{"id":3974,"date":"2021-08-11T16:36:23","date_gmt":"2021-08-11T16:36:23","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3974"},"modified":"2021-11-26T16:24:57","modified_gmt":"2021-11-26T10:54:57","slug":"inverse-trigonometric-function-formula","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/inverse-trigonometric-function-formula\/","title":{"rendered":"Formulas for Inverse Trigonometric Functions"},"content":{"rendered":"

Here, you will learn formulas for inverse trigonometric functions, equation and inequations involving inverse trigonometric function.<\/p>\n

Let’s begin –<\/p>\n

Simplified Inverse Trigonometric Functions<\/h2>\n

(a)  y = f(x) = \\(sin^{-1}({2x\\over {1+x^2}})\\) = \\(\\begin{cases} 2tan^{-1}x, & \\text{if}\\ |x| \\le 1 \\\\ \\pi – 2tan^{-1}x, & \\text{if}\\ x > 1 \\\\ -(\\pi + 2tan^{-1}x), & \\text{if}\\ x < -1 \\end{cases}\\) <\/p>\n

(b)  y = f(x) = \\(cos^{-1}({{1-x^2}\\over {1+x^2}})\\) = \\(\\begin{cases} 2tan^{-1}x, & \\text{if}\\ |x| \\ge 0 \\\\ – 2tan^{-1}x, & \\text{if}\\ x < 0 \\end{cases}\\)<\/p>\n

(c)  y = f(x) = \\(tan^{-1}({2x\\over {1-x^2}})\\) = \\(\\begin{cases} 2tan^{-1}x, & \\text{if}\\ |x| < 1 \\\\ \\pi + 2tan^{-1}x, & \\text{if}\\ x < -1 \\\\ -(\\pi – 2tan^{-1}x), & \\text{if}\\ x > 1 \\end{cases}\\)<\/p>\n

(d)  y = f(x) = \\(sin^{-1}({3x – 4x^3})\\) = \\(\\begin{cases} -(\\pi + 3sin^{-1}x), & \\text{if}\\ -1 \\le x \\le {-1\\over 2} \\\\ 3sin^{-1}x, & \\text{if}\\ {-1\\over 2} \\le x \\le {1\\over 2} \\\\ \\pi – 3sin^{-1}x, & \\text{if}\\ {1\\over 2} \\le x \\le 1 \\end{cases}\\)<\/p>\n

(e)  y = f(x) = \\(cos^{-1}({4x^3 – 3x})\\) = \\(\\begin{cases} 3cos^{-1}x – 2\\pi, & \\text{if}\\ -1 \\le x \\le {-1\\over 2} \\\\ 2\\pi – 3cos^{-1}x, & \\text{if}\\ {-1\\over 2} \\le x \\le {1\\over 2} \\\\ 3cos^{-1}x, & \\text{if}\\ {1\\over 2} \\le x \\le 1 \\end{cases}\\)<\/p>\n

(f)  y = f(x) = \\(sin^{-1}({2x\\sqrt{1-x^2}})\\) = \\(\\begin{cases} -(\\pi + 2sin^{-1}x), & \\text{if}\\ -1 \\le x \\le {-1\\over \\sqrt{2}} \\\\ 2sin^{-1}x, & \\text{if}\\ {-1\\over \\sqrt{2}} \\le x \\le {1\\over \\sqrt{2}} \\\\ \\pi – 2sin^{-1}x, & \\text{if}\\ {1\\over \\sqrt{2}} \\le x \\le 1 \\end{cases}\\)<\/p>\n

(g)  y = f(x) = \\(cos^{-1}({2x^2-1})\\) = \\(\\begin{cases} 2cos^{-1}x, & \\text{if}\\ 0 \\le x \\le 1 \\\\ 2\\pi – 2cos^{-1}x, & \\text{if}\\ -1 \\le x \\le 0 \\end{cases}\\)<\/p>\n\n\n

Example : <\/span> Prove that : \\(2tan^{-1}{1\\over 2}\\) + \\(tan^{-1}{1\\over 7}\\) = \\(tan^{-1}{31\\over 17}\\)<\/p>\n

