{"id":1040,"date":"2024-04-10T16:30:10","date_gmt":"2024-04-10T16:30:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1040"},"modified":"2025-08-17T15:59:44","modified_gmt":"2025-08-17T15:59:44","slug":"trigonometric-functions-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/trigonometric-functions-fresh-take\/","title":{"raw":"Trigonometric Functions: Fresh Take","rendered":"Trigonometric Functions: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li>Convert angle measures between degrees and radians<\/li>\r\n\t<li>Interpret and apply basic trigonometric definitions and identities<\/li>\r\n\t<li>Analyze trigonometric functions by identifying graphs and periods, and describing shifts in sine or cosine graphs.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Degrees versus Radians<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Degrees:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">One degree is [latex]\\frac{1}{360}[\/latex] of a full circular rotation<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Full circle = [latex]360\u00b0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Symbol: [latex]\u00b0[\/latex] (e.g., [latex]45\u00b0[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Radians:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Based on the radius of a circle<\/li>\r\n\t<li class=\"whitespace-normal break-words\">One radian is the angle subtended by an arc length equal to the radius<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Full circle = [latex]2\u03c0[\/latex] radians<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Symbol: rad (e.g., [latex]\\frac{\u03c0}{2}[\/latex] rad)<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Conversion:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\u03c0[\/latex] radians [latex]= 180\u00b0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]1[\/latex] radian [latex]\u2248 57.3\u00b0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Conversion formulas:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Degrees to Radians: [latex]\\text{Angle in Degrees} = \\text{Angle in Radians }\\times\\frac{180}{\u03c0}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Radians to Degrees: [latex]\\text{Angle in Radians} = \\text{Angle in Degrees }\\times\\frac{\u03c0}{180}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Usage:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Degrees: common in everyday situations, navigation, construction<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Radians: preferred in advanced mathematics, physics, engineering<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li id=\"fs-id1170572137475\">Express [latex]210\u00b0[\/latex] using radians.<\/li>\r\n\t<li>Express [latex]\\dfrac{11\\pi}{6}[\/latex] rad using degrees.<\/li>\r\n<\/ol>\r\n<p>[reveal-answer q=\"668822\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"668822\"]<\/p>\r\n<div id=\"fs-id1165041814727\" class=\"commentary\">\r\n<p id=\"fs-id1165041962818\">[latex]\\pi [\/latex] radians is equal to [latex]180^{\\circ}[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/div>\r\n<p>[reveal-answer q=\"fs-id1170572482409\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572482409\"]<\/p>\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n\t<li id=\"fs-id1170572482409\">[latex]\\dfrac{7\\pi}{6}[\/latex] rad<\/li>\r\n\t<li>[latex]330\u00b0[\/latex]<\/li>\r\n<\/ol>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<div id=\"fs-id1170572212258\" class=\"textbook key-takeaways\">\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]196627[\/ohm_question]<\/p>\r\n<\/section>\r\n<h2>The Six Basic Trigonometric Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">The Six Functions:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Sine ([latex]\\sin[\/latex]): [latex]\\frac{\\text{opposite}}{\\text{hypotenuse}}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Cosine ([latex]\\cos[\/latex]): [latex]\\frac{\\text{adjacent}}{\\text{hypotenuse}}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Tangent ([latex]\\tan[\/latex]): [latex]\\frac{\\text{opposite}}{\\text{adjacent}}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Cosecant ([latex]\\csc[\/latex]): [latex]\\frac{\\text{hypotenuse}}{\\text{opposite}}[\/latex] (reciprocal of sine)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Secant ([latex]\\sec[\/latex]): [latex]\\frac{\\text{hypotenuse}}{\\text{adjacent}}[\/latex] (reciprocal of cosine)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Cotangent ([latex]\\cot[\/latex]): [latex]\\frac{\\text{adjacent}}{\\text{opposite}}[\/latex] (reciprocal of tangent)<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Mnemonic