{"id":3743,"date":"2021-08-08T06:37:02","date_gmt":"2021-08-08T06:37:02","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3743"},"modified":"2021-11-30T19:35:03","modified_gmt":"2021-11-30T14:05:03","slug":"formula-and-graph-of-trigonometric-functions","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/formula-and-graph-of-trigonometric-functions\/","title":{"rendered":"Graph of Trigonometric Functions &#8211; Domain &#038; Range"},"content":{"rendered":"<p>Here, you will learn graph of trigonometric functions and domain &amp; range of trigonometric functions.<\/p>\n<h2 id=\"subheading4\" class=\"_mPS2id-t mPS2id-target mPS2id-target-first mPS2id-target-last\">Graph of Trigonometric Functions :<\/h2>\n\n\n<div class=\"row1\">\n        <div class=\"column1\">\n          <img decoding=\"async\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/08\/sinx.png?ssl=1\" style=\"max-width:90%;height:150px;\" data-recalc-dims=\"1\">\n          <p style=\"font-size:18px;font-family:lucida;text-align:center;\"><b>y = sinx<\/b><\/p>\n        <\/div>\n        <div class=\"column1\">\n          <img decoding=\"async\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/08\/cosx.png?ssl=1\" style=\"max-width:90%;height:150px;\" data-recalc-dims=\"1\">\n          <p style=\"font-size:18px;font-family:lucida;text-align:center;\"><b>y = cosx<\/b><\/p>\n        <\/div>\n      <\/div><br>\n\n      <div class=\"row1\">\n        <div class=\"column1\">\n          <img decoding=\"async\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/08\/tanx.png?ssl=1\" style=\"max-width:90%;height:390px;\" data-recalc-dims=\"1\">\n          <p style=\"font-size:18px;font-family:lucida;text-align:center;\"><b>y = tanx<\/b><\/p>\n        <\/div>\n        <div class=\"column1\">\n          <img decoding=\"async\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/08\/cotx.png?ssl=1\" style=\"max-width:90%;height:390px;\" data-recalc-dims=\"1\">\n          <p style=\"font-size:18px;font-family:lucida;text-align:center;\"><b>y = cotx<\/b><\/p>\n        <\/div>\n      <\/div><br>\n\n      <div class=\"row1\">\n        <div class=\"column1\">\n          <img decoding=\"async\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/08\/secx.png?ssl=1\" style=\"max-width:90%;height:220px;\" data-recalc-dims=\"1\">\n          <p style=\"font-size:18px;font-family:lucida;text-align:center;\"><b>y = secx<\/b><\/p>\n        <\/div>\n        <div class=\"column1\">\n          <img decoding=\"async\" src=\"https:\/\/i0.wp.com\/mathemerize.com\/wp-content\/uploads\/2021\/08\/cosecx.png?ssl=1\" style=\"max-width:90%;height:220px;\" data-recalc-dims=\"1\">\n          <p style=\"font-size:18px;font-family:lucida;text-align:center;\"><b>y = cosecx<\/b><\/p>\n        <\/div>\n      <\/div><br>\n\n\n<h2 id=\"subheading3\" class=\"_mPS2id-t mPS2id-target mPS2id-target-first mPS2id-target-last\">Values of T-Ratio of some standard angles<\/h2>\n\n\n<p>\n               <\/p><table style=\"width:100%;font-size:16px;font-family:lucida;text-align:center;\">\n                <tbody><tr>\n                  <td>Angles<br>T-Ratio<\/td>\n                  <td>0<\/td>\n                  <td>\\(\\pi\\over 6\\)<\/td>\n                  <td>\\(\\pi\\over 4\\)<\/td>\n                  <td>\\(\\pi\\over 3\\)<\/td>\n                  <td>\\(\\pi\\over 2\\)<\/td>\n                  <td>\\(\\pi\\)<\/td>\n                <\/tr>\n                <tr>\n                  <td>\\(sin\\theta\\)<\/td>\n                  <td>0<\/td>\n                  <td>\\(1\\over 2\\)<\/td>\n                  <td>\\(1\\over \\sqrt{2}\\)<\/td>\n                  <td>\\(\\sqrt{3}\\over 2\\)<\/td>\n                  <td>1<\/td>\n                  <td>0<\/td>\n                <\/tr>\n                <tr>\n                  <td>\\(cos\\theta\\)<\/td>\n                  <td>1<\/td>\n                  <td>\\(\\sqrt{3}\\over 2\\)<\/td>\n                  <td>\\(1\\over \\sqrt{2}\\)<\/td>\n                  <td>\\(1\\over 2\\)<\/td>\n                  <td>0<\/td>\n                  <td>-1<\/td>\n                <\/tr>\n                <tr>\n                  <td>\\(tan\\theta\\)<\/td>\n                  <td>0<\/td>\n                  <td>\\(1\\over \\sqrt{3}\\)<\/td>\n                  <td>1<\/td>\n                  <td>\\(\\sqrt{3}\\)<\/td>\n                  <td>N.D<\/td>\n                  <td>0<\/td>\n                <\/tr>\n                <tr>\n                  <td>\\(cot\\theta\\)<\/td>\n                  <td>N.D<\/td>\n                  <td>\\(\\sqrt{3}\\)<\/td>\n                  <td>1<\/td>\n                  <td>\\(1\\over \\sqrt{3}\\)<\/td>\n                  <td>0<\/td>\n                  <td>N.D<\/td>\n                <\/tr>\n                <tr>\n                  <td>\\(sec\\theta\\)<\/td>\n                  <td>1<\/td>\n                  <td>\\(2\\over \\sqrt{3}\\)<\/td>\n                  <td>\\(\\sqrt{2}\\)<\/td>\n                  <td>2<\/td>\n                  <td>N.D<\/td>\n                  <td>-1<\/td>\n                <\/tr>\n                <tr>\n                  <td>\\(cosec\\theta\\)<\/td>\n                  <td>N.D<\/td>\n                  <td>2<\/td>\n                  <td>\\(\\sqrt{2}\\)<\/td>\n                  <td>\\(2\\over \\sqrt{3}\\)<\/td>\n                  <td>1<\/td>\n                  <td>N.D<\/td>\n                <\/tr>\n                <\/tbody><\/table><br> N.D = Not defined<p><\/p>\n\n\n<h2>Domain, Ranges and Periodicity of Trigonometric function<\/h2>\n\n\n<table style=\"width:100%;font-size:16px;font-family:lucida;text-align:center;\">\n               <tbody><tr>\n                  <td>T-Ratio<\/td>\n                  <td>Domain<\/td>\n                  <td>Range<\/td>\n                  <td>Period<\/td>\n                <\/tr>\n                <tr>\n                  <td>sin x<\/td>\n                  <td>R<\/td>\n                  <td>[-1, 1]<\/td>\n                  <td>\\(2\\pi\\)<\/td>\n                <\/tr>\n                <tr>\n                  <td>cos x<\/td>\n                  <td>R<\/td>\n                  <td>[-1, 1]<\/td>\n                  <td>\\(2\\pi\\)<\/td>\n                <\/tr>\n                <tr>\n                  <td>tan x<\/td>\n                  <td>R &#8211; {(2n+1)\\(\\pi\/2\\); n \\(\\in\\) I}<\/td>\n                  <td>R<\/td>\n                  <td>\\(\\pi\\)<\/td>\n                <\/tr>\n                <tr>\n                  <td>cot x<\/td>\n                  <td>R &#8211; {n\\(\\pi\\) : n \\(\\in\\) I}<\/td>\n                  <td>R<\/td>\n                  <td>\\(\\pi\\)<\/td>\n                <\/tr>\n                <tr>\n                  <td>sec x<\/td>\n                  <td>R &#8211; {(2n+1)\\(\\pi\/2\\); n \\(\\in\\) I}<\/td>\n                  <td>(-\\(\\infty\\), -1] \\(\\cup\\) [1, \\(\\infty\\)]<\/td>\n                  <td>\\(2\\pi\\)<\/td>\n                <\/tr>\n                <tr>\n                  <td>cosec x<\/td>\n                  <td>R &#8211; {n\\(\\pi\\) : n \\(\\in\\) I}<\/td>\n                  <td>(-\\(\\infty\\), -1] \\(\\cup\\) [1, \\(\\infty\\)]<\/td>\n                  <td>\\(2\\pi\\)<\/td>\n                <\/tr>\n                <\/tbody><\/table><br>\n\n\n<h2>Trigonometric ratios of some standard angles :<\/h2>\n<p>(i)&nbsp; sin\\(18^{\\circ}\\) = sin\\(\\pi\\over 10\\) = \\(\\sqrt{5}-1\\over 4\\) = cos\\(72^{\\circ}\\) = cos\\(2\\pi\\over 5\\)<\/p>\n<p>(ii)&nbsp; cos\\(36^{\\circ}\\) = cos\\(\\pi\\over 5\\) = \\(\\sqrt{5}+1\\over 4\\) = sin\\(54^{\\circ}\\) = sin\\(3\\pi\\over 