{"id":1191,"date":"2019-03-07T14:47:36","date_gmt":"2019-03-07T14:47:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/modeling-with-trigonometric-equations\/"},"modified":"2019-06-13T17:02:19","modified_gmt":"2019-06-13T17:02:19","slug":"modeling-with-trigonometric-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/chapter\/modeling-with-trigonometric-equations\/","title":{"raw":"3.9 Modeling with Trigonometric Equations","rendered":"3.9 Modeling with Trigonometric Equations"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Model equations and graph sinusoidal functions.<\/li>\r\n \t<li>Model periodic behavior.<\/li>\r\n \t<li>Model simple harmonic motion functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"Figure_07_06_001\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"488\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144618\/CNX_Precalc_Figure_07_06_001.jpg\" alt=\"Photo of the top part of a clock.\" width=\"488\" height=\"325\" \/> The hands on a clock are periodic: they repeat positions every twelve hours. (credit: \u201czoutedrop\u201d\/Flickr)[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1334861\">Suppose we charted the average daily temperatures in New York City over the course of one year. We would expect to find the lowest temperatures in January and February and highest in July and August. This familiar cycle repeats year after year, and if we were to extend the graph over multiple years, it would resemble a periodic function.<\/p>\r\n<p id=\"fs-id1223486\">Many other natural phenomena are also periodic. For example, the phases of the moon have a period of approximately 28 days, and birds know to fly south at about the same time each year.<\/p>\r\nSo how can we model an equation to reflect periodic behavior? First, we must collect and record data. We then find a function that resembles an observed pattern. Finally, we make the necessary alterations to the function to get a model that is dependable. In this section, we will take a deeper look at specific types of periodic behavior and model equations to fit data.\r\n<div id=\"fs-id1223497\" class=\"bc-section section\">\r\n<h3>Modeling Periodic Behavior<\/h3>\r\n<div id=\"Example_07_06_04\" class=\"textbox examples\">\r\n<div id=\"fs-id2462411\">\r\n<div id=\"fs-id1388678\">\r\n<h3>Example 1:\u00a0 Modeling an Equation and Sketching a Sinusoidal Graph to Fit Criteria<\/h3>\r\n<p id=\"fs-id1331394\">The average monthly temperatures for a small town in Oregon are given. Find a sinusoidal function of the form [latex]y=A\\mathrm{sin}\\left(B\\left(t-h\\right)\\right)+k[\/latex] that fits the data (round to the nearest tenth) and sketch the graph.<\/p>\r\n\r\n<table id=\"fs-id1336940\" style=\"height: 143px\" summary=\"Thirteen rows, two columns. The table has ordered pairs of these row values: (Month, Temperature in degrees F), (January, 42.5), (February, 44.5), (March, 48.5), (April, 52.5), (May, 58), (June, 63), (July, 68.5), (August, 69), (September, 64.5), (October, 55.5), (November, 46.5), (December, 43.5).\"><caption>Table 1<\/caption><colgroup> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr style=\"height: 11px\">\r\n<th class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">Month<\/th>\r\n<th class=\"border\" style=\"height: 11px;width: 501.656px;text-align: center\">Temperature,[latex]{}^{\\text{o}}\\text{F}[\/latex]<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">January<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">42.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">February<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">44.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">March<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">48.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">April<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">52.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">May<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">58<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">June<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">63<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">July<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">68.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">August<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">69<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">September<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">64.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">October<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">55.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">November<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">46.5<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">December<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">43.5<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1875851\">[reveal-answer q=\"fs-id1875851\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1875851\"]\r\n<p id=\"fs-id2031063\">Recall that amplitude is found using the formula<\/p>\r\n\r\n<div id=\"fs-id1161629\" class=\"unnumbered\" style=\"text-align: center\">[latex]A=\\frac{\\text{largest value }-\\text{smallest value}}{2}.[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id968691\">Thus, the amplitude is<\/p>\r\n\r\n<div id=\"fs-id1333359\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*} |A|&amp;=\\frac{69-42.5}{2}\\\\ &amp;=13.25.\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id2793443\">The data covers a period of 12 months, so [latex]\\frac{2\\pi }{B}=12[\/latex] which gives [latex]B=\\frac{2\\pi }{12}=\\frac{\\pi }{6}.[\/latex][latex]\\\\[\/latex]<\/p>\r\n<p id=\"fs-id1362205\">The vertical shift is found using the following equation.[latex]\\\\[\/latex]<\/p>\r\n\r\n<div id=\"fs-id2084680\" class=\"unnumbered\" style=\"text-align: center\">[latex]k=\\frac{\\text{highest value}+\\text{lowest value}}{2}.[\/latex]<\/div>\r\n<div>\r\n\r\n&nbsp;\r\n\r\n<\/div>\r\nThus, the vertical shift is[latex]\\\\[\/latex]\r\n<div id=\"fs-id2467697\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}k&amp;=\\frac{69+42.5}{2}\\\\ &amp;=55.8.\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1128874\">So far, we have the equation [latex]y=13.3\\mathrm{sin}\\left(\\frac{\\pi}{6}\\left(x-h\\right)\\right)+55.8.[\/latex][latex]\\\\[\/latex]<\/p>\r\n<p id=\"fs-id2160567\">To find the horizontal shift, we can input the [latex]x[\/latex] and [latex]y[\/latex] values for the first month, which will be [latex]x=1[\/latex] and [latex]y=42.5[\/latex].\u00a0 We can then solve for [latex]h[\/latex] as shown below.[latex]\\\\[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1295023\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}42.5&amp;=13.3\\mathrm{sin}\\left(\\frac{\\pi}{6}\\left(1-h\\right)\\right)+55.8\\\\ -13.3&amp;=13.3\\mathrm{sin}\\left(\\frac{\\pi }{6}\\left(1-h\\right)\\right)&amp;&amp;\\text{Subtracted 55.8 from both sides. }\\\\-1&amp;=\\mathrm{sin}\\left(\\frac{\\pi} {6}\\left(1-h\\right)\\right) &amp;&amp;\\text{Now divide both sides by 13.3.