{"id":702,"date":"2025-07-14T21:14:46","date_gmt":"2025-07-14T21:14:46","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=702"},"modified":"2026-01-15T22:51:32","modified_gmt":"2026-01-15T22:51:32","slug":"module-13-trigonometric-functions-cheat-sheet","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/module-13-trigonometric-functions-cheat-sheet\/","title":{"raw":"Trigonometric Functions: Cheat Sheet","rendered":"Trigonometric Functions: Cheat Sheet"},"content":{"raw":"<h2>Essential Concepts<\/h2>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Angles and Angle Measurement<\/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\">An angle is the union of two rays having a common endpoint<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Greek letters ([latex]\\theta, \\phi, \\alpha, \\beta, \\gamma[\/latex]) are commonly used to represent angle measures<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The measure of an angle is the amount of rotation from the initial side to the terminal side\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\">Counterclockwise rotation produces a positive angle<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Clockwise rotation produces a negative angle<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">An angle is in standard position when its vertex is at the origin and its initial side extends along the positive x-axis<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The coordinate plane is divided into four quadrants, numbered I through IV counterclockwise starting from the positive x-axis<img src=\"https:\/\/openstax.org\/apps\/image-cdn\/v1\/f=webp\/apps\/archive\/20250916.165151\/resources\/b60cd925724939299f52716cbf50b67c9a8ebf83\" alt=\"2.1 The Rectangular Coordinate Systems and Graphs - College Algebra 2e | OpenStax\" \/><\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Degrees and Radians<\/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\">A complete circular rotation contains 360\u00b0<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Always include the degree symbol (\u00b0) or the word \"degrees\"<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">One radian is the measure of a central angle that intercepts an arc equal in length to the radius<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">A full revolution equals [latex]2\\pi[\/latex] radians<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">When an angle is described without units, it refers to radian measure<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Use the proportion: [latex]\\frac{\\theta}{180} = \\frac{\\theta^R}{\\pi}[\/latex]\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\">To convert radians to degrees: multiply by [latex]\\frac{180}{\\pi}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">To convert degrees to radians: multiply by [latex]\\frac{\\pi}{180}[\/latex]<\/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>DMS Form (Degrees, Minutes, Seconds)<\/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\">One degree equals 60 minutes: [latex]1\u00b0 = 60'[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">One minute equals 60 seconds: [latex]1' = 60\"[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Written as [latex]D\u00b0 M' S\"[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Coterminal Angles<\/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\">Two angles in standard position with the same terminal side\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\">To find coterminal angles in degrees: add or subtract multiples of 360\u00b0<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">To find coterminal angles in radians: add or subtract multiples of [latex]2\\pi[\/latex]<\/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>Reference Angles<\/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\">The measure of the smallest positive acute angle formed by the terminal side and the horizontal axis\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\">Always between 0\u00b0 and 90\u00b0 (or 0 and [latex]\\frac{\\pi}{2}[\/latex] radians)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">For Quadrant II or III: [latex]|\\pi - t|[\/latex] or [latex]|180\u00b0 - t|[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">For Quadrant IV: [latex]2\\pi - t[\/latex] or [latex]360\u00b0 - t[\/latex]<\/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>Complementary and Supplementary Angles<\/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\">Complementary angles sum to 90\u00b0 (or [latex]\\frac{\\pi}{2}[\/latex] radians)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Supplementary angles sum to 180\u00b0 (or [latex]\\pi[\/latex] radians)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">An angle can only have a complement if it measures less than 90\u00b0<\/li>\r\n<\/ul>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Arcs and Sectors<\/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\">In a circle of radius [latex]r[\/latex], the length of an arc [latex]s[\/latex] subtended by angle [latex]\\theta[\/latex] (in radians) is: [latex]s = r\\theta[\/latex]. The angle must be in radians for this formula to work<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">The area of