{"id":1094,"date":"2019-03-07T14:34:11","date_gmt":"2019-03-07T14:34:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/back-matter\/basic-functions-and-identities\/"},"modified":"2024-04-17T21:30:43","modified_gmt":"2024-04-17T21:30:43","slug":"basic-functions-and-identities","status":"publish","type":"back-matter","link":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/back-matter\/basic-functions-and-identities\/","title":{"raw":"Basic Functions and Identities","rendered":"Basic Functions and Identities"},"content":{"raw":"<div id=\"fs-id1751942\" class=\"bc-section section\">\r\n<h3>Graphs of the Parent Functions<\/h3>\r\n<div id=\"CNX_Precalc_Figure_APP_001\" class=\"wp-caption aligncenter\"><span id=\"fs-id1360321\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07143343\/CNX_Precalc_Figure_APP_001.jpg\" alt=\"Three graphs side-by-side. From left to right, graph of the identify function, square function, and square root function. All three graphs extend from -4 to 4 on each axis.\" \/><\/span><\/div>\r\n<div id=\"CNX_Precalc_Figure_APP_002\" class=\"wp-caption aligncenter\"><span id=\"fs-id1694056\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07143348\/CNX_Precalc_Figure_APP_002.jpg\" alt=\"Three graphs side-by-side. From left to right, graph of the cubic function, cube root function, and reciprocal function. All three graphs extend from -4 to 4 on each axis.\" \/><\/span><\/div>\r\n<div id=\"CNX_Precalc_Figure_APP_003\" class=\"wp-caption aligncenter\"><span id=\"fs-id2029174\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07143352\/CNX_Precalc_Figure_APP_003.jpg\" alt=\"Three graphs side-by-side. From left to right, graph of the absolute value function, exponential function, and natural logarithm function. All three graphs extend from -4 to 4 on each axis.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1339419\" class=\"bc-section section\">\r\n<h3>Graphs of the Trigonometric Functions<\/h3>\r\n<div id=\"CNX_Precalc_Figure_APP_004\" class=\"wp-caption aligncenter\" style=\"width: 978px;\"><img class=\"aligncenter wp-image-3256\" src=\"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-content\/uploads\/sites\/3896\/2019\/03\/Book-Picture-Back-Matter-Trig-1.png\" alt=\"Three graphs of trigonometric functions side-by-side. From left to right, graph of the sine function, cosine function, and tangent function. Graphs of the sine and cosine functions extend from negative two pi to two pi on the x-axis and two to negative two on the y-axis. Graph of tangent extends from negative pi to pi on the x-axis and four to negative 4 on the y-axis.\" width=\"978\" height=\"478\" \/><\/div>\r\n<div id=\"CNX_Precalc_Figure_APP_005\" class=\"wp-caption aligncenter\"><span id=\"fs-id2138188\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07143401\/CNX_Precalc_Figure_APP_005.jpg\" alt=\"Three graphs of trigonometric functions side-by-side. From left to right, graph of the cosecant function, secant function, and cotangent function. Graphs of the cosecant function and secant function extend from negative two pi to two pi on the x-axis and ten to negative ten on the y-axis. Graph of cotangent extends from negative two pi to two pi on the x-axis and twenty-five to negative twenty-five on the y-axis.