}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<div class=\"unnumbered\" style=\"text-align: left\">We can use the following idea: [latex]\\mathrm{sin}\\left(\\theta\\right) =-1\\to \\theta =-\\frac{\\pi }{2}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<div><\/div>\r\n<div class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}\\frac{\\pi} {6}\\left(1-h\\right)&amp;=-\\frac{\\pi }{2}&amp;&amp;\\text{Set the input expression of sine equal to }-\\frac{\\pi}{2}. \\\\ \\frac{\\pi}{6}-\\frac{\\pi}{6}h&amp;=-\\frac{\\pi}{2}&amp;&amp;\\text{Distribute on the left hand side.}\\\\-\\frac{\\pi}{6}h&amp;=-\\frac{\\pi }{2}-\\frac{\\pi}{6}&amp;&amp;\\text{Subtract }\\frac{\\pi}{6}\\text{ from both sides.}\\\\h&amp;=\\left(-\\frac{\\pi }{2}-\\frac{\\pi}{6}\\right)\\frac{-6}{\\pi}&amp;&amp;\\text{Multiply both sides by }-\\frac{6}{\\pi}.\\\\h&amp;=3+1=4 \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1782945\">We have the equation [latex]y=13.3\\mathrm{sin}\\left(\\frac{\\pi }{6}\\left(x-4\\right)\\right)+55.8.[\/latex] See the graph in <a class=\"autogenerated-content\" href=\"#Figure_07_06_011\">Figure 1<\/a>.<\/p>\r\n&nbsp;\r\n<div id=\"Figure_07_06_011\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"441\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144643\/CNX_Precalc_Figure_07_06_011.jpg\" alt=\"Graph of the equation y=13.3sin(pi\/6 x - 2pi\/3) + 55.8. The average value is a dotted horizontal line y=55.8, and the amplitude is 13.3\" width=\"441\" height=\"300\" \/> Figure 1[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_07_06_06\" class=\"textbox examples\">\r\n<div id=\"fs-id1345601\">\r\n<div id=\"fs-id2723419\">\r\n<h3>Example 2:\u00a0 Describing Periodic Motion<\/h3>\r\n<p id=\"fs-id2773835\">The hour hand of the large clock on the wall in Union Station measures 24 inches in length. At noon, the tip of the hour hand is 30 inches from the ceiling. At 3 PM, the tip is 54 inches from the ceiling, and at 6 PM, 78 inches. At 9 PM, it is again 54 inches from the ceiling, and at midnight, the tip of the hour hand returns to its original position 30 inches from the ceiling. Let [latex]y[\/latex] equal the distance from the tip of the hour hand to the ceiling [latex]x[\/latex] hours after noon. Find the equation that models the motion of the clock and sketch the graph.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1328303\">[reveal-answer q=\"fs-id1328303\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1328303\"]Begin by making a table of values as shown in <a class=\"autogenerated-content\" href=\"#fs-id1992294\">Table<\/a>.\r\n<table id=\"fs-id1992294\" style=\"height: 66px\" summary=\"Six rows, three columns. The table has ordered pairs of these row values: (X, y, Points to plot), (Noon, 30 in, (0,30)), (3 PM, 54 in, (3,54)), (6 PM, 78 in, (6,8)), (9 PM, 54 in, (9,54)), (Midnight, 30 in, (12,30)).\"><caption>Table 2<\/caption><colgroup> <col \/> <col \/> <col \/><\/colgroup>\r\n<thead>\r\n<tr style=\"height: 11px\">\r\n<th class=\"border\" style=\"height: 11px;width: 79.6563px;text-align: center\">[latex]x[\/latex]<\/th>\r\n<th class=\"border\" style=\"height: 11px;width: 79.6563px;text-align: center\">[latex]y[\/latex]<\/th>\r\n<th class=\"border\" style=\"height: 11px;width: 151.656px;text-align: center\">Points to plot<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">Noon<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">30 in<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 151.656px\">[latex]\\left(0,30\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">3 PM<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">54 in<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 151.656px\">[latex]\\left(3,54\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">6 PM<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">78 in<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 151.656px\">[latex]\\left(6,78\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">9 PM<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">54 in<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 151.656px\">[latex]\\left(9,54\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 11px\">\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">Midnight<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">30 in<\/td>\r\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 151.656px\">[latex]\\left(12,30\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p id=\"fs-id697250\">To model an equation, we first need to find the amplitude.<\/p>\r\n\r\n<div id=\"fs-id1247000\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}|A|&amp;=|\\frac{78-30}{2}|\\\\&amp;=24\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1847251\">The clock\u2019s cycle repeats every 12 hours. Thus,[latex]\\\\[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1971097\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}B&amp;=\\frac{2\\pi}{12}\\\\ &amp;=\\frac{\\pi}{6}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id2131557\">The vertical shift is<\/p>\r\n\r\n<div id=\"fs-id1612211\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}k&amp;=\\frac{78+30}{2} \\\\ &amp;=54 \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id2102415\">Since the function begins with the minimum value of [latex]y[\/latex] when [latex]x=0[\/latex] (as opposed to the maximum value), we will use the cosine function with the negative value for [latex]A.[\/latex] There is no horizontal shift, so [latex]h=0.[\/latex] In the form [latex]y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k,[\/latex] the equation is[latex]\\\\[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1340988\" class=\"unnumbered\" style=\"text-align: center\">[latex]y=-24\\text{ }\\mathrm{cos}\\left(\\frac{\\pi }{6}x\\right)+54[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1556363\">See <a class=\"autogenerated-content\" href=\"#Figure_07_06_013\">Figure 2<\/a>.<\/p>\r\n\r\n<div id=\"Figure_07_06_013\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"373\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144646\/CNX_Precalc_Figure_07_06_013-1.jpg\" alt=\"Graph of the function y=-24cos(pi\/6 x)+54 using the five key points: (0,30), (3,54), (6,78), (9,54), (12,30).\" width=\"373\" height=\"300\" \/> <strong>Figure 2<\/strong>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_07_06_07\" class=\"textbox examples\">\r\n<div id=\"fs-id2209186\">\r\n<div id=\"fs-id1877849\">\r\n<h3>Example 3:\u00a0 Determining a Model for Tides<\/h3>\r\n<p id=\"fs-id1099909\">The height of the tide in a small beach town is measured along a seawall. Water levels oscillate between 7 feet at low tide and 15 feet at high tide. On a particular day, low tide occurred at 6 AM and high tide occurred at noon. Approximately every 12 hours, the cycle repeats. Find an equation to model the water levels.