Device:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">SOH-CAH-TOA for sine, cosine, and tangent<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Reciprocal Relationships:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\sin{\\theta} = \\frac{1}{\\csc{\\theta}} [\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\cos{\\theta} = \\frac{1}{\\sec{\\theta}} [\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\\tan{\\theta} = \\frac{1}{\\cot{\\theta}} [\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Applications:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Finding unknown side lengths in right triangles<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Solving real-world problems involving angles and distances<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">Using the triangle shown below ,evaluate [latex]\\sin t[\/latex],[latex]\\cos t[\/latex],[latex]\\tan t[\/latex],[latex]\\sec t[\/latex],[latex]\\csc t[\/latex],and [latex]\\cot t[\/latex].<br \/>\r\n[caption id=\"attachment_1063\" align=\"alignnone\" width=\"416\"]<img class=\"wp-image-1063 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10173307\/Screenshot-2024-04-10-133254.png\" alt=\"Right triangle with sides 33, 56, and 65. Angle t is also labeled which is opposite to the side labeled 33. \" width=\"416\" height=\"198\" \/> Right triangle with sides 33, 56, and 65[\/caption]\r\n<br \/>\r\n[reveal-answer q=\"84249\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"84249\"]<center>[latex]\\begin{array}{ccc} \\sin \\alpha &amp; = &amp; \\frac{\\text{opposite to } t}{\\text{hypotenuse}} = \\frac{33}{65} \\\\ \\cos \\alpha &amp; = &amp; \\frac{\\text{adjacent to } t}{\\text{hypotenuse}} = \\frac{56}{65} \\\\ \\tan \\alpha &amp; = &amp; \\frac{\\text{opposite to } t}{\\text{adjacent to } t} = \\frac{33}{56} \\\\ \\sec \\alpha &amp; = &amp; \\frac{\\text{hypotenuse}}{\\text{adjacent to } t} = \\frac{65}{56} \\\\ \\csc \\alpha &amp; = &amp; \\frac{\\text{hypotenuse}}{\\text{opposite to } t} = \\frac{65}{33} \\\\ \\cot \\alpha &amp; = &amp; \\frac{\\text{adjacent to } t}{\\text{opposite to } t} = \\frac{56}{33} \\end{array} [\/altex]<\/center>[\/hidden-answer]<\/section>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572480688\">A house painter wants to lean a [latex]20[\/latex]-ft ladder against a house. If the angle between the base of the ladder and the ground is to be [latex]60^{\\circ}[\/latex], how far from the house should she place the base of the ladder?<\/p>\r\n<p>[reveal-answer q=\"478821\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"478821\"]<\/p>\r\n<p id=\"fs-id1165041814643\">Draw a right triangle with hypotenuse [latex]20[\/latex] ft.<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<p>[reveal-answer q=\"fs-id1170572450935\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"fs-id1170572450935\"]<\/p>\r\n<p id=\"fs-id1170572450935\">[latex]10 [\/latex] ft<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<\/div>\r\n<h2>Trigonometric Identities<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Trigonometric identities are equations involving trigonometric functions that are true for all values where they are defined.<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Types of Identities:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Reciprocal Identities (e.g., [latex]\\csc \\theta =\\large \\frac{1}{\\sin \\theta}[\/latex])<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Pythagorean Identities (e.g., [latex]\\sin^2 \\theta +\\cos^2 \\theta =1[\/latex])<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Addition and Subtraction Formulas<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Double-Angle Formulas<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Importance:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Crucial for solving trigonometric equations<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Used in proving other mathematical statements<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Frequently applied in calculus and higher mathematics<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Verification Methods:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Manipulate one side of the equation to match the other<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Use known identities to make substitutions<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Convert all terms to sines and cosines if needed<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">\r\n<p id=\"fs-id1170572226907\">Prove the trigonometric identity [latex]1+\\cot^2 \\theta =\\csc^2 \\theta[\/latex].