a sector with radius [latex]r[\/latex] and central angle [latex]\\theta[\/latex] (in radians) is: [latex]A = \\frac{1}{2}\\theta r^2[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Linear and Angular Speed<\/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\">Linear speed [latex]v[\/latex] is the distance traveled per unit time: [latex]v = \\frac{s}{t}[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Angular speed [latex]\\omega[\/latex] is the angular rotation per unit time: [latex]\\omega = \\frac{\\theta}{t}[\/latex]<\/li>\r\n<\/ul>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Sine and Cosine Functions<\/h3>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Unit Circle<\/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\">A circle centered at the origin with radius 1<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Any point on the unit circle can be written as [latex](x, y) = (\\cos t, \\sin t)[\/latex]<a href=\"https:\/\/courses.lumenlearning.com\/precalctwoxmaster\/wp-content\/uploads\/sites\/145\/2015\/11\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\"><img class=\"aligncenter wp-image-12625 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003609\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\" alt=\"f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3+IMAGE+IMAGE\" width=\"800\" height=\"728\" \/><\/a><\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Sine and Cosine Functions<\/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\">For a point [latex](x, y)[\/latex] on the unit circle corresponding to angle [latex]t[\/latex]:\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\">[latex]\\cos t = x[\/latex] (the x-coordinate)<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]\\sin t = y[\/latex] (the y-coordinate)<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Output values are always between -1 and 1<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Pythagorean Identity<\/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\">[latex]\\cos^2 t + \\sin^2 t = 1[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Derived from the unit circle equation [latex]x^2 + y^2 = 1[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Special Angles and Their Values<\/strong><\/p>\r\n\r\n<div class=\"overflow-x-auto w-full px-2 mb-6\">\r\n<table class=\"min-w-full border-collapse text-sm leading-[1.7] whitespace-normal\">\r\n<thead class=\"text-left\">\r\n<tr>\r\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\">Angle<\/th>\r\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\">0<\/th>\r\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\">[latex]\\frac{\\pi}{6}[\/latex] (30\u00b0)<\/th>\r\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\">[latex]\\frac{\\pi}{4}[\/latex] (45\u00b0)<\/th>\r\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\">[latex]\\frac{\\pi}{3}[\/latex] (60\u00b0)<\/th>\r\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\">[latex]\\frac{\\pi}{2}[\/latex] (90\u00b0)<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">Cosine<\/td>\r\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">1<\/td>\r\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">0<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">Sine<\/td>\r\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">0<\/td>\r\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">[latex]\\frac{1}{2}[\/latex]<\/td>\r\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\r\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\r\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">The Other Trigonometric 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\">Tangent: [latex]\\tan t = \\frac{y}{x} = \\frac{\\sin t}{\\cos t}[\/latex] (where [latex]x \\neq 0[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Secant: [latex]\\sec t = \\frac{1}{x} = \\frac{1}{\\cos t}[\/latex] (where [latex]x \\neq 0[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Cosecant: [latex]\\csc t = \\frac{1}{y} = \\frac{1}{\\sin t}[\/latex] (where [latex]y \\neq 0[\/latex])<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Cotangent: [latex]\\cot t = \\frac{x}{y} = \\frac{\\cos t}{\\sin t}[\/latex] (where [latex]y \\neq 0[\/latex])<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Even and Odd Functions<\/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\">Even functions (symmetric about y-axis): [latex]\\cos(-t) = \\cos t[\/latex] and [latex]\\sec(-t) = \\sec t[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Odd functions (symmetric about origin): [latex]\\sin(-t) = -\\sin t[\/latex], [latex]\\tan(-t) = -\\tan t[\/latex], [latex]\\csc(-t) = -\\csc t[\/latex], [latex]\\cot(-t) = -\\cot t[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Alternate Pythagorean Identities<\/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\">[latex]1 + \\tan^2 t = \\sec^2 t[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">[latex]\\cot^2 t + 1 = \\csc^2 t[\/latex]<\/li>\r\n<\/ul>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Period of Trigonometric Functions<\/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\">A period is the shortest interval over which a function completes one full cycle<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Sine, cosine, secant, and