Solution : <\/span>We have, \\(2tan^{-1}{1\\over 2}\\) + \\(tan^{-1}{1\\over 7}\\)

\n = \\(2tan^{-1}({{2\\times {1\\over 2}\\over {1-({1\\over 2})^2}}})\\) + \\(tan^{-1}{1\\over 7}\\)       [\\(\\because\\) \\(2tan^{-1}x\\) = \\(tan^{-1}{2x\\over {1-x^2}}\\)]

\n \\(tan^{-1}{4\\over 3}\\) + \\(tan^{-1}{1\\over 7}\\) = \\(tan^{-1}[{{4\\over 3} + {1\\over 7}\\over {1 – {4\\over 3}\\times {1\\over 7}}}]\\) = \\(tan^{-1}{31\\over 17}\\)

<\/p>\n\n\n

Equations involving Inverse trigonometric functions<\/strong><\/h4>\n\n\n

Example : <\/span> Prove that the equation \\(2cos^{-1}x\\) + \\(sin^{-1}x\\) = \\(11\\pi\\over 6\\) has no solution.<\/p>\n

Solution : <\/span>Given equation is \\(2cos^{-1}x\\) + \\(sin^{-1}x\\) = \\(11\\pi\\over 6\\)

\n \\(\\implies\\) \\(cos^{-1}x\\) + (\\(cos^{-1}x\\) + \\(sin^{-1}x\\)) = \\(11\\pi\\over 6\\)

\n \\(\\implies\\) \\(cos^{-1}x\\) + \\(\\pi\\over 2\\) = \\(11\\pi\\over 6\\)

\n \\(\\implies\\) \\(cos^{-1}x\\) = \\(4\\pi\\over 3\\)

\n which is not possible as \\(cos^{-1}x\\) \\(\\in\\) [0, \\(\\pi\\)]. Hence no solution.

\n <\/p>\n\n\n

Inequations involving Inverse trigonometric functions<\/strong><\/h4>\n\n\n

Example : <\/span> Find the complete solution set of \\(sin^{-1}(sin5)\\) > \\(x^2\\) – 4x.<\/p>\n

Solution : <\/span>\\(sin^{-1}(sin5)\\) > \\(x^2\\) – 4x   \\(\\implies\\)   \\(sin^{-1}[sin(5-2\\pi)]\\) > \\(x^2\\) – 4x

\n \\(\\implies\\) \\(x^2\\) – 4x < 5 – 2\\(\\pi\\)   \\(\\implies\\)   \\(x^2\\) – 4x + 2\\(\\pi\\) – 5< 0

\n \\(\\implies\\) 2 – \\(\\sqrt{9-2\\pi}\\) < x < 2 + \\(\\sqrt{9-2\\pi}\\)   \\(\\implies\\)   x \\(\\in\\) (2 – \\(\\sqrt{9-2\\pi}\\), 2 + \\(\\sqrt{9-2\\pi}\\))

<\/p>\n\n\n

Hope you learnt formulas for inverse trigonometric functions, equation and inequations involving inverse trigonometric function, learn more concepts of inverse trigonometric functions and practice more questions to get ahead in competition. Good Luck!<\/p>\n\n\n

\n
Previous – Properties of Inverse Trigonometric Functions<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn formulas for inverse trigonometric functions, equation and inequations involving inverse trigonometric function. Let’s begin – Simplified Inverse Trigonometric Functions (a)  y = f(x) = \\(sin^{-1}({2x\\over {1+x^2}})\\) = \\(\\begin{cases} 2tan^{-1}x, & \\text{if}\\ |x| \\le 1 \\\\ \\pi – 2tan^{-1}x, & \\text{if}\\ x > 1 \\\\ -(\\pi + 2tan^{-1}x), & \\text{if}\\ x < …<\/p>\n

Formulas for Inverse Trigonometric Functions<\/span> Read More »<\/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":[27],"tags":[444,445],"yoast_head":"\nFormulas for Inverse Trigonometric Functions - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post, you will learn formulas for inverse trigonometric functions, equation and inequations involving inverse trigonometric function.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/inverse-trigonometric-function-formula\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Formulas for Inverse Trigonometric Functions - 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