<\/p>\r\n<p>[reveal-answer q=\"265389\"]Hint[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"265389\"]<\/p>\r\n<p id=\"fs-id1165041809092\">Divide both sides of the identity [latex]\\sin^2 \\theta + \\cos^2 \\theta =1[\/latex] by [latex]\\sin^2 \\theta[\/latex].<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">\r\n<p>Simplify the expression by rewriting and using identities:<\/p>\r\n<center>[latex]\\csc^2{\\theta}-\\cot^2{\\theta}[\/latex]<\/center>[reveal-answer q=\"105961\"]Show Answer[\/reveal-answer] [hidden-answer a=\"105961\"]We can start with the Pythagorean identity.<center>[latex]1+\\cot^2 \\theta =\\csc^2 \\theta[\/latex]<\/center>Now we can simplify by substituting [latex]1+\\cot^2 \\theta [\/latex] for [latex]\\csc^2 \\theta[\/latex]. We have<center>[latex]\\begin{align*}\\csc^2{\\theta}-\\cot^2{\\theta}&amp;= 1+ \\cot^2 \\theta-\\cot^2 \\theta \\\\ &amp;= 1\\end{align*}[\/latex]<\/center>[\/hidden-answer]<\/section>\r\n<h2>Graphs and Periods of the Trigonometric Functions<\/h2>\r\n<div class=\"textbox shaded\">\r\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Periodicity:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Trigonometric functions repeat their values at regular intervals<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Sine, cosine, secant, cosecant: period of [latex]2\u03c0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Tangent, cotangent: period of [latex]\u03c0[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Basic Graphs:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Sine and cosine: smooth wave patterns<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Tangent and cotangent: repeating vertical asymptotes<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Secant and cosecant: repeating reciprocal patterns of cosine and sine<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Transformations: General form: [latex]f(x) = A \\sin{(B(x - \u03b1))} + C [\/latex](similar for cosine)\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">[latex]A[\/latex]: Amplitude (vertical stretch\/compression)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]B[\/latex]: Frequency (horizontal stretch\/compression, affects period)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\u03b1[\/latex]: Phase shift (horizontal shift)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]C[\/latex]: Vertical shift<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Effects of Transformations:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Amplitude: [latex]|A|[\/latex]\u00a0is the height from midline to peak\/trough<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Period: [latex]d\\frac{2 \\pi}{|B|}[\/latex] for sine and cosine<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Phase Shift: Right by [latex]\u03b1[\/latex] if positive, left if negative<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Vertical Shift: Up by [latex]C[\/latex] if positive, down if negative<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p>Describe and sketch the graph of [latex]f(x) = 2 \\cos{(\\frac{1}{2}(x + \\frac{\u03c0}{3}))} - 1[\/latex]<\/p>\r\n<p><br \/>\r\n[reveal-answer q=\"480296\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"480296\"]<\/p>\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">Identify components:\r\n\r\n<ul>\r\n\t<li class=\"whitespace-normal break-words\">[latex]A = 2[\/latex] (amplitude)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]B = \\frac{1}{2}[\/latex] (frequency)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]\u03b1 = \\frac{-\u03c0}{3}[\/latex] (phase shift)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">[latex]C = -1[\/latex] (vertical shift)<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Analyze effects:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Amplitude: Graph stretches vertically to height [latex]2[\/latex] from midline<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Period: [latex]\\dfrac{2\u03c0}{|B|} = \\dfrac{2\u03c0}{\\frac{1}{2}} = 4\u03c0[\/latex] (stretched horizontally)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Phase shift: [latex]\\frac{\u03c0}{3}[\/latex] units left (note the [latex]+[\/latex] inside becomes [latex] -[\/latex] outside)<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Vertical shift: [latex]1[\/latex] unit down<\/li>\r\n<\/ul>\r\n<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Sketch the graph:\r\n\r\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\r\n\t<li class=\"whitespace-normal break-words\">Start with basic cosine graph<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Stretch vertically by factor of [latex]2[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Stretch horizontally to period [latex]4\u03c0[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Shift left by [latex]\\frac{\u03c0}{3}[\/latex]<\/li>\r\n\t<li class=\"whitespace-normal break-words\">Shift down by [latex]1[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p>We can verify this by using graphing software to compare the graphs of [latex]f(x) =\\cos{x}[\/latex] and [latex]f(x) = 2 \\cos{(\\frac{1}{2}(x + \\frac{\u03c0}{3}))} - 1[\/latex].