cosecant have period [latex]2\\pi[\/latex]<\/li>\r\n \t<li class=\"whitespace-normal break-words pl-2\">Tangent and cotangent have period [latex]\\pi[\/latex]<\/li>\r\n<\/ul>\r\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Key Equations<\/h2>\r\n<table style=\"width: 88.983%;\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 26.7956%;\"><strong>Arc length<\/strong><\/td>\r\n<td style=\"width: 125.69%;\">[latex]s = r\\theta[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 26.7956%;\"><strong>Area of a sector<\/strong><\/td>\r\n<td style=\"width: 125.69%;\">[latex]A = \\frac{1}{2}\\theta r^2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 26.7956%;\"><strong>Angular speed<\/strong><\/td>\r\n<td style=\"width: 125.69%;\">[latex]\\omega = \\frac{\\theta}{t}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 26.7956%;\"><strong>Linear speed<\/strong><\/td>\r\n<td style=\"width: 125.69%;\">[latex]v = \\frac{s}{t}[\/latex] or [latex]v = r\\omega[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 26.7956%;\"><strong>Pythagorean Identity<\/strong><\/td>\r\n<td style=\"width: 125.69%;\">[latex]\\cos^2 t + \\sin^2 t = 1[\/latex]\r\n\r\n[latex]1 + \\tan^2 t = \\sec^2 t[\/latex]\r\n\r\n[latex]\\cot^2 t + 1 = \\csc^2 t[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 26.7956%;\"><strong>Tangent function<\/strong><\/td>\r\n<td style=\"width: 125.69%;\">[latex]\\tan t = \\frac{\\sin t}{\\cos t}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 26.7956%;\"><strong>Secant function<\/strong><\/td>\r\n<td style=\"width: 125.69%;\">[latex]\\sec t = \\frac{1}{\\cos t}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 26.7956%;\"><strong>Cosecant function<\/strong><\/td>\r\n<td style=\"width: 125.69%;\">[latex]\\csc t = \\frac{1}{\\sin t}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 26.7956%;\"><strong>Cotangent function<\/strong><\/td>\r\n<td style=\"width: 125.69%;\">[latex]\\cot t = \\frac{\\cos t}{\\sin t}[\/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>angle<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The union of two rays having a common endpoint called the vertex.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>angular speed<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The angular rotation [latex]\\theta[\/latex] per unit time [latex]t[\/latex], denoted [latex]\\omega = \\frac{\\theta}{t}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>arc length<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The length of the arc [latex]s[\/latex] along a circle subtended by a central angle, calculated as [latex]s = r\\theta[\/latex] where [latex]\\theta[\/latex] is in radians.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>complementary angles<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Two angles whose measures add up to 90\u00b0 (or [latex]\\frac{\\pi}{2}[\/latex] radians).<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>cosecant<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The reciprocal of the sine function: [latex]\\csc t = \\frac{1}{\\sin t}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>cosine<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">For a point [latex](x, y)[\/latex] on the unit circle corresponding to angle [latex]t[\/latex], [latex]\\cos t = x[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>cotangent<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The reciprocal of the tangent function: [latex]\\cot t = \\frac{\\cos t}{\\sin t}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>coterminal angles<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Two angles in standard position that have the same terminal side.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>degree<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A unit of angle measurement where one degree is [latex]\\frac{1}{360}[\/latex] of a circular rotation.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>DMS form<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Notation for expressing angles using degrees, minutes, and seconds (D\u00b0 M' S\"), where 1\u00b0 = 60' and 1' = 60\".<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>even function<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A function where [latex]f(-x) = f(x)[\/latex]. Cosine and secant are even trigonometric functions.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>initial side<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The fixed ray from which angle measurement begins, extending along the positive x-axis in standard position.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>linear speed<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The distance traveled per unit time, calculated as [latex]v = \\frac{s}{t}[\/latex] or [latex]v = r\\omega[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>odd function<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A function where [latex]f(-x) = -f(x)[\/latex]. Sine, tangent, cosecant, and cotangent are odd trigonometric functions.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>period<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The shortest interval [latex]P[\/latex] over which a function completes one full cycle, where [latex]f(x + P) = f(x)[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>quadrantal angles<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Angles whose terminal side lies on an axis, including 0\u00b0, 90\u00b0, 180\u00b0, 270\u00b0, and 360\u00b0.