\" \/><\/span><\/div>\r\n<div id=\"CNX_Precalc_Figure_APP_006\" class=\"wp-caption aligncenter\"><span id=\"fs-id2028618\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07143406\/CNX_Precalc_Figure_APP_006n.jpg\" alt=\"Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse sine function, inverse cosine function, and inverse tangent function. Graphs of the inverse sine and inverse tangent extend from negative pi over two to pi over two on the x-axis and pi over two to negative pi over two on the y-axis. Graph of inverse cosine extends from negative pi over two to pi on the x-axis and pi to negative pi over two on the y-axis.\" \/><\/span><\/div>\r\n<div id=\"CNX_Precalc_Figure_APP_007\" class=\"wp-caption aligncenter\"><span id=\"fs-id1609426\"><img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07143410\/CNX_Precalc_Figure_APP_007n.jpg\" alt=\"Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse cosecant function, inverse secant function, and inverse cotangent function.\" \/><\/span><\/div>\r\n<\/div>\r\n<div id=\"fs-id1227092\" class=\"bc-section section\">\r\n<h3>Trigonometric Identities<\/h3>\r\n<table id=\"fs-id2260561\" summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\">Pythagorean Identities<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}{\\mathrm{cos}}^{2}t+{\\mathrm{sin}}^{2}t=1\\\\ 1+{\\mathrm{tan}}^{2}t={\\mathrm{sec}}^{2}t\\\\ 1+{\\mathrm{cot}}^{2}t={\\mathrm{csc}}^{2}t\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Even-Odd Identities<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{cos}\\left(-t\\right)=\\mathrm{cos}\\text{ }t\\hfill \\\\ \\mathrm{sec}\\left(-t\\right)=\\mathrm{sec}\\text{ }t\\hfill \\\\ \\mathrm{sin}\\left(-t\\right)=-\\mathrm{sin}\\text{ }t\\hfill \\\\ \\mathrm{tan}\\left(-t\\right)=-\\mathrm{tan}\\text{ }t\\hfill \\\\ \\mathrm{csc}\\left(-t\\right)=-\\mathrm{csc}\\text{ }t\\hfill \\\\ \\mathrm{cot}\\left(-t\\right)=-\\mathrm{cot}\\text{ }t\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Cofunction Identities<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{cos}\\text{ }t=\\mathrm{sin}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{sin}\\text{ }t=\\mathrm{cos}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{tan}\\text{ }t=\\mathrm{cot}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{cot}\\text{ }t=\\mathrm{tan}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{sec}\\text{ }t=\\mathrm{csc}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{csc}\\text{ }t=\\mathrm{sec}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Fundamental Identities<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{tan}\\text{ }t=\\frac{\\mathrm{sin}\\text{ }t}{\\mathrm{cos}\\text{ }t}\\hfill \\\\ \\mathrm{sec}\\text{ }t=\\frac{1}{\\mathrm{cos}\\text{ }t}\\hfill \\\\ \\mathrm{csc}\\text{ }t=\\frac{1}{\\mathrm{sin}\\text{ }t}\\hfill \\\\ \\text{cot}\\text{ }t=\\frac{1}{\\text{tan}\\text{ }t}=\\frac{\\text{cos}\\text{ }t}{\\text{sin}\\text{ }t}\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Sum and Difference Identities<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{cos}\\left(\\alpha +\\beta \\right)=\\mathrm{cos}\\text{ }\\alpha \\text{ }\\mathrm{cos}\\text{ }\\beta -\\mathrm{sin}\\text{ }\\alpha \\text{ }\\mathrm{sin}\\text{ }\\beta \\hfill \\\\ \\mathrm{cos}\\left(\\alpha -\\beta \\right)=\\mathrm{cos}\\text{ }\\alpha \\text{ }\\mathrm{cos}\\text{ }\\beta +\\mathrm{sin}\\text{ }\\alpha \\text{ }\\mathrm{sin}\\text{ }\\beta \\hfill \\\\ \\mathrm{sin}\\left(\\alpha +\\beta \\right)=\\mathrm{sin}\\text{ }\\alpha \\text{ }\\mathrm{cos}\\text{ }\\beta +\\mathrm{cos}\\text{ }\\alpha \\text{ }\\mathrm{sin}\\text{ }\\beta \\hfill \\\\ \\mathrm{sin}\\left(\\alpha -\\beta \\right)=\\mathrm{sin}\\text{ }\\alpha \\text{ }\\mathrm{cos}\\text{ }\\beta -\\mathrm{cos}\\text{ }\\alpha \\text{ }\\mathrm{sin}\\text{ }\\beta \\hfill \\\\ \\mathrm{tan}\\left(\\alpha +\\beta \\right)=\\frac{\\mathrm{tan}\\text{ }\\alpha +\\mathrm{tan}\\text{ }\\beta }{1-\\mathrm{tan}\\text{ }\\alpha \\text{ }\\mathrm{tan}\\text{ }\\beta }\\hfill \\\\ \\mathrm{tan}\\left(\\alpha -\\beta \\right)=\\frac{\\mathrm{tan}\\text{ }\\alpha -\\mathrm{tan}\\text{ }\\beta }{1+\\mathrm{tan}\\text{ }\\alpha \\text{ }\\mathrm{tan}\\text{ }\\beta }\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Double-Angle Formulas<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{sin}\\left(2\\theta \\right)=2\\text{ }\\mathrm{sin}\\text{ }\\theta \\text{ }\\mathrm{cos}\\text{ }\\theta \\hfill \\\\ \\mathrm{cos}\\left(2\\theta \\right)={\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ \\mathrm{cos}\\left(2\\theta \\right)=1-2\\text{ }{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ \\mathrm{cos}\\left(2\\theta \\right)=2\\text{ }{\\mathrm{cos}}^{2}\\theta -1\\hfill \\\\ \\mathrm{tan}\\left(2\\theta \\right)=\\frac{2\\text{ }\\mathrm{tan}\\text{ }\\theta }{1-{\\mathrm{tan}}^{2}\\theta }\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Half-Angle Formulas<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{sin}\\text{ }\\frac{\\alpha }{2}=\u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\text{ }\\alpha }{2}}\\hfill \\\\ \\mathrm{cos}\\text{ }\\frac{\\alpha }{2}=\u00b1\\sqrt{\\frac{1+\\mathrm{cos}\\text{ }\\alpha }{2}}\\hfill \\\\ \\mathrm{tan}\\text{ }\\frac{\\alpha }{2}=\u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\text{ }\\alpha }{1+\\mathrm{cos}\\text{ }\\alpha }}\\hfill \\\\ \\mathrm{tan}\\text{ }\\frac{\\alpha }{2}=\\frac{\\mathrm{sin}\\text{ }\\alpha }{1+\\mathrm{cos}\\text{ }\\alpha }\\hfill \\\\ \\mathrm{tan}\\text{ }\\frac{\\alpha }{2}=\\frac{1-\\mathrm{cos}\\text{ }\\alpha }{\\mathrm{sin}\\text{ }\\alpha }\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Reduction Formulas<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}{\\mathrm{sin}}^{2}\\theta =\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}\\\\ {\\mathrm{cos}}^{2}\\theta =\\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}\\\\ {\\mathrm{tan}}^{2}\\theta =\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}\\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Product-to-Sum Formulas<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{cos}\\text{ }\\alpha \\text{ }\\mathrm{cos}\\text{ }\\beta =\\frac{1}{2}\\left[\\mathrm{cos}\\left(\\alpha -\\beta \\right)+\\mathrm{cos}\\left(\\alpha +\\beta \\right)\\right]\\hfill \\\\ \\mathrm{sin}\\text{ }\\alpha \\text{ }\\mathrm{cos}\\text{ }\\beta =\\frac{1}{2}\\left[\\mathrm{sin}\\left(\\alpha +\\beta \\right)+\\mathrm{sin}\\left(\\alpha -\\beta \\right)\\right]\\hfill \\\\ \\mathrm{sin}\\text{ }\\alpha \\text{ }\\mathrm{sin}\\text{ }\\beta =\\frac{1}{2}\\left[\\mathrm{cos}\\left(\\alpha -\\beta \\right)-\\mathrm{cos}\\left(\\alpha +\\beta \\right)\\right]\\hfill \\\\ \\mathrm{cos}\\text{ }\\alpha \\text{ }\\mathrm{sin}\\text{ }\\beta =\\frac{1}{2}\\left[\\mathrm{sin}\\left(\\alpha +\\beta \\right)-\\mathrm{sin}\\left(\\alpha -\\beta \\right)\\right]\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Sum-to-Product Formulas<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{sin}\\text{ }\\alpha +\\mathrm{sin}\\text{ }\\beta =2\\text{ }\\mathrm{sin}\\left(\\frac{\\alpha +\\beta }{2}\\right)\\text{ }\\mathrm{cos}\\left(\\frac{\\alpha -\\beta }{2}\\right)\\hfill \\\\ \\mathrm{sin}\\text{ }\\alpha -\\mathrm{sin}\\text{ }\\beta =2\\text{ }\\mathrm{sin}\\left(\\frac{\\alpha -\\beta }{2}\\right)\\text{ }\\mathrm{cos}\\left(\\frac{\\alpha +\\beta }{2}\\right)\\hfill \\\\ \\mathrm{cos}\\text{ }\\alpha -\\mathrm{cos}\\text{ }\\beta =-2\\text{ }\\mathrm{sin}\\left(\\frac{\\alpha +\\beta }{2}\\right)\\text{ }\\mathrm{sin}\\left(\\frac{\\alpha -\\beta }{2}\\right)\\hfill \\\\ \\mathrm{cos}\\text{ }\\alpha +\\mathrm{cos}\\text{ }\\beta =2\\text{ }\\mathrm{cos}\\left(\\frac{\\alpha +\\beta }{2}\\right)\\text{ }\\mathrm{cos}\\left(\\frac{\\alpha -\\beta }{2}\\right)\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Law of Sines<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}\\frac{\\mathrm{sin}\\text{ }\\alpha }{a}=\\frac{\\mathrm{sin}\\text{ }\\beta }{b}=\\frac{\\mathrm{sin}\\text{ }\\gamma }{c}\\hfill \\\\ \\frac{a}{\\mathrm{sin}\\text{ }\\alpha }=\\frac{b}{\\mathrm{sin}\\text{ }\\beta }=\\frac{c}{\\mathrm{sin}\\text{ }\\gamma }\\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\">Law of Cosines<\/td>\r\n<td class=\"border\">[latex]\\begin{array}{l}{a}^{2}={b}^{2}+{c}^{2}-2bc\\text{ }\\mathrm{cos}\\text{ }\\alpha \\hfill \\\\ {b}^{2}={a}^{2}+{c}^{2}-2ac\\text{ }\\mathrm{cos}\\text{ }\\beta \\hfill \\\\ {c}^{2}={a}^{2}+{b}^{2}-2ab\\text{ }\\text{cos}\\text{ }\\gamma \\hfill \\end{array}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>","rendered":"<div id=\"fs-id1751942\" class=\"bc-section section\">\n<h3>Graphs of the Parent Functions<\/h3>\n<div id=\"CNX_Precalc_Figure_APP_001\" class=\"wp-caption aligncenter\"><span id=\"fs-id1360321\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07143343\/CNX_Precalc_Figure_APP_001.jpg\" alt=\"Three graphs side-by-side. From left to right, graph of the identify function, square function, and square root function. All three graphs extend from -4 to 4 on each axis.\" \/><\/span><\/div>\n<div id=\"CNX_Precalc_Figure_APP_002\" class=\"wp-caption aligncenter\"><span id=\"fs-id1694056\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07143348\/CNX_Precalc_Figure_APP_002.jpg\" alt=\"Three graphs side-by-side. From left to right, graph of the cubic function, cube root function, and reciprocal function. All three graphs extend from -4 to 4 on each axis.\" \/><\/span><\/div>\n<div id=\"CNX_Precalc_Figure_APP_003\" class=\"wp-caption aligncenter\"><span id=\"fs-id2029174\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07143352\/CNX_Precalc_Figure_APP_003.jpg\" alt=\"Three graphs side-by-side. From left to right, graph of the absolute value function, exponential function, and natural logarithm function. All three graphs extend from -4 to 4 on each axis.