{"id":704,"date":"2025-07-14T21:16:10","date_gmt":"2025-07-14T21:16:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=704"},"modified":"2026-01-16T17:39:25","modified_gmt":"2026-01-16T17:39:25","slug":"module-14-periodic-functions-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/module-14-periodic-functions-cheat-sheet\/","title":{"raw":"Periodic Functions: Cheat Sheet","rendered":"Periodic Functions: Cheat Sheet"},"content":{"raw":"<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Essential Concepts<\/h2>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Graphs of Sine and Cosine Functions<\/h3>\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Key Features\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Period [latex]2\\pi[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Domain: [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Range: [latex][-1, 1][\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]y = \\sin x[\/latex] is an odd function (symmetric about the origin)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]y = \\cos x[\/latex] is an even function (symmetric about the y-axis)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>General form equation\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li>[latex]y = A\\sin(Bx - C) + D[\/latex]<\/li>\r\n \t<li>[latex]y = A\\cos(Bx - C) + D[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Transformations:\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li><strong>Amplitude:<\/strong> [latex]|A|[\/latex] measures the vertical stretch\/compression<\/li>\r\n \t<li><strong>Period:<\/strong> [latex]P = \\frac{2\\pi}{|B|}[\/latex] is the length of one complete cycle\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">If [latex]|B| &gt; 1[\/latex]: horizontal compression (period less than [latex]2\\pi[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">If [latex]|B| &lt; 1[\/latex]: horizontal stretch (period greater than [latex]2\\pi[\/latex])<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\"><strong>Phase shift:<\/strong> [latex]\\frac{C}{B}[\/latex] represents horizontal displacement\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">If [latex]C &gt; 0[\/latex]: shift right<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">If [latex]C &lt; 0[\/latex]: shift left<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\"><strong>Vertical shift:<\/strong> [latex]D[\/latex] is the midline of the graph\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Positive [latex]D[\/latex]: shift up<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Negative [latex]D[\/latex]: shift down<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\"><strong>Reflection:<\/strong> If [latex]A &lt; 0[\/latex], the graph is reflected across the x-axis<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Graphing sine and cosine:<\/strong><\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Identify amplitude [latex]|A|[\/latex], period [latex]\\frac{2\\pi}{|B|}[\/latex], phase shift [latex]\\frac{C}{B}[\/latex], and midline [latex]y = D[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Mark key points at quarter-period intervals<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Apply transformations in order: horizontal stretch\/compression, horizontal shift, vertical stretch\/compression, vertical shift, reflection<\/li>\r\n<\/ol>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Graphs of the Other Trigonometric Functions<\/h3>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Tangent function [latex]y = A\\tan(Bx - C) + D[\/latex]:<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Period: [latex]P = \\frac{\\pi}{|B|}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Domain: [latex]x \\ne \\frac{C}{B} + \\frac{\\pi}{2|B|}k[\/latex], where [latex]k[\/latex] is an odd integer<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Range: [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Vertical asymptotes: [latex]x = \\frac{C}{B} + \\frac{\\pi}{2|B|}k[\/latex], where [latex]k[\/latex] is an odd integer<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">No amplitude; [latex]|A|[\/latex] is the stretching\/compressing factor<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Odd function<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Cotangent function [latex]y = A\\cot(Bx - C) + D[\/latex]:<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Period: [latex]P = \\frac{\\pi}{|B|}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Domain: [latex]x \\ne \\frac{C}{B} + \\frac{\\pi}{|B|}k[\/latex], where [latex]k[\/latex] is an integer<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Range: [latex](-\\infty, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Vertical asymptotes: [latex]x = \\frac{C}{B} + \\frac{\\pi}{|B|}k[\/latex], where [latex]k[\/latex] is an integer<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">No amplitude; [latex]|A|[\/latex] is the stretching\/compressing factor<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Odd function<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Secant function [latex]y = A\\sec(Bx - C) + D[\/latex]:<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Period: [latex]P = \\frac{2\\pi}{|B|}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Domain: [latex]x \\ne \\frac{C}{B} + \\frac{\\pi}{2|B|}k[\/latex], where [latex]k[\/latex] is an odd integer<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Range: [latex](-\\infty, -|A| + D] \\cup [|A| + D, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Vertical asymptotes: [latex]x = \\frac{C}{B} + \\frac{\\pi}{2|B|}k[\/latex], where [latex]k[\/latex] is an odd integer<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Even function<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Graph using reciprocal relationship: [latex]\\sec x = \\frac{1}{\\cos