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>radian<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The measure of a central angle that intercepts an arc equal in length to the radius of the circle. A full revolution equals [latex]2\\pi[\/latex] radians.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>reference angle<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The measure of the smallest positive acute angle formed by the terminal side of an angle and the horizontal axis.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>secant<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The reciprocal of the cosine function: [latex]\\sec t = \\frac{1}{\\cos t}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>sector<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A region of a circle bounded by two radii and the intercepted arc, with area [latex]A = \\frac{1}{2}\\theta r^2[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>sine<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">For a point [latex](x, y)[\/latex] on the unit circle corresponding to angle [latex]t[\/latex], [latex]\\sin t = y[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>standard position<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">An angle positioned with its vertex at the origin and its initial side extending along the positive x-axis.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>supplementary angles<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Two angles whose measures add up to 180\u00b0 (or [latex]\\pi[\/latex] radians).<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>tangent<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The ratio of sine to cosine: [latex]\\tan t = \\frac{\\sin t}{\\cos t}[\/latex].<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>terminal side<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The rotated ray of an angle after rotation from the initial side.<\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>unit circle<\/strong><\/p>\r\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A circle centered at the origin with radius 1, where [latex]x^2 + y^2 = 1[\/latex].<\/p>","rendered":"<h2>Essential Concepts<\/h2>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Angles and Angle Measurement<\/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\">An angle is the union of two rays having a common endpoint<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Greek letters ([latex]\\theta, \\phi, \\alpha, \\beta, \\gamma[\/latex]) are commonly used to represent angle measures<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The measure of an angle is the amount of rotation from the initial side to the terminal side\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\">Counterclockwise rotation produces a positive angle<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Clockwise rotation produces a negative angle<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words pl-2\">An angle is in standard position when its vertex is at the origin and its initial side extends along the positive x-axis<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The coordinate plane is divided into four quadrants, numbered I through IV counterclockwise starting from the positive x-axis<img decoding=\"async\" src=\"https:\/\/openstax.org\/apps\/image-cdn\/v1\/f=webp\/apps\/archive\/20250916.165151\/resources\/b60cd925724939299f52716cbf50b67c9a8ebf83\" alt=\"2.1 The Rectangular Coordinate Systems and Graphs - College Algebra 2e | OpenStax\" \/><\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Degrees and Radians<\/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\">A complete circular rotation contains 360\u00b0<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Always include the degree symbol (\u00b0) or the word &#8220;degrees&#8221;<\/li>\n<li class=\"whitespace-normal break-words pl-2\">One radian is the measure of a central angle that intercepts an arc equal in length to the radius<\/li>\n<li class=\"whitespace-normal break-words pl-2\">A full revolution equals [latex]2\\pi[\/latex] radians<\/li>\n<li class=\"whitespace-normal break-words pl-2\">When an angle is described without units, it refers to radian measure<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Use the proportion: [latex]\\frac{\\theta}{180} = \\frac{\\theta^R}{\\pi}[\/latex]\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\">To convert radians to degrees: multiply by [latex]\\frac{180}{\\pi}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">To convert degrees to radians: multiply by [latex]\\frac{\\pi}{180}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>DMS Form (Degrees, Minutes, Seconds)<\/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\">One degree equals 60 minutes: [latex]1\u00b0 = 60'[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">One minute equals 60 seconds: [latex]1' = 60\"[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Written as [latex]D\u00b0 M' S\"[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Coterminal