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1339419\" class=\"bc-section section\">\n<h3>Graphs of the Trigonometric Functions<\/h3>\n<div id=\"CNX_Precalc_Figure_APP_004\" class=\"wp-caption aligncenter\" style=\"width: 978px;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3256\" src=\"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-content\/uploads\/sites\/3896\/2019\/03\/Book-Picture-Back-Matter-Trig-1.png\" alt=\"Three graphs of trigonometric functions side-by-side. From left to right, graph of the sine function, cosine function, and tangent function. Graphs of the sine and cosine functions extend from negative two pi to two pi on the x-axis and two to negative two on the y-axis. Graph of tangent extends from negative pi to pi on the x-axis and four to negative 4 on the y-axis.\" width=\"978\" height=\"478\" srcset=\"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-content\/uploads\/sites\/3896\/2019\/03\/Book-Picture-Back-Matter-Trig-1.png 673w, https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-content\/uploads\/sites\/3896\/2019\/03\/Book-Picture-Back-Matter-Trig-1-300x147.png 300w, https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-content\/uploads\/sites\/3896\/2019\/03\/Book-Picture-Back-Matter-Trig-1-65x32.png 65w, https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-content\/uploads\/sites\/3896\/2019\/03\/Book-Picture-Back-Matter-Trig-1-225x110.png 225w, https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-content\/uploads\/sites\/3896\/2019\/03\/Book-Picture-Back-Matter-Trig-1-350x171.png 350w\" sizes=\"auto, (max-width: 978px) 100vw, 978px\" \/><\/div>\n<div id=\"CNX_Precalc_Figure_APP_005\" class=\"wp-caption aligncenter\"><span id=\"fs-id2138188\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07143401\/CNX_Precalc_Figure_APP_005.jpg\" alt=\"Three graphs of trigonometric functions side-by-side. From left to right, graph of the cosecant function, secant function, and cotangent function. Graphs of the cosecant function and secant function extend from negative two pi to two pi on the x-axis and ten to negative ten on the y-axis. Graph of cotangent extends from negative two pi to two pi on the x-axis and twenty-five to negative twenty-five on the y-axis.\" \/><\/span><\/div>\n<div id=\"CNX_Precalc_Figure_APP_006\" class=\"wp-caption aligncenter\"><span id=\"fs-id2028618\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07143406\/CNX_Precalc_Figure_APP_006n.jpg\" alt=\"Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse sine function, inverse cosine function, and inverse tangent function. Graphs of the inverse sine and inverse tangent extend from negative pi over two to pi over two on the x-axis and pi over two to negative pi over two on the y-axis. Graph of inverse cosine extends from negative pi over two to pi on the x-axis and pi to negative pi over two on the y-axis.\" \/><\/span><\/div>\n<div id=\"CNX_Precalc_Figure_APP_007\" class=\"wp-caption aligncenter\"><span id=\"fs-id1609426\"><img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3896\/2019\/03\/07143410\/CNX_Precalc_Figure_APP_007n.jpg\" alt=\"Three graphs of trigonometric functions side-by-side. From left to right, graph of the inverse cosecant function, inverse secant function, and inverse cotangent function.