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id1367707\">[reveal-answer q=\"fs-id1367707\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1367707\"]\r\n<p id=\"fs-id1961795\">As the water level varies from 7 ft to 15 ft, we can calculate the amplitude as[latex]\\\\[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1151331\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}|A|&amp;=|\\frac{\\left(15-7\\right)}{2}|\\\\ &amp;=4.\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1981423\">The cycle repeats every 12 hours; therefore, [latex]B[\/latex] is[latex]\\\\[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1688189\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\frac{2\\pi}{12}=\\frac{\\pi}{6}.[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1853601\">There is a vertical translation of [latex]\\frac{\\left(15+7\\right)}{2}=11.[\/latex] Since the value of the function is at a maximum at [latex]t=0,[\/latex] we will use the cosine function, with the positive value for [latex]A.[\/latex][latex]\\\\[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1692290\" class=\"unnumbered\" style=\"text-align: center\">[latex]y=4\\mathrm{cos}\\left(\\frac{\\pi}{6}t\\right)+11[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1698026\">See <a class=\"autogenerated-content\" href=\"#Figure_07_06_015\">Figure 3<\/a>.<\/p>\r\n\r\n<div id=\"Figure_07_06_015\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"443\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144649\/CNX_Precalc_Figure_07_06_015.jpg\" alt=\"Graph of the function y=4cos(pi\/6 t) + 11 from 0 to 12. The midline is y=11, three key points are (0,15), (6,7), and (12, 15).\" width=\"443\" height=\"300\" \/> <strong>Figure 3<\/strong>[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1248789\" class=\"precalculus tryit\">\r\n<h3>Try it #1<\/h3>\r\n<div id=\"ti_07_06_03\">\r\n<div id=\"fs-id1522753\">\r\n<p id=\"fs-id1522754\">The daily temperature in the month of March in a certain city varies from a low of [latex]24\\text{\u00b0F}[\/latex] to a high of [latex]40\\text{\u00b0F}\\text{.}[\/latex] Find a sinusoidal function to model daily temperature and sketch the graph. Approximate the time when the temperature reaches the freezing point [latex]32\\text{\u00b0F}\\text{.}[\/latex] Let [latex]t=0[\/latex] correspond to noon.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-id2020742\">[reveal-answer q=\"fs-id2020742\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id2020742\"]\r\n<p id=\"fs-id1893405\">[latex]y=8\\mathrm{sin}\\left(\\frac{\\pi }{12}t\\right)+32[\/latex]<\/p>\r\nThe temperature reaches freezing at noon and at midnight.<span id=\"fs-id2322911\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144652\/CNX_Precalc_Figure_07_06_016.jpg\" alt=\"Graph of the function y=8sin(pi\/12 t) + 32 for temperature. The midline is at 32. The times when the temperature is at 32 are midnight and noon.\" width=\"355\" height=\"300\" \/><\/span>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"Example_07_06_08\" class=\"textbox examples\">\r\n<div id=\"fs-id2052518\">\r\n<div id=\"fs-id2251185\">\r\n<h3>Example 4:\u00a0 Interpreting the Periodic Behavior Equation<\/h3>\r\n<p id=\"fs-id1956234\">The average person\u2019s blood pressure is modeled by the function [latex]f\\left(t\\right)=20\\mathrm{sin}\\left(160\\pi t\\right)+100,[\/latex] where [latex]f\\left(t\\right)[\/latex] represents the blood pressure at time [latex]t,[\/latex] measured in minutes.\u00a0 Sketch the graph and find the blood pressure reading.<\/p>\r\n\r\n<div id=\"fs-id1113490\">[reveal-answer q=\"fs-id1113490\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1113490\"]\r\n<p id=\"fs-id2339667\">The period is given by<\/p>\r\n\r\n<div id=\"fs-id1722880\" class=\"unnumbered\">\r\n<p style=\"text-align: center\">[latex]\\begin{align*}\\frac{2\\pi}{B }&amp;=\\frac{2\\pi }{160\\pi } \\\\&amp;=\\frac{1}{80}.\\end{align*}[\/latex][latex]\\\\[\/latex]<\/p>\r\n\r\n<div class=\"unnumbered\">\r\n\r\nSince the period is [latex]\\frac{1}{80},[\/latex] we know it takes [latex]\\frac{1}{80}^{th}[\/latex] of a minute for the blood pressure to cycle through a full range of values.\r\n\r\nSee the graph in <a class=\"autogenerated-content\" href=\"#Figure_07_06_017\">\u00a0Figure 4<\/a>.\r\n\r\n[caption id=\"attachment_2879\" align=\"aligncenter\" width=\"455\"]<img class=\"wp-image-2879 \" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/13165924\/39Ex4.png\" alt=\"\" width=\"455\" height=\"290\" \/> Figure 4[\/caption]\r\n\r\n<div id=\"Figure_07_06_017\" class=\"small\">\r\n<div id=\"fs-id1936106\">\r\n<h3>Analysis<\/h3>\r\n<p id=\"fs-id1535704\">Blood pressure of [latex]\\frac{120}{80}[\/latex] is considered to be normal. The top number is the maximum or systolic reading, which measures the pressure in the arteries when the heart contracts. The bottom number is the minimum or diastolic reading, which measures the pressure in the arteries as the heart relaxes between beats, refilling with blood. Thus, normal blood pressure can be modeled by a periodic function with a maximum of 120 and a minimum of 80.\u00a0 Since the period is\u00a0[latex]\\frac{1}{80}^{th}[\/latex] of a minute, we know there are 80 heartbeats in a minute.<\/p>\r\n<span style=\"font-size: 0.8em\">[\/hidden-answer]<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id1146451\" class=\"bc-section section\">\r\n<h3>Modeling Harmonic Motion Functions<\/h3>\r\n<p id=\"fs-id1782792\">Harmonic motion is a form of periodic motion, but there are factors to consider that differentiate the two types. While general <span class=\"no-emphasis\">periodic motion<\/span> applications cycle through their periods with no outside interference, <span class=\"no-emphasis\">harmonic motion<\/span> requires a restoring force. Examples of harmonic motion include springs, gravitational force, and magnetic force.<\/p>\r\n\r\n<div id=\"fs-id1227506\" class=\"bc-section section\">\r\n<h4>Simple Harmonic Motion<\/h4>\r\n<p id=\"fs-id3038708\">A type of motion described as <span class=\"no-emphasis\">simple harmonic motion<\/span> involves a restoring force but assumes that the motion will continue forever. Imagine a weighted object hanging on a spring, When that object is not disturbed, we say that the object is at rest, or in equilibrium. If the object is pulled down and then released, the force of the spring pulls the object back toward equilibrium and harmonic motion begins. The restoring force is directly proportional to the displacement of the object from its equilibrium point. When [latex]t=0,d=0.