{"id":1542,"date":"2025-07-25T02:40:29","date_gmt":"2025-07-25T02:40:29","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1542"},"modified":"2026-03-12T06:34:13","modified_gmt":"2026-03-12T06:34:13","slug":"modeling-with-trigonometric-equations-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/modeling-with-trigonometric-equations-fresh-take\/","title":{"raw":"Modeling with Trigonometric Equations: Fresh Take","rendered":"Modeling with Trigonometric Equations: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Determine the amplitude and period of a periodic context<\/li>\r\n \t<li>Model periodic behavior with sinusoidal functions<\/li>\r\n \t<li>Write both a sine and cosine function to model the same periodic behavior<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Determine the Amplitude and Period of a Periodic Context<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\nMany real-world situations\u2014like tides, Ferris wheels, or seasonal temperatures\u2014follow predictable cycles. These can be modeled with sine or cosine functions, where the amplitude describes how far the values swing above and below the midline, and the period describes how long it takes for one full cycle to repeat. Identifying amplitude and period from a context helps create accurate mathematical models of cyclical behavior.\r\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Finding Amplitude and Period<\/strong><\/p>\r\n\r\n<ul>\r\n \t<li><strong>Amplitude [latex]a[\/latex]<\/strong>\r\n<ul>\r\n \t<li>Formula: [latex]\\text{Amplitude} = \\dfrac{\\text{Maximum Value} - \\text{Minimum Value}}{2}[\/latex]<\/li>\r\n \t<li>Represents half the distance between the peak and the trough<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Period [latex]P[\/latex]<\/strong>\r\n<ul>\r\n \t<li>In real-world contexts, period corresponds to the time or distance for one complete cycle<\/li>\r\n \t<li>Formula: [latex]P = \\dfrac{2\\pi}{b}[\/latex] where [latex]b[\/latex] is the coefficient of [latex]x[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Midline [latex]d[\/latex]<\/strong>\r\n<ul>\r\n \t<li>Formula: [latex]\\text{Midline} = \\dfrac{\\text{Maximum Value} + \\text{Minimum Value}}{2}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">The temperature in a city varies from a low of 30\u00b0F to a high of 70\u00b0F over a 12-month period. Find the amplitude, period, and midline for a model of temperature over time.[reveal-answer q=\"period-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"period-001\"]\r\n[latex]\\text{Amplitude} = \\frac{70 - 30}{2} = \\frac{40}{2} = 20[\/latex][latex]\\text{Midline} = \\frac{70 + 30}{2} = \\frac{100}{2} = 50[\/latex][\/hidden-answer]<\/section>\r\n<h2>Model Periodic Behavior with Sinusoidal Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\nMany real-world patterns repeat in cycles, such as tides, sound waves, daylight hours, or seasonal temperatures. These can be modeled with sine or cosine functions, called sinusoids. A sinusoidal model captures the maximum and minimum values, the average (midline), how long the cycle lasts (period), and whether the curve starts at a peak, trough, or midline.\r\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Building a Sinusoidal Model<\/strong><\/p>\r\n\r\n<ol>\r\n \t<li><strong>Identify Key Features<\/strong>\r\n<ul>\r\n \t<li>Amplitude: [latex]\\dfrac{\\text{max} - \\text{min}}{2}[\/latex]<\/li>\r\n \t<li>Midline: [latex]\\dfrac{\\text{max} + \\text{min}}{2}[\/latex]<\/li>\r\n \t<li>Period: length of one full cycle<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Pick Sine or Cosine<\/strong>\r\n<ul>\r\n \t<li>Use cosine if the graph starts at a maximum or minimum<\/li>\r\n \t<li>Use sine if the graph starts at the midline<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Write the General Formula<\/strong>\r\n<ul>\r\n \t<li>[latex]y = a\\sin(bx-c)+d[\/latex] or [latex]y = a\\cos(bx-c)+d[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<section class=\"textbox example\">The depth of water at a dock varies between 4 feet and 12 feet. High tide occurs at noon, and the water returns to high tide every 12 hours. Write a cosine function to model the water depth [latex]h[\/latex] as a function of time [latex]t[\/latex] in hours after noon.