<\/p>\r\n\r\n[caption id=\"attachment_3639\" align=\"aligncenter\" width=\"500\"]<img class=\"wp-image-3639\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01192231\/Screenshot-2024-07-01-150623.png\" alt=\"Here's the alt text for the image: &quot;A graph showing two trigonometric functions on a coordinate plane. The red curve represents f(x) = cos(x), a standard cosine function with amplitude 1 and period 2\u03c0. The blue curve represents g(x) = 2cos(1\/2(x + \u03c0\/3)) - 1, which has a larger amplitude of 2, a longer period, and is shifted vertically down by 1 unit.\" width=\"500\" height=\"193\" \/> Comparison of cosine functions: f(x) = cos(x) (red) and g(x) = 2cos(1\/2(x + \u03c0\/3)) - 1 (blue)[\/caption]\r\n\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<p>&nbsp;<\/p>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Convert angle measures between degrees and radians<\/li>\n<li>Interpret and apply basic trigonometric definitions and identities<\/li>\n<li>Analyze trigonometric functions by identifying graphs and periods, and describing shifts in sine or cosine graphs.<\/li>\n<\/ul>\n<\/section>\n<h2>Degrees versus Radians<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Degrees:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">One degree is [latex]\\frac{1}{360}[\/latex] of a full circular rotation<\/li>\n<li class=\"whitespace-normal break-words\">Full circle = [latex]360\u00b0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Symbol: [latex]\u00b0[\/latex] (e.g., [latex]45\u00b0[\/latex])<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Radians:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Based on the radius of a circle<\/li>\n<li class=\"whitespace-normal break-words\">One radian is the angle subtended by an arc length equal to the radius<\/li>\n<li class=\"whitespace-normal break-words\">Full circle = [latex]2\u03c0[\/latex] radians<\/li>\n<li class=\"whitespace-normal break-words\">Symbol: rad (e.g., [latex]\\frac{\u03c0}{2}[\/latex] rad)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Conversion:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\u03c0[\/latex] radians [latex]= 180\u00b0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]1[\/latex] radian [latex]\u2248 57.3\u00b0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Conversion formulas:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Degrees to Radians: [latex]\\text{Angle in Degrees} = \\text{Angle in Radians }\\times\\frac{180}{\u03c0}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Radians to Degrees: [latex]\\text{Angle in Radians} = \\text{Angle in Degrees }\\times\\frac{\u03c0}{180}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Usage:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Degrees: common in everyday situations, navigation, construction<\/li>\n<li class=\"whitespace-normal break-words\">Radians: preferred in advanced mathematics, physics, engineering<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li id=\"fs-id1170572137475\">Express [latex]210\u00b0[\/latex] using radians.<\/li>\n<li>Express [latex]\\dfrac{11\\pi}{6}[\/latex] rad using degrees.<\/li>\n<\/ol>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q668822\">Hint<\/button><\/p>\n<div id=\"q668822\" class=\"hidden-answer\" style=\"display: none\">\n<div id=\"fs-id1165041814727\" class=\"commentary\">\n<p id=\"fs-id1165041962818\">[latex]\\pi[\/latex] radians is equal to [latex]180^{\\circ}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572482409\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572482409\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: lower-alpha;\">\n<li id=\"fs-id1170572482409\">[latex]\\dfrac{7\\pi}{6}[\/latex] rad<\/li>\n<li>[latex]330\u00b0[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<div id=\"fs-id1170572212258\" class=\"textbook key-takeaways\">\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm196627\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=196627&theme=lumen&iframe_resize_id=ohm196627&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<h2>The Six Basic Trigonometric Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">The Six Functions:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Sine ([latex]\\sin[\/latex]): [latex]\\frac{\\text{opposite}}{\\text{hypotenuse}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Cosine ([latex]\\cos[\/latex]): [latex]\\frac{\\text{adjacent}}{\\text{hypotenuse}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Tangent ([latex]\\tan[\/latex]): [latex]\\frac{\\text{opposite}}{\\text{adjacent}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Cosecant ([latex]\\csc[\/latex]): [latex]\\frac{\\text{hypotenuse}}{\\text{opposite}}[\/latex] (reciprocal of sine)<\/li>\n<li class=\"whitespace-normal break-words\">Secant ([latex]\\sec[\/latex]): [latex]\\frac{\\text{hypotenuse}}{\\text{adjacent}}[\/latex] (reciprocal of cosine)<\/li>\n<li class=\"whitespace-normal break-words\">Cotangent ([latex]\\cot[\/latex]): [latex]\\frac{\\text{adjacent}}{\\text{opposite}}[\/latex] (reciprocal of tangent)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Mnemonic Device:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">SOH-CAH-TOA for sine, cosine, and tangent<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Reciprocal Relationships:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]\\sin{\\theta} = \\frac{1}{\\csc{\\theta}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\cos{\\theta} = \\frac{1}{\\sec{\\theta}}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\\tan{\\theta} = \\frac{1}{\\cot{\\theta}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Applications:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Finding unknown side lengths in right triangles<\/li>\n<li class=\"whitespace-normal break-words\">Solving real-world problems involving angles and distances<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">Using the triangle shown below ,evaluate [latex]\\sin t[\/latex],[latex]\\cos t[\/latex],[latex]\\tan t[\/latex],[latex]\\sec t[\/latex],[latex]\\csc t[\/latex],and [latex]\\cot t[\/latex].<\/p>\n<figure id=\"attachment_1063\" aria-describedby=\"caption-attachment-1063\" style=\"width: 416px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1063 size-full\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10173307\/Screenshot-2024-04-10-133254.png\" alt=\"Right triangle with sides 33, 56, and 65. Angle t is also labeled which is opposite to the side labeled 33.\" width=\"416\" height=\"198\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10173307\/Screenshot-2024-04-10-133254.png 416w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10173307\/Screenshot-2024-04-10-133254-300x143.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10173307\/Screenshot-2024-04-10-133254-65x31.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10173307\/Screenshot-2024-04-10-133254-225x107.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/10173307\/Screenshot-2024-04-10-133254-350x167.png 350w\" sizes=\"(max-width: 416px) 100vw, 416px\" \/><figcaption id=\"caption-attachment-1063\" class=\"wp-caption-text\">Right triangle with sides 33, 56, and 65<\/figcaption><\/figure>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q84249\">Show Answer<\/button><\/p>\n<div id=\"q84249\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex][\/latex]\\begin{array}{ccc} \\sin \\alpha &amp; = &amp; \\frac{\\text{opposite to } t}{\\text{hypotenuse}} = \\frac{33}{65} \\\\ \\cos \\alpha &amp; = &amp; \\frac{\\text{adjacent to } t}{\\text{hypotenuse}} = \\frac{56}{65} \\\\ \\tan \\alpha &amp; = &amp; \\frac{\\text{opposite to } t}{\\text{adjacent to } t} = \\frac{33}{56} \\\\ \\sec \\alpha &amp; = &amp; \\frac{\\text{hypotenuse}}{\\text{adjacent to } t} = \\frac{65}{56} \\\\ \\csc \\alpha &amp; = &amp; \\frac{\\text{hypotenuse}}{\\text{opposite to } t} = \\frac{65}{33} \\\\ \\cot \\alpha &amp; = &amp; \\frac{\\text{adjacent to } t}{\\text{opposite to } t} = \\frac{56}{33} \\end{array} [\/altex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572480688\">A house painter wants to lean a [latex]20[\/latex]-ft ladder against a house. If the angle between the base of the ladder and the ground is to be [latex]60^{\\circ}[\/latex], how far from the house should she place the base of the ladder?<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q478821\">Hint<\/button><\/p>\n<div id=\"q478821\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165041814643\">Draw a right triangle with hypotenuse [latex]20[\/latex] ft.<\/p>\n<\/div>\n<\/div>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"qfs-id1170572450935\">Show Solution<\/button><\/p>\n<div id=\"qfs-id1170572450935\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1170572450935\">[latex]10[\/latex] ft<\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<h2>Trigonometric Identities<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Trigonometric identities are equations involving trigonometric functions that are true for all values where they are defined.