[\/latex]<\/p>\r\n\r\n<div id=\"fs-id1167808\">\r\n<h3>Simple Harmonic Motion<\/h3>\r\n<p id=\"fs-id2248340\">We see that simple harmonic motion equations are given in terms of displacement:<\/p>\r\n\r\n<div id=\"Equation_07_06_02\" style=\"text-align: center\">[latex]d=A\\mathrm{cos}\\left(Bt\\right)\\text{ or }d=A\\mathrm{sin}\\left(Bt\\right)[\/latex][latex]\\\\[\/latex]<\/div>\r\n<p id=\"fs-id1480838\" style=\"text-align: center\">where [latex]|A|[\/latex] is the amplitude, and [latex]\\frac{2\\pi }{B }[\/latex] is the period.<\/p>\r\n\r\n<\/div>\r\n<div id=\"Example_07_06_09\" class=\"textbox examples\">\r\n<div id=\"fs-id2867398\">\r\n<div id=\"fs-id2867400\">\r\n<h3>Example 5:\u00a0 Finding the Displacement, Period, and Frequency, and Graphing a Function<\/h3>\r\n<p id=\"fs-id2614724\">For the given functions,<\/p>\r\n\r\n<ol id=\"fs-id2614727\" type=\"1\">\r\n \t<li>Find the maximum displacement of an object.<\/li>\r\n \t<li>Find the period or the time required for one vibration.<\/li>\r\n \t<li>Sketch the graph.\r\n<ol id=\"fs-id1611913\" type=\"a\">\r\n \t<li>[latex]y=5\\mathrm{sin}\\left(3t\\right)[\/latex]<\/li>\r\n \t<li>[latex]y=6\\mathrm{cos}\\left(\\pi t\\right)[\/latex]<\/li>\r\n \t<li>[latex]y=5\\mathrm{cos}\\left(\\frac{\\pi }{2}t\\right)[\/latex]<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"fs-id1805776\">[reveal-answer q=\"fs-id1805776\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1805776\"]\r\n<ol id=\"fs-id1378735\" type=\"a\">\r\n \t<li>[latex]y=5\\mathrm{sin}\\left(3t\\right)[\/latex]\r\n<ol id=\"fs-id1961806\" type=\"1\">\r\n \t<li>The maximum displacement is equal to the amplitude, [latex]|A|,[\/latex] which is 5.<\/li>\r\n \t<li>The period is [latex]\\frac{2\\pi }{B}=\\frac{2\\pi }{3}.[\/latex]<\/li>\r\n \t<li>See <a class=\"autogenerated-content\" href=\"#Figure_07_07_006\">\u00a0Figure 5<\/a>. The graph indicates the five key points. \u00a0Remember, you take [latex]\\frac{1}{4}[\/latex] of the period and starting at 0, add this value repeatedly 4 times. \u00a0For this problem, you would use an interval of [latex]\\frac{1}{4}\\times\\frac{2\\pi}{3}=\\frac{\\pi}{6}.[\/latex]\r\n<div id=\"Figure_07_07_006\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"310\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144658\/CNX_Precalc_Figure_07_07_006.jpg\" alt=\"Graph of the function y=5sin(3t) from 0 to 2pi\/3. The five key points are (0,0), (pi\/6, 5), (pi\/3, 0), (pi\/2, -5), (2pi\/3, 0).\" width=\"310\" height=\"282\" \/> Figure 5[\/caption]\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>[latex]y=6\\mathrm{cos}\\left(\\pi t\\right)[\/latex]\r\n<ol id=\"fs-id1240167\" type=\"1\">\r\n \t<li>The maximum displacement is [latex]6.[\/latex]<\/li>\r\n \t<li>The period is [latex]\\frac{2\\pi }{B}=\\frac{2\\pi }{\\pi }=2.[\/latex]<\/li>\r\n \t<li>See <a class=\"autogenerated-content\" href=\"#Figure_07_07_007\">\u00a0Figure 6<\/a>.\r\n<div id=\"Figure_07_07_007\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"354\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144700\/CNX_Precalc_Figure_07_07_007.jpg\" alt=\"Graph of the function y=6cos(pi t) from 0 to 3.\" width=\"354\" height=\"211\" \/> Figure 6[\/caption]\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>[latex]y=5\\mathrm{cos}\\left(\\frac{\\pi }{2}t\\right)[\/latex]\r\n<ol id=\"fs-id1793844\" type=\"1\">\r\n \t<li>The maximum displacement is [latex]5.[\/latex]<\/li>\r\n \t<li>The period is [latex]\\frac{2\\pi }{B}=\\frac{2\\pi }{\\frac{\\pi }{2}}=4.[\/latex]<\/li>\r\n \t<li>See <a class=\"autogenerated-content\" href=\"#Figure_07_07_008\">\u00a0Figure 7<\/a>.\r\n<div id=\"Figure_07_07_008\" class=\"small\">\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"287\"]<img class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144704\/CNX_Precalc_Figure_07_07_008.jpg\" alt=\"Graph of the function y=5cos(pi\/2 t) from 0 to 4.\" width=\"287\" height=\"306\" \/> Figure 7[\/caption]\r\n\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id2957014\" class=\"bc-section section\">\r\n<div id=\"fs-id1403864\" class=\"precalculus media\">\r\n<p id=\"fs-id2838123\">Access these online resources for additional instruction and practice with trigonometric applications.<\/p>\r\n\r\n<ul id=\"fs-id2838126\">\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/solvetrigprob\">Solving Problems Using Trigonometry<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/ferriswheel\">Ferris Wheel Trigonometry<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/dailytemp\">Daily Temperatures and Trigonometry<\/a><\/li>\r\n \t<li><a href=\"http:\/\/openstax.org\/l\/simpleharm\">Simple Harmonic Motion<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-id3483144\" class=\"key-equations\">\r\n<h3>Key Equations<\/h3>\r\n<table id=\"fs-id2608518\" summary=\"..\"><caption>\u00a0<\/caption>\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">Standard form of sinusoidal equation<\/td>\r\n<td class=\"border\">[latex]y=A\\mathrm{sin}\\left(B\\left(t-h\\right)\\right)+k\\text{ or }y=A\\mathrm{cos}\\left(B\\left(t-h\\right)\\right)+k[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Simple harmonic motion<\/td>\r\n<td class=\"border\">[latex]d=a\\mathrm{cos}\\left(B t\\right)\\text{ or }d=a\\mathrm{sin}\\left(B t\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\"><\/td>\r\n<td class=\"border\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<div id=\"fs-id1157947\" class=\"textbox key-takeaways\">\r\n<h3>Key Concepts<\/h3>\r\n<ul id=\"fs-id1793027\">\r\n \t<li>Sinusoidal functions are represented by the sine and cosine graphs. In standard form, we can find the amplitude, period, and horizontal and vertical shifts.<\/li>\r\n \t<li>Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values.<\/li>\r\n \t<li>Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year.<\/li>\r\n \t<li>Harmonic motion functions are modeled from given data. Similar to periodic motion applications, harmonic motion requires a restoring force. Examples include gravitational force and spring motion activated by weight.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-id1817053\" class=\"review-exercises\"><\/div>\r\n<div class=\"textbox shaded\">\r\n<h3>Glossary<\/h3>\r\n<dl id=\"fs-id2617951\">\r\n \t<dt>simple harmonic motion<\/dt>\r\n \t<dd id=\"fs-id2617956\">a repetitive motion that can be modeled by periodic sinusoidal oscillation<\/dd>\r\n<\/dl>\r\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>Model equations and graph sinusoidal functions.<\/li>\n<li>Model periodic behavior.<\/li>\n<li>Model simple harmonic motion functions.<\/li>\n<\/ul>\n<\/div>\n<div id=\"Figure_07_06_001\" class=\"small\">\n<div style=\"width: 498px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144618\/CNX_Precalc_Figure_07_06_001.jpg\" alt=\"Photo of the top part of a clock.