4[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">\r\n<div class=\"bcc-box bcc-success\">\r\n\r\nA weight is attached to a spring that is then hung from a board. As the spring oscillates up and down, the position <em>y<\/em> of the weight relative to the board ranges from \u20131 in. (at time <em>x<\/em> = 0) to \u20137in. (at time <em>x<\/em> = \u03c0) below the board. Assume the position of <em>y<\/em> is given as a sinusoidal function of <em>x<\/em>. Sketch a graph of the function, and then find a cosine function that gives the position <em>y<\/em> in terms of x.\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004006\/CNX_Precalc_Figure_06_01_029.jpg\" alt=\"An illustration of a spring with length y.\" width=\"487\" height=\"351\" \/>\r\n\r\n[reveal-answer q=\"518116\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"518116\"]\r\n\r\n[latex]y=3\\cos(x)\u22124[\/latex]\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004032\/CNX_Precalc_Figure_06_01_027.jpg\" alt=\"A cosine graph with range [-1,-7]. Period is 2 pi. Local maximums at (0,-1), (2pi,-1), and (4pi, -1). Local minimums at (pi,-7) and (3pi, -7).\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>\r\n<div class=\"bcc-box bcc-success\"><section class=\"textbox example\" aria-label=\"Example\">The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider\u2019s height above ground as a function of time in minutes.\r\n\r\n[reveal-answer q=\"304167\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"304167\"]\r\n\r\nWith a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center.\r\n\r\nPassengers board 2 m above ground level, so the center of the wheel must be located 67.5 + 2 = 69.5 m above ground level. The midline of the oscillation will be at 69.5 m.\r\n\r\nThe wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.\r\n\r\nLastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.\r\n<ul>\r\n \t<li>Amplitude: 67.5, so <em>A\u00a0<\/em>= 67.5<\/li>\r\n \t<li>Midline: 69.5, so <em>D<\/em> = 69.5<\/li>\r\n \t<li>Period: 30, so [latex]B=\\frac{2\\pi}{30}=\\frac{\\pi}{15}[\/latex]<\/li>\r\n \t<li>Shape: \u2212cos(<em>t<\/em>)<\/li>\r\n<\/ul>\r\nAn equation for the rider\u2019s height would be\r\n<p style=\"text-align: center;\">[latex]y=\u221267.5\\cos\\left(\\frac{\\pi}{15}t\\right)+69.5[\/latex]<\/p>\r\nwhere <em>t<\/em> is in minutes and <em>y<\/em> is measured in meters.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]127257[\/ohm_question]<\/section><\/div>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Determine amplitude, period, phase shift, and vertical shift of a sine or cosine graph from its equation.<\/li>\n<li>Graph transformations of y=cos x and y=sin x .<\/li>\n<li>Determine a function formula that would have a given sinusoidal graph.<\/li>\n<li>Determine functions that model circular and periodic motion.<\/li>\n<\/ul>\n<\/section>\n<h2>Using Transformations of Sine and Cosine Functions<\/h2>\n<p>We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, <strong>circular motion<\/strong> can be modeled using either the sine or <strong>cosine function<\/strong>.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">A point rotates around a circle of radius 3 centered at the origin. Sketch a graph of the <em>y<\/em>-coordinate of the point as a function of the angle of rotation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q140255\">Show Solution<\/button><\/p>\n<div id=\"q140255\" class=\"hidden-answer\" style=\"display: none\">\n<p>Recall that, for a point on a circle of radius r, the y-coordinate of the point is [latex]y=r\\sin(x)[\/latex], so in this case, we get the equation [latex]y(x)=3\\sin(x)[\/latex]. The constant 3 causes a vertical stretch of the y-values of the function by a factor of 3, which we can see in the graph.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003959\/CNX_Precalc_Figure_06_01_023.jpg\" alt=\"A graph of 3sin(x). Graph has period of 2pi, amplitude of 3, and range of &#091;-3,3&#093;.\" width=\"487\" height=\"319\" \/><\/p>\n<h4>Analysis of the Solution<\/h4>\n<p>Notice that the period of the function is still 2\u03c0; as we travel around the circle, we return to the point (3,0) for [latex]x=2\\pi,4\\pi,6\\pi,\\dots[\/latex] Because the outputs of the graph will now oscillate between \u20133 and 3, the amplitude of the sine wave is 3.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"bcc-box bcc-success\">\n<p>What is the amplitude of the function [latex]f(x)=7\\cos(x)[\/latex]? Sketch a graph of this function.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q317443\">Show Solution<\/button><\/p>\n<div id=\"q317443\" class=\"hidden-answer\" style=\"display: none\">\n<p><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004029\/CNX_Precalc_Figure_06_01_024.jpg\" alt=\"A graph of 7cos(x). Graph has amplitude of 7, period of 2pi, and range of [-7,7].