<\/li>\n<li class=\"whitespace-normal break-words\">Types of Identities:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Reciprocal Identities (e.g., [latex]\\csc \\theta =\\large \\frac{1}{\\sin \\theta}[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Pythagorean Identities (e.g., [latex]\\sin^2 \\theta +\\cos^2 \\theta =1[\/latex])<\/li>\n<li class=\"whitespace-normal break-words\">Addition and Subtraction Formulas<\/li>\n<li class=\"whitespace-normal break-words\">Double-Angle Formulas<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Importance:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Crucial for solving trigonometric equations<\/li>\n<li class=\"whitespace-normal break-words\">Used in proving other mathematical statements<\/li>\n<li class=\"whitespace-normal break-words\">Frequently applied in calculus and higher mathematics<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Verification Methods:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Manipulate one side of the equation to match the other<\/li>\n<li class=\"whitespace-normal break-words\">Use known identities to make substitutions<\/li>\n<li class=\"whitespace-normal break-words\">Convert all terms to sines and cosines if needed<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">\n<p id=\"fs-id1170572226907\">Prove the trigonometric identity [latex]1+\\cot^2 \\theta =\\csc^2 \\theta[\/latex].<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q265389\">Hint<\/button><\/p>\n<div id=\"q265389\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1165041809092\">Divide both sides of the identity [latex]\\sin^2 \\theta + \\cos^2 \\theta =1[\/latex] by [latex]\\sin^2 \\theta[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">\n<p>Simplify the expression by rewriting and using identities:<\/p>\n<div style=\"text-align: center;\">[latex]\\csc^2{\\theta}-\\cot^2{\\theta}[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q105961\">Show Answer<\/button> <\/p>\n<div id=\"q105961\" class=\"hidden-answer\" style=\"display: none\">We can start with the Pythagorean identity.<\/p>\n<div style=\"text-align: center;\">[latex]1+\\cot^2 \\theta =\\csc^2 \\theta[\/latex]<\/div>\n<p>Now we can simplify by substituting [latex]1+\\cot^2 \\theta[\/latex] for [latex]\\csc^2 \\theta[\/latex]. We have<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align*}\\csc^2{\\theta}-\\cot^2{\\theta}&= 1+ \\cot^2 \\theta-\\cot^2 \\theta \\\\ &= 1\\end{align*}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<h2>Graphs and Periods of the Trigonometric Functions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<ul>\n<li class=\"whitespace-normal break-words\">Periodicity:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Trigonometric functions repeat their values at regular intervals<\/li>\n<li class=\"whitespace-normal break-words\">Sine, cosine, secant, cosecant: period of [latex]2\u03c0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Tangent, cotangent: period of [latex]\u03c0[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Basic Graphs:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Sine and cosine: smooth wave patterns<\/li>\n<li class=\"whitespace-normal break-words\">Tangent and cotangent: repeating vertical asymptotes<\/li>\n<li class=\"whitespace-normal break-words\">Secant and cosecant: repeating reciprocal patterns of cosine and sine<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Transformations: General form: [latex]f(x) = A \\sin{(B(x - \u03b1))} + C[\/latex](similar for cosine)\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">[latex]A[\/latex]: Amplitude (vertical stretch\/compression)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]B[\/latex]: Frequency (horizontal stretch\/compression, affects period)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\u03b1[\/latex]: Phase shift (horizontal shift)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]C[\/latex]: Vertical shift<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Effects of Transformations:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Amplitude: [latex]|A|[\/latex]\u00a0is the height from midline to peak\/trough<\/li>\n<li class=\"whitespace-normal break-words\">Period: [latex]d\\frac{2 \\pi}{|B|}[\/latex] for sine and cosine<\/li>\n<li class=\"whitespace-normal break-words\">Phase Shift: Right by [latex]\u03b1[\/latex] if positive, left if negative<\/li>\n<li class=\"whitespace-normal break-words\">Vertical Shift: Up by [latex]C[\/latex] if positive, down if