\" width=\"488\" height=\"325\" \/><\/p>\n<p class=\"wp-caption-text\">The hands on a clock are periodic: they repeat positions every twelve hours. (credit: \u201czoutedrop\u201d\/Flickr)<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1334861\">Suppose we charted the average daily temperatures in New York City over the course of one year. We would expect to find the lowest temperatures in January and February and highest in July and August. This familiar cycle repeats year after year, and if we were to extend the graph over multiple years, it would resemble a periodic function.<\/p>\n<p id=\"fs-id1223486\">Many other natural phenomena are also periodic. For example, the phases of the moon have a period of approximately 28 days, and birds know to fly south at about the same time each year.<\/p>\n<p>So how can we model an equation to reflect periodic behavior? First, we must collect and record data. We then find a function that resembles an observed pattern. Finally, we make the necessary alterations to the function to get a model that is dependable. In this section, we will take a deeper look at specific types of periodic behavior and model equations to fit data.<\/p>\n<div id=\"fs-id1223497\" class=\"bc-section section\">\n<h3>Modeling Periodic Behavior<\/h3>\n<div id=\"Example_07_06_04\" class=\"textbox examples\">\n<div id=\"fs-id2462411\">\n<div id=\"fs-id1388678\">\n<h3>Example 1:\u00a0 Modeling an Equation and Sketching a Sinusoidal Graph to Fit Criteria<\/h3>\n<p id=\"fs-id1331394\">The average monthly temperatures for a small town in Oregon are given. Find a sinusoidal function of the form [latex]y=A\\mathrm{sin}\\left(B\\left(t-h\\right)\\right)+k[\/latex] that fits the data (round to the nearest tenth) and sketch the graph.<\/p>\n<table id=\"fs-id1336940\" style=\"height: 143px\" summary=\"Thirteen rows, two columns. The table has ordered pairs of these row values: (Month, Temperature in degrees F), (January, 42.5), (February, 44.5), (March, 48.5), (April, 52.5), (May, 58), (June, 63), (July, 68.5), (August, 69), (September, 64.5), (October, 55.5), (November, 46.5), (December, 43.5).\">\n<caption>Table 1<\/caption>\n<colgroup>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr style=\"height: 11px\">\n<th class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">Month<\/th>\n<th class=\"border\" style=\"height: 11px;width: 501.656px;text-align: center\">Temperature,[latex]{}^{\\text{o}}\\text{F}[\/latex]<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">January<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">42.5<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">February<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">44.5<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">March<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">48.5<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">April<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">52.5<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">May<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">58<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">June<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">63<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">July<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">68.5<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">August<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">69<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">September<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">64.5<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">October<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">55.5<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">November<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">46.5<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"height: 11px;width: 122.656px;text-align: center\">December<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 501.656px\">43.5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1875851\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1875851\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1875851\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id2031063\">Recall that amplitude is found using the formula<\/p>\n<div id=\"fs-id1161629\" class=\"unnumbered\" style=\"text-align: center\">[latex]A=\\frac{\\text{largest value }-\\text{smallest value}}{2}.[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id968691\">Thus, the amplitude is<\/p>\n<div id=\"fs-id1333359\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*} |A|&=\\frac{69-42.5}{2}\\\\ &=13.25.\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id2793443\">The data covers a period of 12 months, so [latex]\\frac{2\\pi }{B}=12[\/latex] which gives [latex]B=\\frac{2\\pi }{12}=\\frac{\\pi }{6}.[\/latex][latex]\\\\[\/latex]<\/p>\n<p id=\"fs-id1362205\">The vertical shift is found using the following equation.[latex]\\\\[\/latex]<\/p>\n<div id=\"fs-id2084680\" class=\"unnumbered\" style=\"text-align: center\">[latex]k=\\frac{\\text{highest value}+\\text{lowest value}}{2}.[\/latex]<\/div>\n<div>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Thus, the vertical shift is[latex]\\\\[\/latex]<\/p>\n<div id=\"fs-id2467697\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}k&=\\frac{69+42.5}{2}\\\\ &=55.8.\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1128874\">So far, we have the equation [latex]y=13.3\\mathrm{sin}\\left(\\frac{\\pi}{6}\\left(x-h\\right)\\right)+55.8.[\/latex][latex]\\\\[\/latex]<\/p>\n<p id=\"fs-id2160567\">To find the horizontal shift, we can input the [latex]x[\/latex] and [latex]y[\/latex] values for the first month, which will be [latex]x=1[\/latex] and [latex]y=42.5[\/latex].\u00a0 We can then solve for [latex]h[\/latex] as shown below.[latex]\\\\[\/latex]<\/p>\n<div id=\"fs-id1295023\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}42.5&=13.3\\mathrm{sin}\\left(\\frac{\\pi}{6}\\left(1-h\\right)\\right)+55.8\\\\ -13.3&=13.3\\mathrm{sin}\\left(\\frac{\\pi }{6}\\left(1-h\\right)\\right)&&\\text{Subtracted 55.8 from both sides. }\\\\-1&=\\mathrm{sin}\\left(\\frac{\\pi} {6}\\left(1-h\\right)\\right) &&\\text{Now divide both sides by 13.3.}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<div class=\"unnumbered\" style=\"text-align: left\">We can use the following idea: [latex]\\mathrm{sin}\\left(\\theta\\right) =-1\\to \\theta =-\\frac{\\pi }{2}[\/latex][latex]\\\\[\/latex]<\/div>\n<div><\/div>\n<div class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}\\frac{\\pi} {6}\\left(1-h\\right)&=-\\frac{\\pi }{2}&&\\text{Set the input expression of sine equal to }-\\frac{\\pi}{2}. \\\\ \\frac{\\pi}{6}-\\frac{\\pi}{6}h&=-\\frac{\\pi}{2}&&\\text{Distribute on the left hand side.}\\\\-\\frac{\\pi}{6}h&=-\\frac{\\pi }{2}-\\frac{\\pi}{6}&&\\text{Subtract }\\frac{\\pi}{6}\\text{ from both sides.}\\\\h&=\\left(-\\frac{\\pi }{2}-\\frac{\\pi}{6}\\right)\\frac{-6}{\\pi}&&\\text{Multiply both sides by }-\\frac{6}{\\pi}.\\\\h&=3+1=4 \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1782945\">We have the equation [latex]y=13.3\\mathrm{sin}\\left(\\frac{\\pi }{6}\\left(x-4\\right)\\right)+55.8.