[reveal-answer q=\"model-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"model-001\"]Amplitude:[latex]a = \\frac{12 - 4}{2} = \\frac{8}{2} = 4 \\\\[\/latex]Midline:[latex]d = \\frac{12 + 4}{2} = \\frac{16}{2} = 8[\/latex]Period:[latex]P = 12[\/latex] hours, so\r\n[latex]\\\\ 12 = \\frac{2\\pi}{b}[\/latex]\r\n[latex]b = \\frac{2\\pi}{12} = \\frac{\\pi}{6}[\/latex]Starting point:Since the water is at high tide (maximum) when [latex]t = 0[\/latex], use cosine with no phase shift.Model:[latex]h(t) = 4\\cos\\left(\\frac{\\pi}{6}t\\right) + 8[\/latex]\r\n\r\nwhere [latex]t[\/latex] is hours after noon and [latex]h[\/latex] is depth in feet.\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbdahbec-KOOORDz_AYw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/KOOORDz_AYw?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-bbdahbec-KOOORDz_AYw\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14661388&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-bbdahbec-KOOORDz_AYw&vembed=0&video_id=KOOORDz_AYw&video_target=tpm-plugin-bbdahbec-KOOORDz_AYw'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Sinusoidal+Function+Word+Problems+-+Ferris+Wheels+and+Temperature_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSinusoidal Function Word Problems: Ferris Wheels and Temperature\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Write Both a Sine and Cosine Function to Model the Same Periodic Behavior<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea<\/strong>\r\n\r\nAny periodic behavior can be written using either a sine or a cosine function, because the two are just phase-shifted versions of each other. This means that if a cosine model starts at a peak, you can write an equivalent sine model by shifting it horizontally, and vice versa. Writing both functions for the same context helps show that the choice between sine and cosine is flexible\u2014it depends on which starting point makes the model easier to describe.\r\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Writing Both Sine and Cosine Models<\/strong><\/p>\r\n\r\n<ul>\r\n \t<li><strong>Start with Key Features<\/strong> (amplitude, midline, period)<\/li>\r\n \t<li><strong>Cosine Model:<\/strong> Use when the cycle starts at a maximum or minimum<\/li>\r\n \t<li><strong>Sine Model:<\/strong> Use when the cycle starts at the midline<\/li>\r\n \t<li><strong>Relating the Two:<\/strong> A sine curve shifted by [latex]\\dfrac{\\pi}{2b}[\/latex] equals a cosine curve<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">A Ferris wheel has a diameter of 50 feet with its center 30 feet above the ground. It completes one rotation every 60 seconds. At [latex]t = 0[\/latex], a rider is at the lowest point. Write both a sine and a cosine function to model the rider's height [latex]h[\/latex] above the ground as a function of time [latex]t[\/latex] in seconds.[reveal-answer q=\"both-001\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"both-001\"]Key features:Maximum height: 30 + 25 = 55 feet\r\nMinimum height: 30 - 25 = 5 feet\r\nAmplitude: [latex]a = 25[\/latex]\r\nMidline: [latex]d = 30[\/latex]\r\nPeriod: [latex]P = 60[\/latex], so [latex]b = \\frac{2\\pi}{60} = \\frac{\\pi}{30}[\/latex]Cosine model:\r\nSince the rider starts at the minimum (not maximum), we need a reflection or phase shift:\r\n[latex]h(t) = -25\\cos\\left(\\frac{\\pi}{30}t\\right) + 30[\/latex]or equivalently with a phase shift:\r\n[latex]h(t) = 25\\cos\\left(\\frac{\\pi}{30}t - \\pi\\right) + 30[\/latex]Sine model:\r\nThe rider starts at the minimum and rises, which matches a sine curve shifted down from the midline:\r\n[latex]h(t) = 25\\sin\\left(\\frac{\\pi}{30}t - \\frac{\\pi}{2}\\right) + 30[\/latex]or using a negative amplitude:\r\n[latex]h(t) = -25\\sin\\left(\\frac{\\pi}{30}t + \\frac{\\pi}{2}\\right) + 30[\/latex]Both models produce the same values for the rider's height at any time [latex]t[\/latex].\r\n[\/hidden-answer]<\/section><section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fdgdbddh-Vp6uOaOxUS0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Vp6uOaOxUS0?