negative<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p>Describe and sketch the graph of [latex]f(x) = 2 \\cos{(\\frac{1}{2}(x + \\frac{\u03c0}{3}))} - 1[\/latex]<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q480296\">Show Answer<\/button><\/p>\n<div id=\"q480296\" class=\"hidden-answer\" style=\"display: none\">\n<ul>\n<li class=\"whitespace-normal break-words\">Identify components:\n<ul>\n<li class=\"whitespace-normal break-words\">[latex]A = 2[\/latex] (amplitude)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]B = \\frac{1}{2}[\/latex] (frequency)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]\u03b1 = \\frac{-\u03c0}{3}[\/latex] (phase shift)<\/li>\n<li class=\"whitespace-normal break-words\">[latex]C = -1[\/latex] (vertical shift)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Analyze effects:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Amplitude: Graph stretches vertically to height [latex]2[\/latex] from midline<\/li>\n<li class=\"whitespace-normal break-words\">Period: [latex]\\dfrac{2\u03c0}{|B|} = \\dfrac{2\u03c0}{\\frac{1}{2}} = 4\u03c0[\/latex] (stretched horizontally)<\/li>\n<li class=\"whitespace-normal break-words\">Phase shift: [latex]\\frac{\u03c0}{3}[\/latex] units left (note the [latex]+[\/latex] inside becomes [latex]-[\/latex] outside)<\/li>\n<li class=\"whitespace-normal break-words\">Vertical shift: [latex]1[\/latex] unit down<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words\">Sketch the graph:\n<ul class=\"-mt-1 list-disc space-y-2 pl-8\">\n<li class=\"whitespace-normal break-words\">Start with basic cosine graph<\/li>\n<li class=\"whitespace-normal break-words\">Stretch vertically by factor of [latex]2[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Stretch horizontally to period [latex]4\u03c0[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Shift left by [latex]\\frac{\u03c0}{3}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words\">Shift down by [latex]1[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p>We can verify this by using graphing software to compare the graphs of [latex]f(x) =\\cos{x}[\/latex] and [latex]f(x) = 2 \\cos{(\\frac{1}{2}(x + \\frac{\u03c0}{3}))} - 1[\/latex].<\/p>\n<figure id=\"attachment_3639\" aria-describedby=\"caption-attachment-3639\" style=\"width: 500px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3639\" src=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01192231\/Screenshot-2024-07-01-150623.png\" alt=\"Here's the alt text for the image: &quot;A graph showing two trigonometric functions on a coordinate plane. The red curve represents f(x) = cos(x), a standard cosine function with amplitude 1 and period 2\u03c0. The blue curve represents g(x) = 2cos(1\/2(x + \u03c0\/3)) - 1, which has a larger amplitude of 2, a longer period, and is shifted vertically down by 1 unit.\" width=\"500\" height=\"193\" srcset=\"https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01192231\/Screenshot-2024-07-01-150623.png 996w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01192231\/Screenshot-2024-07-01-150623-300x116.png 300w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01192231\/Screenshot-2024-07-01-150623-768x296.png 768w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01192231\/Screenshot-2024-07-01-150623-65x25.png 65w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01192231\/Screenshot-2024-07-01-150623-225x87.png 225w, https:\/\/content-cdn.one.lumenlearning.com\/wp-content\/uploads\/sites\/34\/2024\/04\/01192231\/Screenshot-2024-07-01-150623-350x135.png 350w\" sizes=\"(max-width: 500px) 100vw, 500px\" \/><figcaption id=\"caption-attachment-3639\" class=\"wp-caption-text\">Comparison of cosine functions: f(x) = cos(x) (red) and g(x) = 2cos(1\/2(x + \u03c0\/3)) - 1 (blue)<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<p>&nbsp;<\/p>\n","protected":false},"author":15,"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":759,"module-header":"fresh_take","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\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1040"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":20,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1040\/revisions"}],"predecessor-version":[{"id":4854,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1040\/revisions\/4854"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/759"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1040\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1040"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1040"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1040"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1040"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}