[\/latex] See the graph in <a class=\"autogenerated-content\" href=\"#Figure_07_06_011\">Figure 1<\/a>.<\/p>\n<p>&nbsp;<\/p>\n<div id=\"Figure_07_06_011\" class=\"small\">\n<div style=\"width: 451px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144643\/CNX_Precalc_Figure_07_06_011.jpg\" alt=\"Graph of the equation y=13.3sin(pi\/6 x - 2pi\/3) + 55.8. The average value is a dotted horizontal line y=55.8, and the amplitude is 13.3\" width=\"441\" height=\"300\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 1<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_07_06_06\" class=\"textbox examples\">\n<div id=\"fs-id1345601\">\n<div id=\"fs-id2723419\">\n<h3>Example 2:\u00a0 Describing Periodic Motion<\/h3>\n<p id=\"fs-id2773835\">The hour hand of the large clock on the wall in Union Station measures 24 inches in length. At noon, the tip of the hour hand is 30 inches from the ceiling. At 3 PM, the tip is 54 inches from the ceiling, and at 6 PM, 78 inches. At 9 PM, it is again 54 inches from the ceiling, and at midnight, the tip of the hour hand returns to its original position 30 inches from the ceiling. Let [latex]y[\/latex] equal the distance from the tip of the hour hand to the ceiling [latex]x[\/latex] hours after noon. Find the equation that models the motion of the clock and sketch the graph.<\/p>\n<\/div>\n<div id=\"fs-id1328303\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1328303\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1328303\" class=\"hidden-answer\" style=\"display: none\">Begin by making a table of values as shown in <a class=\"autogenerated-content\" href=\"#fs-id1992294\">Table<\/a>.<\/p>\n<table id=\"fs-id1992294\" style=\"height: 66px\" summary=\"Six rows, three columns. The table has ordered pairs of these row values: (X, y, Points to plot), (Noon, 30 in, (0,30)), (3 PM, 54 in, (3,54)), (6 PM, 78 in, (6,8)), (9 PM, 54 in, (9,54)), (Midnight, 30 in, (12,30)).\">\n<caption>Table 2<\/caption>\n<colgroup>\n<col \/>\n<col \/>\n<col \/><\/colgroup>\n<thead>\n<tr style=\"height: 11px\">\n<th class=\"border\" style=\"height: 11px;width: 79.6563px;text-align: center\">[latex]x[\/latex]<\/th>\n<th class=\"border\" style=\"height: 11px;width: 79.6563px;text-align: center\">[latex]y[\/latex]<\/th>\n<th class=\"border\" style=\"height: 11px;width: 151.656px;text-align: center\">Points to plot<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">Noon<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">30 in<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 151.656px\">[latex]\\left(0,30\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">3 PM<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">54 in<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 151.656px\">[latex]\\left(3,54\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">6 PM<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">78 in<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 151.656px\">[latex]\\left(6,78\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">9 PM<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">54 in<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 151.656px\">[latex]\\left(9,54\\right)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 11px\">\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">Midnight<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 79.6563px\">30 in<\/td>\n<td class=\"border\" style=\"text-align: center;height: 11px;width: 151.656px\">[latex]\\left(12,30\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id697250\">To model an equation, we first need to find the amplitude.<\/p>\n<div id=\"fs-id1247000\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}|A|&=|\\frac{78-30}{2}|\\\\&=24\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1847251\">The clock\u2019s cycle repeats every 12 hours. Thus,[latex]\\\\[\/latex]<\/p>\n<div id=\"fs-id1971097\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}B&=\\frac{2\\pi}{12}\\\\ &=\\frac{\\pi}{6}\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id2131557\">The vertical shift is<\/p>\n<div id=\"fs-id1612211\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}k&=\\frac{78+30}{2} \\\\ &=54 \\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id2102415\">Since the function begins with the minimum value of [latex]y[\/latex] when [latex]x=0[\/latex] (as opposed to the maximum value), we will use the cosine function with the negative value for [latex]A.[\/latex] There is no horizontal shift, so [latex]h=0.[\/latex] In the form [latex]y=A\\mathrm{cos}\\left(B\\left(x-h\\right)\\right)+k,[\/latex] the equation is[latex]\\\\[\/latex]<\/p>\n<div id=\"fs-id1340988\" class=\"unnumbered\" style=\"text-align: center\">[latex]y=-24\\text{ }\\mathrm{cos}\\left(\\frac{\\pi }{6}x\\right)+54[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1556363\">See <a class=\"autogenerated-content\" href=\"#Figure_07_06_013\">Figure 2<\/a>.<\/p>\n<div id=\"Figure_07_06_013\" class=\"small\">\n<div style=\"width: 383px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144646\/CNX_Precalc_Figure_07_06_013-1.jpg\" alt=\"Graph of the function y=-24cos(pi\/6 x)+54 using the five key points: (0,30), (3,54), (6,78), (9,54), (12,30).\" width=\"373\" height=\"300\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 2<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_07_06_07\" class=\"textbox examples\">\n<div id=\"fs-id2209186\">\n<div id=\"fs-id1877849\">\n<h3>Example 3:\u00a0 Determining a Model for Tides<\/h3>\n<p id=\"fs-id1099909\">The height of the tide in a small beach town is measured along a seawall. Water levels oscillate between 7 feet at low tide and 15 feet at high tide. On a particular day, low tide occurred at 6 AM and high tide occurred at noon. Approximately every 12 hours, the cycle repeats. Find an equation to model the water levels.<\/p>\n<\/div>\n<div id=\"fs-id1367707\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1367707\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1367707\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1961795\">As the water level varies from 7 ft to 15 ft, we can calculate the amplitude as[latex]\\\\[\/latex]<\/p>\n<div id=\"fs-id1151331\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\begin{align*}|A|&=|\\frac{\\left(15-7\\right)}{2}|\\\\ &=4.\\end{align*}[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1981423\">The cycle repeats every 12 hours; therefore, [latex]B[\/latex] is[latex]\\\\[\/latex]<\/p>\n<div id=\"fs-id1688189\" class=\"unnumbered\" style=\"text-align: center\">[latex]\\frac{2\\pi}{12}=\\frac{\\pi}{6}.[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1853601\">There is a vertical translation of [latex]\\frac{\\left(15+7\\right)}{2}=11.[\/latex] Since the value of the function is at a maximum at [latex]t=0,[\/latex] we will use the cosine function, with the positive value for [latex]A.