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fdgdbddh-Vp6uOaOxUS0\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=14660584&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fdgdbddh-Vp6uOaOxUS0&vembed=0&video_id=Vp6uOaOxUS0&video_target=tpm-plugin-fdgdbddh-Vp6uOaOxUS0'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Determining+the+Equation+of+a+Sine+and+Cosine+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining the Equation of a Sine and Cosine Graph\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Determine the amplitude and period of a periodic context<\/li>\n<li>Model periodic behavior with sinusoidal functions<\/li>\n<li>Write both a sine and cosine function to model the same periodic behavior<\/li>\n<\/ul>\n<\/section>\n<h2>Determine the Amplitude and Period of a Periodic Context<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>Many real-world situations\u2014like tides, Ferris wheels, or seasonal temperatures\u2014follow predictable cycles. These can be modeled with sine or cosine functions, where the amplitude describes how far the values swing above and below the midline, and the period describes how long it takes for one full cycle to repeat. Identifying amplitude and period from a context helps create accurate mathematical models of cyclical behavior.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Finding Amplitude and Period<\/strong><\/p>\n<ul>\n<li><strong>Amplitude [latex]a[\/latex]<\/strong>\n<ul>\n<li>Formula: [latex]\\text{Amplitude} = \\dfrac{\\text{Maximum Value} - \\text{Minimum Value}}{2}[\/latex]<\/li>\n<li>Represents half the distance between the peak and the trough<\/li>\n<\/ul>\n<\/li>\n<li><strong>Period [latex]P[\/latex]<\/strong>\n<ul>\n<li>In real-world contexts, period corresponds to the time or distance for one complete cycle<\/li>\n<li>Formula: [latex]P = \\dfrac{2\\pi}{b}[\/latex] where [latex]b[\/latex] is the coefficient of [latex]x[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Midline [latex]d[\/latex]<\/strong>\n<ul>\n<li>Formula: [latex]\\text{Midline} = \\dfrac{\\text{Maximum Value} + \\text{Minimum Value}}{2}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">The temperature in a city varies from a low of 30\u00b0F to a high of 70\u00b0F over a 12-month period. Find the amplitude, period, and midline for a model of temperature over time.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qperiod-001\">Show Solution<\/button><\/p>\n<div id=\"qperiod-001\" class=\"hidden-answer\" style=\"display: none\">\n[latex]\\text{Amplitude} = \\frac{70 - 30}{2} = \\frac{40}{2} = 20[\/latex][latex]\\text{Midline} = \\frac{70 + 30}{2} = \\frac{100}{2} = 50[\/latex]<\/div>\n<\/div>\n<\/section>\n<h2>Model Periodic Behavior with Sinusoidal Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>Many real-world patterns repeat in cycles, such as tides, sound waves, daylight hours, or seasonal temperatures. These can be modeled with sine or cosine functions, called sinusoids. A sinusoidal model captures the maximum and minimum values, the average (midline), how long the cycle lasts (period), and whether the curve starts at a peak, trough, or midline.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Building a Sinusoidal Model<\/strong><\/p>\n<ol>\n<li><strong>Identify Key Features<\/strong>\n<ul>\n<li>Amplitude: [latex]\\dfrac{\\text{max} - \\text{min}}{2}[\/latex]<\/li>\n<li>Midline: [latex]\\dfrac{\\text{max} + \\text{min}}{2}[\/latex]<\/li>\n<li>Period: length of one full cycle<\/li>\n<\/ul>\n<\/li>\n<li><strong>Pick Sine or Cosine<\/strong>\n<ul>\n<li>Use cosine if the graph starts at a maximum or minimum<\/li>\n<li>Use sine if the graph starts at the midline<\/li>\n<\/ul>\n<\/li>\n<li><strong>Write the General Formula<\/strong>\n<ul>\n<li>[latex]y = a\\sin(bx-c)+d[\/latex] or [latex]y = a\\cos(bx-c)+d[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n<\/div>\n<section class=\"textbox example\">The depth of water at a dock varies between 4 feet and 12 feet. High tide occurs at noon, and the water returns to high tide every 12 hours. Write a cosine function to model the water depth [latex]h[\/latex] as a function of time [latex]t[\/latex] in hours after noon.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qmodel-001\">Show Solution<\/button><\/p>\n<div id=\"qmodel-001\" class=\"hidden-answer\" style=\"display: none\">Amplitude:[latex]a = \\frac{12 - 4}{2} = \\frac{8}{2} = 4 \\\\[\/latex]Midline:[latex]d = \\frac{12 + 4}{2} = \\frac{16}{2} = 8[\/latex]Period:[latex]P = 12[\/latex] hours, so<br \/>\n[latex]\\\\ 12 = \\frac{2\\pi}{b}[\/latex]<br \/>\n[latex]b = \\frac{2\\pi}{12} = \\frac{\\pi}{6}[\/latex]Starting point:Since the water is at high tide (maximum) when [latex]t = 0[\/latex], use cosine with no phase shift.Model:[latex]h(t) = 4\\cos\\left(\\frac{\\pi}{6}t\\right) + 8[\/latex]<\/p>\n<p>where [latex]t[\/latex] is hours after noon and [latex]h[\/latex] is depth in feet.