Angles<\/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\">Two angles in standard position with the same terminal side\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\">To find coterminal angles in degrees: add or subtract multiples of 360\u00b0<\/li>\n<li class=\"whitespace-normal break-words pl-2\">To find coterminal angles in radians: add or subtract multiples of [latex]2\\pi[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Reference Angles<\/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\">The measure of the smallest positive acute angle formed by the terminal side and the horizontal axis\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\">Always between 0\u00b0 and 90\u00b0 (or 0 and [latex]\\frac{\\pi}{2}[\/latex] radians)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">For Quadrant II or III: [latex]|\\pi - t|[\/latex] or [latex]|180\u00b0 - t|[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">For Quadrant IV: [latex]2\\pi - t[\/latex] or [latex]360\u00b0 - t[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Complementary and Supplementary Angles<\/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\">Complementary angles sum to 90\u00b0 (or [latex]\\frac{\\pi}{2}[\/latex] radians)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Supplementary angles sum to 180\u00b0 (or [latex]\\pi[\/latex] radians)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">An angle can only have a complement if it measures less than 90\u00b0<\/li>\n<\/ul>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Arcs and Sectors<\/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\">In a circle of radius [latex]r[\/latex], the length of an arc [latex]s[\/latex] subtended by angle [latex]\\theta[\/latex] (in radians) is: [latex]s = r\\theta[\/latex]. The angle must be in radians for this formula to work<\/li>\n<li class=\"whitespace-normal break-words pl-2\">The area of a sector with radius [latex]r[\/latex] and central angle [latex]\\theta[\/latex] (in radians) is: [latex]A = \\frac{1}{2}\\theta r^2[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Linear and Angular Speed<\/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\">Linear speed [latex]v[\/latex] is the distance traveled per unit time: [latex]v = \\frac{s}{t}[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Angular speed [latex]\\omega[\/latex] is the angular rotation per unit time: [latex]\\omega = \\frac{\\theta}{t}[\/latex]<\/li>\n<\/ul>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">Sine and Cosine Functions<\/h3>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Unit Circle<\/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\">A circle centered at the origin with radius 1<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Any point on the unit circle can be written as [latex](x, y) = (\\cos t, \\sin t)[\/latex]<a href=\"https:\/\/courses.lumenlearning.com\/precalctwoxmaster\/wp-content\/uploads\/sites\/145\/2015\/11\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-12625 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003609\/f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3-IMAGE-IMAGE.png\" alt=\"f-d-43392176e093fa07f39e1f3687226d4d751b809be928c16abac5dcb3+IMAGE+IMAGE\" width=\"800\" height=\"728\" \/><\/a><\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Sine and Cosine Functions<\/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\">For a point [latex](x, y)[\/latex] on the unit circle corresponding to angle [latex]t[\/latex]:\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\">[latex]\\cos t = x[\/latex] (the x-coordinate)<\/li>\n<li class=\"whitespace-normal break-words pl-2\">[latex]\\sin t = y[\/latex] (the y-coordinate)<\/li>\n<\/ul>\n<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Output values are always between -1 and 1<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>The Pythagorean Identity<\/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\">[latex]\\cos^2 t + \\sin^2 t = 1[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Derived from the unit circle equation [latex]x^2 + y^2 = 1[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Special Angles and Their Values<\/strong><\/p>\n<div class=\"overflow-x-auto w-full px-2 mb-6\">\n<table class=\"min-w-full border-collapse text-sm leading-[1.7] whitespace-normal\">\n<thead class=\"text-left\">\n<tr>\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\">Angle<\/th>\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\">0<\/th>\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\">[latex]\\frac{\\pi}{6}[\/latex] (30\u00b0)<\/th>\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\">[latex]\\frac{\\pi}{4}[\/latex] (45\u00b0)<\/th>\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\">[latex]\\frac{\\pi}{3}[\/latex] (60\u00b0)<\/th>\n<th class=\"text-text-100 border-b-0.5 border-border-300\/60 py-2 pr-4 align-top font-bold\">[latex]\\frac{\\pi}{2}[\/latex] (90\u00b0)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">Cosine<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">1<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">0<\/td>\n<\/tr>\n<tr>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">Sine<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">0<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">[latex]\\frac{\\sqrt{2}}{2}[\/latex]<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">[latex]\\frac{\\sqrt{3}}{2}[\/latex]<\/td>\n<td