x}[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Cosecant function [latex]y = A\\csc(Bx - C) + D[\/latex]:<\/strong><\/p>\r\n\r\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Period: [latex]P = \\frac{2\\pi}{|B|}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Domain: [latex]x \\ne \\frac{C}{B} + \\frac{\\pi}{|B|}k[\/latex], where [latex]k[\/latex] is an integer<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Range: [latex](-\\infty, -|A| + D] \\cup [|A| + D, \\infty)[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Vertical asymptotes: [latex]x = \\frac{C}{B} + \\frac{\\pi}{|B|}k[\/latex], where [latex]k[\/latex] is an integer<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Odd function<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Graph using reciprocal relationship: [latex]\\csc x = \\frac{1}{\\sin x}[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Graphing tangent and cotangent:<\/strong><\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">Identify stretching factor [latex]|A|[\/latex] and period [latex]\\frac{\\pi}{|B|}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Locate vertical asymptotes<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Plot reference points including the center point and quarter-period points<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Draw the curve approaching asymptotes correctly<\/li>\r\n<\/ol>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Graphing secant and cosecant:<\/strong><\/p>\r\n\r\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\r\n \t<li class=\"whitespace-normal break-words pl-2\">First sketch the corresponding cosine or sine function<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Use reciprocal relationship to draw secant or cosecant<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Vertical asymptotes occur where cosine or sine equals zero<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Local extrema of secant\/cosecant occur at extrema of cosine\/sine<\/li>\r\n<\/ol>\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Key Equations<\/h2>\r\n<table style=\"width: 92.0541%; height: 506px;\">\r\n<tbody>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 25.6352%; height: 22px;\"><strong>General sine function<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 22px;\">[latex]y = A\\sin(Bx - C) + D[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 25.6352%; height: 22px;\"><strong>General cosine function<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 22px;\">[latex]y = A\\cos(Bx - C) + D[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 25.6352%; height: 22px;\"><strong>Period of sine\/cosine<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 22px;\">[latex]P = \\frac{2\\pi}{|B|}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 25.6352%; height: 22px;\"><strong>Phase shift<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 22px;\">[latex]\\frac{C}{B}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 25.6352%; height: 22px;\"><strong>Midline<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 22px;\">[latex]y = D[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 25.6352%; height: 22px;\"><strong>General tangent function<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 22px;\">[latex]y = A\\tan(Bx - C) + D[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 25.6352%; height: 22px;\"><strong>Tangent asymptotes<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 22px;\">[latex]x = \\frac{C}{B} + \\frac{\\pi}{2|B|}k[\/latex], [latex]k[\/latex] odd integer<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td style=\"width: 25.6352%; height: 44px;\"><strong>General cotangent function<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 44px;\">[latex]y = A\\cot(Bx - C) + D[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 25.6352%; height: 22px;\"><strong>Cotangent asymptotes<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 22px;\">[latex]x = \\frac{C}{B} + \\frac{\\pi}{|B|}k[\/latex], [latex]k[\/latex] integer<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td style=\"width: 25.6352%; height: 44px;\"><strong>Period of tangent\/cotangent<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 44px;\">[latex]P = \\frac{\\pi}{|B|}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 22px;\">\r\n<td style=\"width: 25.6352%; height: 22px;\"><strong>General secant function<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 22px;\">[latex]y = A\\sec(Bx - C) + D[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td style=\"width: 25.6352%; height: 44px;\"><strong>General cosecant function<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 44px;\">[latex]y = A\\csc(Bx - C) + D[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 44px;\">\r\n<td style=\"width: 25.6352%; height: 44px;\"><strong>Period of secant\/cosecant<\/strong><\/td>\r\n<td style=\"width: 102.861%; height: 44px;\">[latex]P = \\frac{2\\pi}{|B|}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Glossary<\/h2>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>amplitude<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The distance from the midline to the maximum or minimum of a sinusoidal function; calculated as [latex]|A|[\/latex] in the general forms [latex]y = A\\sin(Bx - C) + D[\/latex] or [latex]y = A\\cos(Bx - C) + D[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>cosecant function<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The function [latex]\\csc x = \\frac{1}{\\sin x}[\/latex], which has period [latex]2\\pi[\/latex], vertical asymptotes where [latex]\\sin x = 0[\/latex], and is undefined at those points<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>cotangent function<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The function [latex]\\cot x = \\frac{1}{\\tan x} = \\frac{\\cos x}{\\sin x}[\/latex], which has period [latex]\\pi[\/latex], vertical asymptotes where [latex]\\sin x = 0[\/latex], and range [latex](-\\infty, \\infty)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>midline<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The horizontal line [latex]y = D[\/latex] that runs through the middle of a sinusoidal graph, located halfway between the maximum and minimum values<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>period of a function<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The length of one complete cycle of a periodic function's graph; for sine and cosine, [latex]P = \\frac{2\\pi}{|B|}[\/latex]; for tangent and cotangent, [latex]P = \\frac{\\pi}{|B|}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>phase shift<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The horizontal displacement of a periodic function from its standard position; calculated as [latex]\\frac{C}{B}[\/latex] in the general form [latex]y = A\\sin(Bx - C) + D[\/latex] or similar<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>reciprocal identity<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">An identity expressing one trigonometric function as the reciprocal of another, such as [latex]\\sec x = \\frac{1}{\\cos x}[\/latex] or [latex]\\csc x = \\frac{1}{\\sin x}[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>secant function<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The function [latex]\\sec x = \\frac{1}{\\cos x}[\/latex], which has period [latex]2\\pi[\/latex], vertical asymptotes where [latex]\\cos x = 0[\/latex], and is undefined at those points<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>sinusoidal function<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A function that has the same general shape as a sine or cosine function<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>stretching\/compressing factor<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The coefficient [latex]|A|[\/latex] in trigonometric functions that do not have amplitude (tangent, cotangent, secant, cosecant); indicates vertical stretch or compression<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>tangent function<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The function [latex]\\tan x = \\frac{\\sin x}{\\cos x}[\/latex], which has period [latex]\\pi[\/latex], vertical asymptotes where [latex]\\cos x = 0[\/latex], and range [latex](-\\infty, \\infty)[\/latex]<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>vertical shift<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The vertical displacement [latex]D[\/latex] that moves the midline of a periodic function up or down from [latex]y = 0[\/latex]<\/p>","rendered":"<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Essential Concepts<\/h2>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Graphs of Sine and Cosine Functions<\/h3>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Key Features\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Period [latex]2\\pi[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Domain: [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Range: [latex][-1, 1][\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">[latex]y = \\sin x[\/latex] is an odd function (symmetric about the origin)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">[latex]y = \\cos x[\/latex] is an even function (symmetric about the y-axis)<\/li>\n<\/ul>\n<\/li>\n<li>General form equation\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li>[latex]y = A\\sin(Bx - C) + D[\/latex]<\/li>\n<li>[latex]y = A\\cos(Bx - C) + D[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li>Transformations:\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li><strong>Amplitude:<\/strong> [latex]|A|[\/latex] measures the vertical stretch\/compression<\/li>\n<li><strong>Period:<\/strong> [latex]P = \\frac{2\\pi}{|B|}[\/latex] is the length of one complete cycle\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">If [latex]|B| > 1[\/latex]: horizontal compression (period less than [latex]2\\pi[\/latex])<\/li>\n<li class=\"whitespace-normal break-words pl-2\">If [latex]|B| < 1[\/latex]: horizontal stretch (period greater than [latex]2\\pi[\/latex])<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words pl-2\"><strong>Phase shift:<\/strong> [latex]\\frac{C}{B}[\/latex] represents horizontal displacement\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">If [latex]C > 0[\/latex]: shift right<\/li>\n<li class=\"whitespace-normal break-words pl-2\">If [latex]C < 0[\/latex]: shift left<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words pl-2\"><strong>Vertical shift:<\/strong> [latex]D[\/latex] is the midline of the graph\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Positive [latex]D[\/latex]: shift up<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Negative [latex]D[\/latex]: shift down<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words pl-2\"><strong>Reflection:<\/strong> If [latex]A < 0[\/latex], the graph is reflected across the x-axis<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Graphing sine and cosine:<\/strong><\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Identify amplitude [latex]|A|[\/latex], period [latex]\\frac{2\\pi}{|B|}[\/latex], phase shift [latex]\\frac{C}{B}[\/latex], and midline [latex]y = D[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Mark key points at quarter-period intervals<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Apply transformations in order: horizontal stretch\/compression, horizontal shift, vertical stretch\/compression, vertical shift, reflection<\/li>\n<\/ol>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Graphs of the Other Trigonometric Functions<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Tangent function [latex]y = A\\tan(Bx - C) + D[\/latex]:<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Period: [latex]P = \\frac{\\pi}{|B|}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Domain: [latex]x \\ne \\frac{C}{B} + \\frac{\\pi}{2|B|}k[\/latex], where [latex]k[\/latex] is an odd integer<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Range: [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Vertical asymptotes: [latex]x = \\frac{C}{B} + \\frac{\\pi}{2|B|}k[\/latex], where [latex]k[\/latex] is an odd integer<\/li>\n<li class=\"whitespace-normal break-words pl-2\">No amplitude; [latex]|A|[\/latex] is the stretching\/compressing factor<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Odd function<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Cotangent function [latex]y = A\\cot(Bx - C) + D[\/latex]:<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Period: [latex]P = \\frac{\\pi}{|B|}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Domain: [latex]x \\ne \\frac{C}{B} + \\frac{\\pi}{|B|}k[\/latex], where [latex]k[\/latex] is an integer<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Range: [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Vertical asymptotes: [latex]x = \\frac{C}{B} + \\frac{\\pi}{|B|}k[\/latex], where [latex]k[\/latex] is an integer<\/li>\n<li class=\"whitespace-normal break-words pl-2\">No amplitude; [latex]|A|[\/latex] is the stretching\/compressing factor<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Odd function<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Secant function [latex]y = A\\sec(Bx - C) + D[\/latex]:<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Period: [latex]P = \\frac{2\\pi}{|B|}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Domain: [latex]x \\ne \\frac{C}{B} + \\frac{\\pi}{2|B|}k[\/latex], where [latex]k[\/latex] is an odd integer<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Range: [latex](-\\infty, -|A| + D] \\cup [|A| + D, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Vertical asymptotes: [latex]x = \\frac{C}{B} + \\frac{\\pi}{2|B|}k[\/latex], where [latex]k[\/latex] is an odd integer<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Even function<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Graph using reciprocal relationship: [latex]\\sec x = \\frac{1}{\\cos x}[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Cosecant function [latex]y = A\\csc(Bx - C) + D[\/latex]:<\/strong><\/p>\n<ul class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-disc flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Period: [latex]P = \\frac{2\\pi}{|B|}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Domain: [latex]x \\ne \\frac{C}{B} + \\frac{\\pi}{|B|}k[\/latex], where [latex]k[\/latex] is an integer<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Range: [latex](-\\infty, -|A| + D] \\cup [|A| + D, \\infty)[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Vertical asymptotes: [latex]x = \\frac{C}{B} + \\frac{\\pi}{|B|}k[\/latex], where [latex]k[\/latex] is an integer<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Odd function<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Graph using reciprocal relationship: [latex]\\csc x = \\frac{1}{\\sin x}[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Graphing tangent and cotangent:<\/strong><\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">Identify stretching factor [latex]|A|[\/latex] and period [latex]\\frac{\\pi}{|B|}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Locate vertical asymptotes<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Plot reference points including the center point and quarter-period points<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Draw the curve approaching asymptotes correctly<\/li>\n<\/ol>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Graphing secant and cosecant:<\/strong><\/p>\n<ol class=\"[li_&amp;]:mb-0 [li_&amp;]:mt-1 [li_&amp;]:gap-1 [&amp;:not(:last-child)_ul]:pb-1 [&amp;:not(:last-child)_ol]:pb-1 list-decimal flex flex-col gap-1 pl-8 mb-3\">\n<li class=\"whitespace-normal break-words pl-2\">First sketch the corresponding cosine or sine function<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Use reciprocal relationship to draw secant or cosecant<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Vertical asymptotes occur where cosine or sine equals zero<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Local extrema of secant\/cosecant occur at extrema of cosine\/sine<\/li>\n<\/ol>\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Key Equations<\/h2>\n<table style=\"width: 92.0541%; height: 506px;\">\n<tbody>\n<tr style=\"height: 22px;\">\n<td style=\"width: 25.6352%; height: 22px;\"><strong>General sine function<\/strong><\/td>\n<td style=\"width: 102.861%; height: 22px;\">[latex]y = A\\sin(Bx - C) + D[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 25.6352%; height: 22px;\"><strong>General cosine function<\/strong><\/td>\n<td style=\"width: 102.861%; height: 22px;\">[latex]y = A\\cos(Bx - C) + D[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 25.6352%; height: 22px;\"><strong>Period of sine\/cosine<\/strong><\/td>\n<td style=\"width: 102.861%; height: 22px;\">[latex]P = \\frac{2\\pi}{|B|}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 25.6352%; height: 22px;\"><strong>Phase shift<\/strong><\/td>\n<td style=\"width: 102.861%; height: 22px;\">[latex]\\frac{C}{B}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 25.6352%; height: 22px;\"><strong>Midline<\/strong><\/td>\n<td style=\"width: 102.861%; height: 22px;\">[latex]y = D[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 25.6352%; height: 22px;\"><strong>General tangent function<\/strong><\/td>\n<td style=\"width: 102.861%; height: 22px;\">[latex]y = A\\tan(Bx - C) + D[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 25.6352%; height: 22px;\"><strong>Tangent asymptotes<\/strong><\/td>\n<td style=\"width: 102.861%; height: 22px;\">[latex]x = \\frac{C}{B} + \\frac{\\pi}{2|B|}k[\/latex], [latex]k[\/latex] odd integer<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"width: 25.6352%; height: 44px;\"><strong>General cotangent function<\/strong><\/td>\n<td style=\"width: 102.861%; height: 44px;\">[latex]y = A\\cot(Bx - C) + D[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 25.6352%; height: 22px;\"><strong>Cotangent asymptotes<\/strong><\/td>\n<td style=\"width: 102.861%; height: 22px;\">[latex]x = \\frac{C}{B} + \\frac{\\pi}{|B|}k[\/latex], [latex]k[\/latex] integer<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"width: 25.6352%; height: 44px;\"><strong>Period of tangent\/cotangent<\/strong><\/td>\n<td style=\"width: 102.861%; height: 44px;\">[latex]P = \\frac{\\pi}{|B|}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 22px;\">\n<td style=\"width: 25.6352%; height: 22px;\"><strong>General secant function<\/strong><\/td>\n<td style=\"width: 102.861%; height: 22px;\">[latex]y = A\\sec(Bx - C) + D[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"width: 25.6352%; height: 44px;\"><strong>General cosecant function<\/strong><\/td>\n<td style=\"width: 102.861%; height: 44px;\">[latex]y = A\\csc(Bx - C) + D[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 44px;\">\n<td style=\"width: 25.6352%; height: 44px;\"><strong>Period of secant\/cosecant<\/strong><\/td>\n<td style=\"width: 102.861%; height: 44px;\">[latex]P = \\frac{2\\pi}{|B|}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Glossary<\/h2>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>amplitude<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The distance from the midline to the maximum or minimum of a sinusoidal function; calculated as [latex]|A|[\/latex] in the general forms [latex]y = A\\sin(Bx - C) + D[\/latex] or [latex]y = A\\cos(Bx - C) + D[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>cosecant function<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The function [latex]\\csc x = \\frac{1}{\\sin x}[\/latex], which has period [latex]2\\pi[\/latex], vertical asymptotes where [latex]\\sin x = 0[\/latex], and is undefined at those points<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>cotangent function<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The function [latex]\\cot x = \\frac{1}{\\tan x} = \\frac{\\cos x}{\\sin x}[\/latex], which has period [latex]\\pi[\/latex], vertical asymptotes where [latex]\\sin x = 0[\/latex], and range [latex](-\\infty, \\infty)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>midline<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The horizontal line [latex]y = D[\/latex] that runs through the middle of a sinusoidal graph, located halfway between the maximum and minimum values<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>period of a function<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The length of one complete cycle of a periodic function&#8217;s graph; for sine and cosine, [latex]P = \\frac{2\\pi}{|B|}[\/latex]; for tangent and cotangent, [latex]P = \\frac{\\pi}{|B|}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>phase shift<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The horizontal displacement of a periodic function from its standard position; calculated as [latex]\\frac{C}{B}[\/latex] in the general form [latex]y = A\\sin(Bx - C) + D[\/latex] or similar<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>reciprocal identity<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">An identity expressing one trigonometric function as the reciprocal of another, such as [latex]\\sec x = \\frac{1}{\\cos x}[\/latex] or [latex]\\csc x = \\frac{1}{\\sin x}[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>secant function<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The function [latex]\\sec x = \\frac{1}{\\cos x}[\/latex], which has period [latex]2\\pi[\/latex], vertical asymptotes where [latex]\\cos x = 0[\/latex], and is undefined at those points<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>sinusoidal function<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A function that has the same general shape as a sine or cosine function<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>stretching\/compressing factor<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The coefficient [latex]|A|[\/latex] in trigonometric functions that do not have amplitude (tangent, cotangent, secant, cosecant); indicates vertical stretch or compression<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>tangent function<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The function [latex]\\tan x = \\frac{\\sin x}{\\cos x}[\/latex], which has period [latex]\\pi[\/latex], vertical asymptotes where [latex]\\cos x = 0[\/latex], and range [latex](-\\infty, \\infty)[\/latex]<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>vertical shift<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The vertical displacement [latex]D[\/latex] that moves the midline of a periodic function up or down from [latex]y = 0[\/latex]<\/p>\n","protected":false},"author":67,"menu_order":1,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":191,"module-header":"cheat_sheet","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/704"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":8,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/704\/revisions"}],"predecessor-version":[{"id":5405,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/704\/revisions\/5405"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/191"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/704\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=704"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=704"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=704"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=704"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}