\n<\/p><\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-bbdahbec-KOOORDz_AYw\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/KOOORDz_AYw?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-bbdahbec-KOOORDz_AYw\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14661388&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-bbdahbec-KOOORDz_AYw&#38;vembed=0&#38;video_id=KOOORDz_AYw&#38;video_target=tpm-plugin-bbdahbec-KOOORDz_AYw\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Sinusoidal+Function+Word+Problems+-+Ferris+Wheels+and+Temperature_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cSinusoidal Function Word Problems: Ferris Wheels and Temperature\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Write Both a Sine and Cosine Function to Model the Same Periodic Behavior<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<\/strong><\/p>\n<p>Any periodic behavior can be written using either a sine or a cosine function, because the two are just phase-shifted versions of each other. This means that if a cosine model starts at a peak, you can write an equivalent sine model by shifting it horizontally, and vice versa. Writing both functions for the same context helps show that the choice between sine and cosine is flexible\u2014it depends on which starting point makes the model easier to describe.<\/p>\n<p class=\"font-600 text-xl font-bold\"><strong>Quick Tips: Writing Both Sine and Cosine Models<\/strong><\/p>\n<ul>\n<li><strong>Start with Key Features<\/strong> (amplitude, midline, period)<\/li>\n<li><strong>Cosine Model:<\/strong> Use when the cycle starts at a maximum or minimum<\/li>\n<li><strong>Sine Model:<\/strong> Use when the cycle starts at the midline<\/li>\n<li><strong>Relating the Two:<\/strong> A sine curve shifted by [latex]\\dfrac{\\pi}{2b}[\/latex] equals a cosine curve<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">A Ferris wheel has a diameter of 50 feet with its center 30 feet above the ground. It completes one rotation every 60 seconds. At [latex]t = 0[\/latex], a rider is at the lowest point. Write both a sine and a cosine function to model the rider&#8217;s height [latex]h[\/latex] above the ground as a function of time [latex]t[\/latex] in seconds.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qboth-001\">Show Solution<\/button><\/p>\n<div id=\"qboth-001\" class=\"hidden-answer\" style=\"display: none\">Key features:Maximum height: 30 + 25 = 55 feet<br \/>\nMinimum height: 30 &#8211; 25 = 5 feet<br \/>\nAmplitude: [latex]a = 25[\/latex]<br \/>\nMidline: [latex]d = 30[\/latex]<br \/>\nPeriod: [latex]P = 60[\/latex], so [latex]b = \\frac{2\\pi}{60} = \\frac{\\pi}{30}[\/latex]Cosine model:<br \/>\nSince the rider starts at the minimum (not maximum), we need a reflection or phase shift:<br \/>\n[latex]h(t) = -25\\cos\\left(\\frac{\\pi}{30}t\\right) + 30[\/latex]or equivalently with a phase shift:<br \/>\n[latex]h(t) = 25\\cos\\left(\\frac{\\pi}{30}t - \\pi\\right) + 30[\/latex]Sine model:<br \/>\nThe rider starts at the minimum and rises, which matches a sine curve shifted down from the midline:<br \/>\n[latex]h(t) = 25\\sin\\left(\\frac{\\pi}{30}t - \\frac{\\pi}{2}\\right) + 30[\/latex]or using a negative amplitude:<br \/>\n[latex]h(t) = -25\\sin\\left(\\frac{\\pi}{30}t + \\frac{\\pi}{2}\\right) + 30[\/latex]Both models produce the same values for the rider&#8217;s height at any time [latex]t[\/latex].\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fdgdbddh-Vp6uOaOxUS0\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/Vp6uOaOxUS0?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fdgdbddh-Vp6uOaOxUS0\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=14660584&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fdgdbddh-Vp6uOaOxUS0&#38;vembed=0&#38;video_id=Vp6uOaOxUS0&#38;video_target=tpm-plugin-fdgdbddh-Vp6uOaOxUS0\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Precalculus\/Transcripts\/Determining+the+Equation+of+a+Sine+and+Cosine+Graph_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cDetermining the Equation of a Sine and Cosine Graph\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n","protected":false},"author":67,"menu_order":39,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Sinusoidal Function Word Problems: Ferris Wheels and 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