[\/latex][latex]\\\\[\/latex]<\/p>\n<div id=\"fs-id1692290\" class=\"unnumbered\" style=\"text-align: center\">[latex]y=4\\mathrm{cos}\\left(\\frac{\\pi}{6}t\\right)+11[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1698026\">See <a class=\"autogenerated-content\" href=\"#Figure_07_06_015\">Figure 3<\/a>.<\/p>\n<div id=\"Figure_07_06_015\" class=\"small\">\n<div style=\"width: 453px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144649\/CNX_Precalc_Figure_07_06_015.jpg\" alt=\"Graph of the function y=4cos(pi\/6 t) + 11 from 0 to 12. The midline is y=11, three key points are (0,15), (6,7), and (12, 15).\" width=\"443\" height=\"300\" \/><\/p>\n<p class=\"wp-caption-text\"><strong>Figure 3<\/strong><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1248789\" class=\"precalculus tryit\">\n<h3>Try it #1<\/h3>\n<div id=\"ti_07_06_03\">\n<div id=\"fs-id1522753\">\n<p id=\"fs-id1522754\">The daily temperature in the month of March in a certain city varies from a low of [latex]24\\text{\u00b0F}[\/latex] to a high of [latex]40\\text{\u00b0F}\\text{.}[\/latex] Find a sinusoidal function to model daily temperature and sketch the graph. Approximate the time when the temperature reaches the freezing point [latex]32\\text{\u00b0F}\\text{.}[\/latex] Let [latex]t=0[\/latex] correspond to noon.<\/p>\n<\/div>\n<div id=\"fs-id2020742\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id2020742\">Show Solution<\/span><\/p>\n<div id=\"qfs-id2020742\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1893405\">[latex]y=8\\mathrm{sin}\\left(\\frac{\\pi }{12}t\\right)+32[\/latex]<\/p>\n<p>The temperature reaches freezing at noon and at midnight.<span id=\"fs-id2322911\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144652\/CNX_Precalc_Figure_07_06_016.jpg\" alt=\"Graph of the function y=8sin(pi\/12 t) + 32 for temperature. The midline is at 32. The times when the temperature is at 32 are midnight and noon.\" width=\"355\" height=\"300\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"Example_07_06_08\" class=\"textbox examples\">\n<div id=\"fs-id2052518\">\n<div id=\"fs-id2251185\">\n<h3>Example 4:\u00a0 Interpreting the Periodic Behavior Equation<\/h3>\n<p id=\"fs-id1956234\">The average person\u2019s blood pressure is modeled by the function [latex]f\\left(t\\right)=20\\mathrm{sin}\\left(160\\pi t\\right)+100,[\/latex] where [latex]f\\left(t\\right)[\/latex] represents the blood pressure at time [latex]t,[\/latex] measured in minutes.\u00a0 Sketch the graph and find the blood pressure reading.<\/p>\n<div id=\"fs-id1113490\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1113490\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1113490\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id2339667\">The period is given by<\/p>\n<div id=\"fs-id1722880\" class=\"unnumbered\">\n<p style=\"text-align: center\">[latex]\\begin{align*}\\frac{2\\pi}{B }&=\\frac{2\\pi }{160\\pi } \\\\&=\\frac{1}{80}.\\end{align*}[\/latex][latex]\\\\[\/latex]<\/p>\n<div class=\"unnumbered\">\n<p>Since the period is [latex]\\frac{1}{80},[\/latex] we know it takes [latex]\\frac{1}{80}^{th}[\/latex] of a minute for the blood pressure to cycle through a full range of values.<\/p>\n<p>See the graph in <a class=\"autogenerated-content\" href=\"#Figure_07_06_017\">\u00a0Figure 4<\/a>.<\/p>\n<div id=\"attachment_2879\" style=\"width: 465px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-2879\" class=\"wp-image-2879\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/13165924\/39Ex4.png\" alt=\"\" width=\"455\" height=\"290\" \/><\/p>\n<p id=\"caption-attachment-2879\" class=\"wp-caption-text\">Figure 4<\/p>\n<\/div>\n<div id=\"Figure_07_06_017\" class=\"small\">\n<div id=\"fs-id1936106\">\n<h3>Analysis<\/h3>\n<p id=\"fs-id1535704\">Blood pressure of [latex]\\frac{120}{80}[\/latex] is considered to be normal. The top number is the maximum or systolic reading, which measures the pressure in the arteries when the heart contracts. The bottom number is the minimum or diastolic reading, which measures the pressure in the arteries as the heart relaxes between beats, refilling with blood. Thus, normal blood pressure can be modeled by a periodic function with a maximum of 120 and a minimum of 80.\u00a0 Since the period is\u00a0[latex]\\frac{1}{80}^{th}[\/latex] of a minute, we know there are 80 heartbeats in a minute.<\/p>\n<p><span style=\"font-size: 0.8em\"><\/div>\n<\/div>\n<p><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id1146451\" class=\"bc-section section\">\n<h3>Modeling Harmonic Motion Functions<\/h3>\n<p id=\"fs-id1782792\">Harmonic motion is a form of periodic motion, but there are factors to consider that differentiate the two types. While general <span class=\"no-emphasis\">periodic motion<\/span> applications cycle through their periods with no outside interference, <span class=\"no-emphasis\">harmonic motion<\/span> requires a restoring force. Examples of harmonic motion include springs, gravitational force, and magnetic force.<\/p>\n<div id=\"fs-id1227506\" class=\"bc-section section\">\n<h4>Simple Harmonic Motion<\/h4>\n<p id=\"fs-id3038708\">A type of motion described as <span class=\"no-emphasis\">simple harmonic motion<\/span> involves a restoring force but assumes that the motion will continue forever. Imagine a weighted object hanging on a spring, When that object is not disturbed, we say that the object is at rest, or in equilibrium. If the object is pulled down and then released, the force of the spring pulls the object back toward equilibrium and harmonic motion begins. The restoring force is directly proportional to the displacement of the object from its equilibrium point. When [latex]t=0,d=0.[\/latex]<\/p>\n<div id=\"fs-id1167808\">\n<h3>Simple Harmonic Motion<\/h3>\n<p id=\"fs-id2248340\">We see that simple harmonic motion equations are given in terms of displacement:<\/p>\n<div id=\"Equation_07_06_02\" style=\"text-align: center\">[latex]d=A\\mathrm{cos}\\left(Bt\\right)\\text{ or }d=A\\mathrm{sin}\\left(Bt\\right)[\/latex][latex]\\\\[\/latex]<\/div>\n<p id=\"fs-id1480838\" style=\"text-align: center\">where [latex]|A|[\/latex] is the amplitude, and [latex]\\frac{2\\pi }{B }[\/latex] is the period.<\/p>\n<\/div>\n<div id=\"Example_07_06_09\" class=\"textbox examples\">\n<div id=\"fs-id2867398\">\n<div id=\"fs-id2867400\">\n<h3>Example 5:\u00a0 Finding the Displacement, Period, and Frequency, and Graphing a Function<\/h3>\n<p id=\"fs-id2614724\">For the given functions,<\/p>\n<ol id=\"fs-id2614727\" type=\"1\">\n<li>Find the maximum displacement of an object.<\/li>\n<li>Find the period or the time required for one vibration.<\/li>\n<li>Sketch the graph.