\" \/><\/span><\/div>\n<\/div>\n<div id=\"fs-id1227092\" class=\"bc-section section\">\n<h3>Trigonometric Identities<\/h3>\n<table id=\"fs-id2260561\" summary=\"..\">\n<tbody>\n<tr>\n<td class=\"border\">Pythagorean Identities<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}{\\mathrm{cos}}^{2}t+{\\mathrm{sin}}^{2}t=1\\\\ 1+{\\mathrm{tan}}^{2}t={\\mathrm{sec}}^{2}t\\\\ 1+{\\mathrm{cot}}^{2}t={\\mathrm{csc}}^{2}t\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Even-Odd Identities<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{cos}\\left(-t\\right)=\\mathrm{cos}\\text{ }t\\hfill \\\\ \\mathrm{sec}\\left(-t\\right)=\\mathrm{sec}\\text{ }t\\hfill \\\\ \\mathrm{sin}\\left(-t\\right)=-\\mathrm{sin}\\text{ }t\\hfill \\\\ \\mathrm{tan}\\left(-t\\right)=-\\mathrm{tan}\\text{ }t\\hfill \\\\ \\mathrm{csc}\\left(-t\\right)=-\\mathrm{csc}\\text{ }t\\hfill \\\\ \\mathrm{cot}\\left(-t\\right)=-\\mathrm{cot}\\text{ }t\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Cofunction Identities<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{cos}\\text{ }t=\\mathrm{sin}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{sin}\\text{ }t=\\mathrm{cos}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{tan}\\text{ }t=\\mathrm{cot}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{cot}\\text{ }t=\\mathrm{tan}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{sec}\\text{ }t=\\mathrm{csc}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\\\ \\mathrm{csc}\\text{ }t=\\mathrm{sec}\\left(\\frac{\\pi }{2}-t\\right)\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Fundamental Identities<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{tan}\\text{ }t=\\frac{\\mathrm{sin}\\text{ }t}{\\mathrm{cos}\\text{ }t}\\hfill \\\\ \\mathrm{sec}\\text{ }t=\\frac{1}{\\mathrm{cos}\\text{ }t}\\hfill \\\\ \\mathrm{csc}\\text{ }t=\\frac{1}{\\mathrm{sin}\\text{ }t}\\hfill \\\\ \\text{cot}\\text{ }t=\\frac{1}{\\text{tan}\\text{ }t}=\\frac{\\text{cos}\\text{ }t}{\\text{sin}\\text{ }t}\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Sum and Difference Identities<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{cos}\\left(\\alpha +\\beta \\right)=\\mathrm{cos}\\text{ }\\alpha \\text{ }\\mathrm{cos}\\text{ }\\beta -\\mathrm{sin}\\text{ }\\alpha \\text{ }\\mathrm{sin}\\text{ }\\beta \\hfill \\\\ \\mathrm{cos}\\left(\\alpha -\\beta \\right)=\\mathrm{cos}\\text{ }\\alpha \\text{ }\\mathrm{cos}\\text{ }\\beta +\\mathrm{sin}\\text{ }\\alpha \\text{ }\\mathrm{sin}\\text{ }\\beta \\hfill \\\\ \\mathrm{sin}\\left(\\alpha +\\beta \\right)=\\mathrm{sin}\\text{ }\\alpha \\text{ }\\mathrm{cos}\\text{ }\\beta +\\mathrm{cos}\\text{ }\\alpha \\text{ }\\mathrm{sin}\\text{ }\\beta \\hfill \\\\ \\mathrm{sin}\\left(\\alpha -\\beta \\right)=\\mathrm{sin}\\text{ }\\alpha \\text{ }\\mathrm{cos}\\text{ }\\beta -\\mathrm{cos}\\text{ }\\alpha \\text{ }\\mathrm{sin}\\text{ }\\beta \\hfill \\\\ \\mathrm{tan}\\left(\\alpha +\\beta \\right)=\\frac{\\mathrm{tan}\\text{ }\\alpha +\\mathrm{tan}\\text{ }\\beta }{1-\\mathrm{tan}\\text{ }\\alpha \\text{ }\\mathrm{tan}\\text{ }\\beta }\\hfill \\\\ \\mathrm{tan}\\left(\\alpha -\\beta \\right)=\\frac{\\mathrm{tan}\\text{ }\\alpha -\\mathrm{tan}\\text{ }\\beta }{1+\\mathrm{tan}\\text{ }\\alpha \\text{ }\\mathrm{tan}\\text{ }\\beta }\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Double-Angle Formulas<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{sin}\\left(2\\theta \\right)=2\\text{ }\\mathrm{sin}\\text{ }\\theta \\text{ }\\mathrm{cos}\\text{ }\\theta \\hfill \\\\ \\mathrm{cos}\\left(2\\theta \\right)={\\mathrm{cos}}^{2}\\theta -{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ \\mathrm{cos}\\left(2\\theta \\right)=1-2\\text{ }{\\mathrm{sin}}^{2}\\theta \\hfill \\\\ \\mathrm{cos}\\left(2\\theta \\right)=2\\text{ }{\\mathrm{cos}}^{2}\\theta -1\\hfill \\\\ \\mathrm{tan}\\left(2\\theta \\right)=\\frac{2\\text{ }\\mathrm{tan}\\text{ }\\theta }{1-{\\mathrm{tan}}^{2}\\theta }\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Half-Angle Formulas<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{sin}\\text{ }\\frac{\\alpha }{2}=\u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\text{ }\\alpha }{2}}\\hfill \\\\ \\mathrm{cos}\\text{ }\\frac{\\alpha }{2}=\u00b1\\sqrt{\\frac{1+\\mathrm{cos}\\text{ }\\alpha }{2}}\\hfill \\\\ \\mathrm{tan}\\text{ }\\frac{\\alpha }{2}=\u00b1\\sqrt{\\frac{1-\\mathrm{cos}\\text{ }\\alpha }{1+\\mathrm{cos}\\text{ }\\alpha }}\\hfill \\\\ \\mathrm{tan}\\text{ }\\frac{\\alpha }{2}=\\frac{\\mathrm{sin}\\text{ }\\alpha }{1+\\mathrm{cos}\\text{ }\\alpha }\\hfill \\\\ \\mathrm{tan}\\text{ }\\frac{\\alpha }{2}=\\frac{1-\\mathrm{cos}\\text{ }\\alpha }{\\mathrm{sin}\\text{ }\\alpha }\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Reduction Formulas<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}{\\mathrm{sin}}^{2}\\theta =\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{2}\\\\ {\\mathrm{cos}}^{2}\\theta =\\frac{1+\\mathrm{cos}\\left(2\\theta \\right)}{2}\\\\ {\\mathrm{tan}}^{2}\\theta =\\frac{1-\\mathrm{cos}\\left(2\\theta \\right)}{1+\\mathrm{cos}\\left(2\\theta \\right)}\\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Product-to-Sum Formulas<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{cos}\\text{ }\\alpha \\text{ }\\mathrm{cos}\\text{ }\\beta =\\frac{1}{2}\\left[\\mathrm{cos}\\left(\\alpha -\\beta \\right)+\\mathrm{cos}\\left(\\alpha +\\beta \\right)\\right]\\hfill \\\\ \\mathrm{sin}\\text{ }\\alpha \\text{ }\\mathrm{cos}\\text{ }\\beta =\\frac{1}{2}\\left[\\mathrm{sin}\\left(\\alpha +\\beta \\right)+\\mathrm{sin}\\left(\\alpha -\\beta \\right)\\right]\\hfill \\\\ \\mathrm{sin}\\text{ }\\alpha \\text{ }\\mathrm{sin}\\text{ }\\beta =\\frac{1}{2}\\left[\\mathrm{cos}\\left(\\alpha -\\beta \\right)-\\mathrm{cos}\\left(\\alpha +\\beta \\right)\\right]\\hfill \\\\ \\mathrm{cos}\\text{ }\\alpha \\text{ }\\mathrm{sin}\\text{ }\\beta =\\frac{1}{2}\\left[\\mathrm{sin}\\left(\\alpha +\\beta \\right)-\\mathrm{sin}\\left(\\alpha -\\beta \\right)\\right]\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Sum-to-Product Formulas<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}\\mathrm{sin}\\text{ }\\alpha +\\mathrm{sin}\\text{ }\\beta =2\\text{ }\\mathrm{sin}\\left(\\frac{\\alpha +\\beta }{2}\\right)\\text{ }\\mathrm{cos}\\left(\\frac{\\alpha -\\beta }{2}\\right)\\hfill \\\\ \\mathrm{sin}\\text{ }\\alpha -\\mathrm{sin}\\text{ }\\beta =2\\text{ }\\mathrm{sin}\\left(\\frac{\\alpha -\\beta }{2}\\right)\\text{ }\\mathrm{cos}\\left(\\frac{\\alpha +\\beta }{2}\\right)\\hfill \\\\ \\mathrm{cos}\\text{ }\\alpha -\\mathrm{cos}\\text{ }\\beta =-2\\text{ }\\mathrm{sin}\\left(\\frac{\\alpha +\\beta }{2}\\right)\\text{ }\\mathrm{sin}\\left(\\frac{\\alpha -\\beta }{2}\\right)\\hfill \\\\ \\mathrm{cos}\\text{ }\\alpha +\\mathrm{cos}\\text{ }\\beta =2\\text{ }\\mathrm{cos}\\left(\\frac{\\alpha +\\beta }{2}\\right)\\text{ }\\mathrm{cos}\\left(\\frac{\\alpha -\\beta }{2}\\right)\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Law of Sines<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}\\frac{\\mathrm{sin}\\text{ }\\alpha }{a}=\\frac{\\mathrm{sin}\\text{ }\\beta }{b}=\\frac{\\mathrm{sin}\\text{ }\\gamma }{c}\\hfill \\\\ \\frac{a}{\\mathrm{sin}\\text{ }\\alpha }=\\frac{b}{\\mathrm{sin}\\text{ }\\beta }=\\frac{c}{\\mathrm{sin}\\text{ }\\gamma }\\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td class=\"border\">Law of Cosines<\/td>\n<td class=\"border\">[latex]\\begin{array}{l}{a}^{2}={b}^{2}+{c}^{2}-2bc\\text{ }\\mathrm{cos}\\text{ }\\alpha \\hfill \\\\ {b}^{2}={a}^{2}+{c}^{2}-2ac\\text{ }\\mathrm{cos}\\text{ }\\beta \\hfill \\\\ {c}^{2}={a}^{2}+{b}^{2}-2ab\\text{ }\\text{cos}\\text{ }\\gamma \\hfill \\end{array}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1094\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Basic Functions and Identities. <strong>Authored by<\/strong>: Douglas Hoffman. <strong>Provided by<\/strong>: Openstax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:EyqJ6FuE@4\/Basic-Functions-and-Identities\">https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:EyqJ6FuE@4\/Basic-Functions-and-Identities<\/a>. <strong>Project<\/strong>: Essential Precalcus, Part 2. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":311,"menu_order":1,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Basic Functions and Identities\",\"author\":\"Douglas Hoffman\",\"organization\":\"Openstax\",\"url\":\"https:\/\/cnx.org\/contents\/8si1Yf2B@2.21:EyqJ6FuE@4\/Basic-Functions-and-Identities\",\"project\":\"Essential Precalcus, Part 2\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"back-matter-type":[],"contributor":[],"license":[],"class_list":["post-1094","back-matter","type-back-matter","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/back-matter\/1094","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/back-matter"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/types\/back-matter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/users\/311"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/back-matter\/1094\/revisions"}],"predecessor-version":[{"id":3258,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/back-matter\/1094\/revisions\/3258"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/back-matter\/1094\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/media?parent=1094"}],"wp:term":[{"taxonomy":"back-matter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/pressbooks\/v2\/back-matter-type?post=1094"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/contributor?post=1094"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/suny-dutchess-precalculus\/wp-json\/wp\/v2\/license?post=1094"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}