class=\"border-b-0.5 border-border-300\/30 py-2 pr-4 align-top\">1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3 class=\"text-text-100 mt-2 -mb-1 text-base font-bold\">The Other Trigonometric 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\">Tangent: [latex]\\tan t = \\frac{y}{x} = \\frac{\\sin t}{\\cos t}[\/latex] (where [latex]x \\neq 0[\/latex])<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Secant: [latex]\\sec t = \\frac{1}{x} = \\frac{1}{\\cos t}[\/latex] (where [latex]x \\neq 0[\/latex])<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Cosecant: [latex]\\csc t = \\frac{1}{y} = \\frac{1}{\\sin t}[\/latex] (where [latex]y \\neq 0[\/latex])<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Cotangent: [latex]\\cot t = \\frac{x}{y} = \\frac{\\cos t}{\\sin t}[\/latex] (where [latex]y \\neq 0[\/latex])<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Even and Odd Functions<\/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\">Even functions (symmetric about y-axis): [latex]\\cos(-t) = \\cos t[\/latex] and [latex]\\sec(-t) = \\sec t[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Odd functions (symmetric about origin): [latex]\\sin(-t) = -\\sin t[\/latex], [latex]\\tan(-t) = -\\tan t[\/latex], [latex]\\csc(-t) = -\\csc t[\/latex], [latex]\\cot(-t) = -\\cot t[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Alternate Pythagorean Identities<\/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\">[latex]1 + \\tan^2 t = \\sec^2 t[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">[latex]\\cot^2 t + 1 = \\csc^2 t[\/latex]<\/li>\n<\/ul>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>Period of Trigonometric Functions<\/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\">A period is the shortest interval over which a function completes one full cycle<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Sine, cosine, secant, and cosecant have period [latex]2\\pi[\/latex]<\/li>\n<li class=\"whitespace-normal break-words pl-2\">Tangent and cotangent have period [latex]\\pi[\/latex]<\/li>\n<\/ul>\n<h2 class=\"text-text-100 mt-3 -mb-1 text-[1.125rem] font-bold\">Key Equations<\/h2>\n<table style=\"width: 88.983%;\">\n<tbody>\n<tr>\n<td style=\"width: 26.7956%;\"><strong>Arc length<\/strong><\/td>\n<td style=\"width: 125.69%;\">[latex]s = r\\theta[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 26.7956%;\"><strong>Area of a sector<\/strong><\/td>\n<td style=\"width: 125.69%;\">[latex]A = \\frac{1}{2}\\theta r^2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 26.7956%;\"><strong>Angular speed<\/strong><\/td>\n<td style=\"width: 125.69%;\">[latex]\\omega = \\frac{\\theta}{t}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 26.7956%;\"><strong>Linear speed<\/strong><\/td>\n<td style=\"width: 125.69%;\">[latex]v = \\frac{s}{t}[\/latex] or [latex]v = r\\omega[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 26.7956%;\"><strong>Pythagorean Identity<\/strong><\/td>\n<td style=\"width: 125.69%;\">[latex]\\cos^2 t + \\sin^2 t = 1[\/latex]<\/p>\n<p>[latex]1 + \\tan^2 t = \\sec^2 t[\/latex]<\/p>\n<p>[latex]\\cot^2 t + 1 = \\csc^2 t[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 26.7956%;\"><strong>Tangent function<\/strong><\/td>\n<td style=\"width: 125.69%;\">[latex]\\tan t = \\frac{\\sin t}{\\cos t}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 26.7956%;\"><strong>Secant function<\/strong><\/td>\n<td style=\"width: 125.69%;\">[latex]\\sec t = \\frac{1}{\\cos t}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 26.7956%;\"><strong>Cosecant function<\/strong><\/td>\n<td style=\"width: 125.69%;\">[latex]\\csc t = \\frac{1}{\\sin t}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 26.7956%;\"><strong>Cotangent function<\/strong><\/td>\n<td style=\"width: 125.69%;\">[latex]\\cot t = \\frac{\\cos t}{\\sin t}[\/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>angle<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The union of two rays having a common endpoint called the vertex.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>angular speed<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The angular rotation [latex]\\theta[\/latex] per unit time [latex]t[\/latex], denoted [latex]\\omega = \\frac{\\theta}{t}[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>arc length<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The length of the arc [latex]s[\/latex] along a circle subtended by a central angle, calculated as [latex]s = r\\theta[\/latex] where [latex]\\theta[\/latex] is in radians.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>complementary angles<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Two angles whose measures add up to 90\u00b0 (or [latex]\\frac{\\pi}{2}[\/latex] radians).