\n<ol id=\"fs-id1611913\" type=\"a\">\n<li>[latex]y=5\\mathrm{sin}\\left(3t\\right)[\/latex]<\/li>\n<li>[latex]y=6\\mathrm{cos}\\left(\\pi t\\right)[\/latex]<\/li>\n<li>[latex]y=5\\mathrm{cos}\\left(\\frac{\\pi }{2}t\\right)[\/latex]<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<div id=\"fs-id1805776\">\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"qfs-id1805776\">Show Solution<\/span><\/p>\n<div id=\"qfs-id1805776\" class=\"hidden-answer\" style=\"display: none\">\n<ol id=\"fs-id1378735\" type=\"a\">\n<li>[latex]y=5\\mathrm{sin}\\left(3t\\right)[\/latex]\n<ol id=\"fs-id1961806\" type=\"1\">\n<li>The maximum displacement is equal to the amplitude, [latex]|A|,[\/latex] which is 5.<\/li>\n<li>The period is [latex]\\frac{2\\pi }{B}=\\frac{2\\pi }{3}.[\/latex]<\/li>\n<li>See <a class=\"autogenerated-content\" href=\"#Figure_07_07_006\">\u00a0Figure 5<\/a>. The graph indicates the five key points. \u00a0Remember, you take [latex]\\frac{1}{4}[\/latex] of the period and starting at 0, add this value repeatedly 4 times. \u00a0For this problem, you would use an interval of [latex]\\frac{1}{4}\\times\\frac{2\\pi}{3}=\\frac{\\pi}{6}.[\/latex]\n<div id=\"Figure_07_07_006\" class=\"small\">\n<div style=\"width: 320px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144658\/CNX_Precalc_Figure_07_07_006.jpg\" alt=\"Graph of the function y=5sin(3t) from 0 to 2pi\/3. The five key points are (0,0), (pi\/6, 5), (pi\/3, 0), (pi\/2, -5), (2pi\/3, 0).\" width=\"310\" height=\"282\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 5<\/p>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/li>\n<li>[latex]y=6\\mathrm{cos}\\left(\\pi t\\right)[\/latex]\n<ol id=\"fs-id1240167\" type=\"1\">\n<li>The maximum displacement is [latex]6.[\/latex]<\/li>\n<li>The period is [latex]\\frac{2\\pi }{B}=\\frac{2\\pi }{\\pi }=2.[\/latex]<\/li>\n<li>See <a class=\"autogenerated-content\" href=\"#Figure_07_07_007\">\u00a0Figure 6<\/a>.\n<div id=\"Figure_07_07_007\" class=\"small\">\n<div style=\"width: 364px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144700\/CNX_Precalc_Figure_07_07_007.jpg\" alt=\"Graph of the function y=6cos(pi t) from 0 to 3.\" width=\"354\" height=\"211\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 6<\/p>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/li>\n<li>[latex]y=5\\mathrm{cos}\\left(\\frac{\\pi }{2}t\\right)[\/latex]\n<ol id=\"fs-id1793844\" type=\"1\">\n<li>The maximum displacement is [latex]5.[\/latex]<\/li>\n<li>The period is [latex]\\frac{2\\pi }{B}=\\frac{2\\pi }{\\frac{\\pi }{2}}=4.[\/latex]<\/li>\n<li>See <a class=\"autogenerated-content\" href=\"#Figure_07_07_008\">\u00a0Figure 7<\/a>.\n<div id=\"Figure_07_07_008\" class=\"small\">\n<div style=\"width: 297px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07144704\/CNX_Precalc_Figure_07_07_008.jpg\" alt=\"Graph of the function y=5cos(pi\/2 t) from 0 to 4.\" width=\"287\" height=\"306\" \/><\/p>\n<p class=\"wp-caption-text\">Figure 7<\/p>\n<\/div>\n<\/div>\n<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-id2957014\" class=\"bc-section section\">\n<div id=\"fs-id1403864\" class=\"precalculus media\">\n<p id=\"fs-id2838123\">Access these online resources for additional instruction and practice with trigonometric applications.<\/p>\n<ul id=\"fs-id2838126\">\n<li><a href=\"http:\/\/openstax.org\/l\/solvetrigprob\">Solving Problems Using Trigonometry<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/ferriswheel\">Ferris Wheel Trigonometry<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/dailytemp\">Daily Temperatures and Trigonometry<\/a><\/li>\n<li><a href=\"http:\/\/openstax.org\/l\/simpleharm\">Simple Harmonic Motion<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div id=\"fs-id3483144\" class=\"key-equations\">\n<h3>Key Equations<\/h3>\n<table id=\"fs-id2608518\" summary=\"..\">\n<caption>\u00a0<\/caption>\n<tbody>\n<tr>\n<td class=\"border\">Standard form of sinusoidal equation<\/td>\n<td class=\"border\">[latex]y=A\\mathrm{sin}\\left(B\\left(t-h\\right)\\right)+k\\text{ or }y=A\\mathrm{cos}\\left(B\\left(t-h\\right)\\right)+k[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Simple harmonic motion<\/td>\n<td class=\"border\">[latex]d=a\\mathrm{cos}\\left(B t\\right)\\text{ or }d=a\\mathrm{sin}\\left(B t\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\"><\/td>\n<td class=\"border\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div id=\"fs-id1157947\" class=\"textbox key-takeaways\">\n<h3>Key Concepts<\/h3>\n<ul id=\"fs-id1793027\">\n<li>Sinusoidal functions are represented by the sine and cosine graphs. In standard form, we can find the amplitude, period, and horizontal and vertical shifts.<\/li>\n<li>Use key points to graph a sinusoidal function. The five key points include the minimum and maximum values and the midline values.<\/li>\n<li>Periodic functions can model events that reoccur in set cycles, like the phases of the moon, the hands on a clock, and the seasons in a year.<\/li>\n<li>Harmonic motion functions are modeled from given data. Similar to periodic motion applications, harmonic motion requires a restoring force. Examples include gravitational force and spring motion activated by weight.<\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-id1817053\" class=\"review-exercises\"><\/div>\n<div class=\"textbox shaded\">\n<h3>Glossary<\/h3>\n<dl id=\"fs-id2617951\">\n<dt>simple harmonic motion<\/dt>\n<dd id=\"fs-id2617956\">a repetitive motion that can be modeled by periodic sinusoidal oscillation<\/dd>\n<\/dl>\n<\/div>\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-1191\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Modeling with Trigonometric Equations. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:IrVR2zv_@5\/Modeling-with-Trigonometric-Equations\">https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:IrVR2zv_@5\/Modeling-with-Trigonometric-Equations<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 2. <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":311,"menu_order":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Modeling with Trigonometric Equations\",\"author\":\"Douglas Hoffman\",\"organization\":\"Openstax\",\"url\":\"https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:IrVR2zv_@5\/Modeling-with-Trigonometric-Equations\",\"project\":\"Essential Precalcus, Part 2\",\"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-1191","chapter","type-chapter","status-publish","hentry"],"part":478,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1191","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":41,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1191\/revisions"}],"predecessor-version":[{"id":2882,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1191\/revisions\/2882"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/parts\/478"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapters\/1191\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=1191"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1191"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=1191"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=1191"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}