\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled <em>P<\/em>. Sketch a graph of the height above the ground of the point <em>P<\/em> as the circle is rotated; then find a function that gives the height in terms of the angle of rotation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004001\/CNX_Precalc_Figure_06_01_025.jpg\" alt=\"An illustration of a circle lifted 4 feet off the ground. Circle has radius of 3 ft. There is a point P labeled on the circle's circumference.\" width=\"487\" height=\"300\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q367979\">Show Solution<\/button><\/p>\n<div id=\"q367979\" class=\"hidden-answer\" style=\"display: none\">\n<p>Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004004\/CNX_Precalc_Figure_06_01_026.jpg\" alt=\"A graph of -3cox(x)+4. Graph has midline at y=4, amplitude of 3, and period of 2pi.\" width=\"487\" height=\"521\" \/><\/p>\n<p>Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other. Let\u2019s use a cosine function because it starts at the highest or lowest value, while a <strong>sine function<\/strong> starts at the middle value. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection.<\/p>\n<p>Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example.<\/p>\n<p>Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these transformations together, we find that<\/p>\n<p style=\"text-align: center;\">[latex]y=\u22123\\cos(x)+4[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">\n<div class=\"bcc-box bcc-success\">\n<p>A weight is attached to a spring that is then hung from a board. As the spring oscillates up and down, the position <em>y<\/em> of the weight relative to the board ranges from \u20131 in. (at time <em>x<\/em> = 0) to \u20137in. (at time <em>x<\/em> = \u03c0) below the board. Assume the position of <em>y<\/em> is given as a sinusoidal function of <em>x<\/em>. Sketch a graph of the function, and then find a cosine function that gives the position <em>y<\/em> in terms of x.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004006\/CNX_Precalc_Figure_06_01_029.jpg\" alt=\"An illustration of a spring with length y.\" width=\"487\" height=\"351\" \/><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q518116\">Show Solution<\/button><\/p>\n<div id=\"q518116\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]y=3\\cos(x)\u22124[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27004032\/CNX_Precalc_Figure_06_01_027.jpg\" alt=\"A cosine graph with range &#091;-1,-7&#093;. Period is 2 pi. Local maximums at (0,-1), (2pi,-1), and (4pi, -1). Local minimums at (pi,-7) and (3pi, -7).\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"bcc-box bcc-success\">\n<section class=\"textbox example\" aria-label=\"Example\">The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider\u2019s height above ground as a function of time in minutes.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q304167\">Show Solution<\/button><\/p>\n<div id=\"q304167\" class=\"hidden-answer\" style=\"display: none\">\n<p>With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center.<\/p>\n<p>Passengers board 2 m above ground level, so the center of the wheel must be located 67.5 + 2 = 69.5 m above ground level. The midline of the oscillation will be at 69.5 m.<\/p>\n<p>The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.<\/p>\n<p>Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.<\/p>\n<ul>\n<li>Amplitude: 67.5, so <em>A\u00a0<\/em>= 67.5<\/li>\n<li>Midline: 69.5, so <em>D<\/em> = 69.5<\/li>\n<li>Period: 30, so [latex]B=\\frac{2\\pi}{30}=\\frac{\\pi}{15}[\/latex]<\/li>\n<li>Shape: \u2212cos(<em>t<\/em>)<\/li>\n<\/ul>\n<p>An equation for the rider\u2019s height would be<\/p>\n<p style=\"text-align: center;\">[latex]y=\u221267.5\\cos\\left(\\frac{\\pi}{15}t\\right)+69.5[\/latex]<\/p>\n<p>where <em>t<\/em> is in minutes and <em>y<\/em> is measured in meters.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm127257\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=127257&theme=lumen&iframe_resize_id=ohm127257&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<\/div>\n","protected":false},"author":13,"menu_order":10,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":191,"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\/1910"}],"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\/13"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1910\/revisions"}],"predecessor-version":[{"id":4622,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1910\/revisions\/4622"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/191"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1910\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1910"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1910"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1910"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1910"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}