10\\)<\/p>\n<p>(iii)&nbsp; sin\\(72^{\\circ}\\) = sin\\(2\\pi\\over 5\\) = \\(\\sqrt{10 + 2\\sqrt{5}}\\over 4\\) = cos\\(18^{\\circ}\\) = cos\\(\\pi\\over 10\\)<\/p>\n<p>(iv)&nbsp; sin\\(36^{\\circ}\\) = sin\\(\\pi\\over 5\\) = \\(\\sqrt{10 &#8211; 2\\sqrt{5}}\\over 4\\) = cos\\(54^{\\circ}\\) = cos\\(3\\pi\\over 10\\)<\/p>\n<p>(v)&nbsp; sin\\(15^{\\circ}\\) = sin\\(\\pi\\over 12\\) = \\(\\sqrt{3}-1\\over {2\\sqrt{2}}\\) = cos\\(75^{\\circ}\\) = cos\\(5\\pi\\over 12\\)<\/p>\n<p>(vi)&nbsp; cos\\(15^{\\circ}\\) = sin\\(\\pi\\over 12\\) = \\(\\sqrt{3}+1\\over {2\\sqrt{2}}\\) = sin\\(75^{\\circ}\\) = sin\\(5\\pi\\over 12\\)<\/p>\n<p>(vii)&nbsp; tan\\(15^{\\circ}\\) = tan\\(\\pi\\over 12\\) = \\(2 &#8211; \\sqrt{3}\\) = \\(\\sqrt{3}-1\\over {\\sqrt{3}+1}\\) = cot\\(75^{\\circ}\\) = cot\\(5\\pi\\over 12\\)<\/p>\n<p>(viii)&nbsp; tan\\(75^{\\circ}\\) = tan\\(5\\pi\\over 12\\) = \\(2 + \\sqrt{3}\\) = \\(\\sqrt{3}+1\\over {\\sqrt{3}-1}\\) = cot\\(15^{\\circ}\\) = cot\\(\\pi\\over 12\\)<\/p>\n<p>(ix)&nbsp; tan(\\(22.5^{\\circ}\\)) = tan\\(\\pi\\over 8\\) = \\(\\sqrt{2}-1\\) = cot(\\(67.5^{\\circ}\\)) = cot\\(3\\pi\\over 8\\)<\/p>\n<p>(x)&nbsp; tan(\\(67.5^{\\circ}\\)) = tan\\(3\\pi\\over 8\\) = \\(\\sqrt{2}+1\\) = cot(\\(22.5^{\\circ}\\)) = cot\\(\\pi\\over 8\\)<\/p>\n\n\n<p><span style=\"font-size:18px;font-family:georgia;text-align:left;color:#3498DB;\">Example : <\/span> Evaluate sin\\(78^{\\circ}\\) &#8211; sin\\(66^{\\circ}\\) &#8211; sin\\(42^{\\circ}\\) + sin\\(6^{\\circ}\\)<\/p>\n              <p><span style=\"font-size:18px;font-family:georgia;text-align:left; color:#3498DB;\">Solution : <\/span>(sin\\(78^{\\circ}\\) &#8211; sin\\(66^{\\circ}\\)) &#8211; (sin\\(42^{\\circ}\\) &#8211; sin\\(6^{\\circ}\\)-)<br><br>\n              = 2cos(\\(60^{\\circ}\\))sin(\\(18^{\\circ}\\)) &#8211; 2cos(\\(36^{\\circ}\\))sin(\\(30^{\\circ}\\))<br><br>\n              = sin\\(18^{\\circ}\\) &#8211; cos\\(36^{\\circ}\\)<br><br>\n              = (\\(\\sqrt{5}-1\\over 4\\)) &#8211; (\\(\\sqrt{5}+1\\over 4\\)) = \\(-1\\over 2\\)<br><br><\/p>\n\n\n\n<div class=\"wp-block-buttons is-content-justification-right is-layout-flex\">\n<div class=\"wp-block-button has-custom-font-size has-small-font-size\"><a class=\"wp-block-button__link has-background\" href=\"https:\/\/mathemerize.com\/conditional-trigonometric-identities\/\" style=\"background-color:#3498db\">Next &#8211; Conditional Trigonometric Identities \u2013 Maximum &amp; Minimum Value<\/a><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-buttons is-content-justification-left is-layout-flex\">\n<div class=\"wp-block-button has-custom-font-size has-small-font-size\"><a class=\"wp-block-button__link has-background\" href=\"https:\/\/mathemerize.com\/trigonometric-identities-for-class-10th-formulas\/\" style=\"background-color:#3498db\">Previous &#8211; Trigonometric Identities for Class 10th \u2013 Formulas <\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Here, you will learn graph of trigonometric functions and domain &amp; range of trigonometric functions. Graph of Trigonometric Functions : y = sinx y = cosx y = tanx y = cotx y = secx y = cosecx Values of T-Ratio of some standard angles AnglesT-Ratio 0 \\(\\pi\\over 6\\) \\(\\pi\\over 4\\) \\(\\pi\\over 3\\) \\(\\pi\\over 2\\) &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"\" href=\"https:\/\/mathemerize.com\/formula-and-graph-of-trigonometric-functions\/\"> <span class=\"screen-reader-text\">Graph of Trigonometric Functions &#8211; Domain &#038; Range<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[28],"tags":[688,686],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v20.12 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Graph of Trigonometric Functions - Domain &amp; 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