<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>cosecant<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The reciprocal of the sine function: [latex]\\csc t = \\frac{1}{\\sin t}[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>cosine<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">For a point [latex](x, y)[\/latex] on the unit circle corresponding to angle [latex]t[\/latex], [latex]\\cos t = x[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>cotangent<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The reciprocal of the tangent function: [latex]\\cot t = \\frac{\\cos t}{\\sin t}[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>coterminal angles<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Two angles in standard position that have the same terminal side.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>degree<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A unit of angle measurement where one degree is [latex]\\frac{1}{360}[\/latex] of a circular rotation.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>DMS form<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Notation for expressing angles using degrees, minutes, and seconds (D\u00b0 M&#8217; S&#8221;), where 1\u00b0 = 60&#8242; and 1&#8242; = 60&#8243;.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>even function<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A function where [latex]f(-x) = f(x)[\/latex]. Cosine and secant are even trigonometric functions.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>initial side<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The fixed ray from which angle measurement begins, extending along the positive x-axis in standard position.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>linear speed<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The distance traveled per unit time, calculated as [latex]v = \\frac{s}{t}[\/latex] or [latex]v = r\\omega[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>odd function<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A function where [latex]f(-x) = -f(x)[\/latex]. Sine, tangent, cosecant, and cotangent are odd trigonometric functions.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>period<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The shortest interval [latex]P[\/latex] over which a function completes one full cycle, where [latex]f(x + P) = f(x)[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>quadrantal angles<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Angles whose terminal side lies on an axis, including 0\u00b0, 90\u00b0, 180\u00b0, 270\u00b0, and 360\u00b0.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>radian<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The measure of a central angle that intercepts an arc equal in length to the radius of the circle. A full revolution equals [latex]2\\pi[\/latex] radians.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>reference angle<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The measure of the smallest positive acute angle formed by the terminal side of an angle and the horizontal axis.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>secant<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The reciprocal of the cosine function: [latex]\\sec t = \\frac{1}{\\cos t}[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>sector<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A region of a circle bounded by two radii and the intercepted arc, with area [latex]A = \\frac{1}{2}\\theta r^2[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>sine<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">For a point [latex](x, y)[\/latex] on the unit circle corresponding to angle [latex]t[\/latex], [latex]\\sin t = y[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>standard position<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">An angle positioned with its vertex at the origin and its initial side extending along the positive x-axis.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>supplementary angles<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">Two angles whose measures add up to 180\u00b0 (or [latex]\\pi[\/latex] radians).<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>tangent<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The ratio of sine to cosine: [latex]\\tan t = \\frac{\\sin t}{\\cos t}[\/latex].<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>terminal side<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">The rotated ray of an angle after rotation from the initial side.<\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\"><strong>unit circle<\/strong><\/p>\n<p class=\"font-claude-response-body break-words whitespace-normal leading-[1.7]\">A circle centered at the origin with radius 1, where [latex]x^2 + y^2 = 1[\/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":178,"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\/702"}],"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":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/702\/revisions"}],"predecessor-version":[{"id":5400,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/702\/revisions\/5400"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/178"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/702\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=702"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=702"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=702"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=702"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}