{"id":3851,"date":"2021-08-09T13:20:07","date_gmt":"2021-08-09T13:20:07","guid":{"rendered":"https:\/\/mathemerize.com\/?p=3851"},"modified":"2021-11-26T16:28:23","modified_gmt":"2021-11-26T10:58:23","slug":"properties-of-inverse-trigonometric-functions","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/properties-of-inverse-trigonometric-functions\/","title":{"rendered":"Properties of Inverse Trigonometric Functions with Example"},"content":{"rendered":"

Here, you will learn all the properties of inverse trigonometric functions class 12 with examples.<\/p>\n

Let’s begin –<\/p>\n

Properties of Inverse Trigonometric functions<\/h2>\n

Property – 1<\/strong><\/p>\n

\n

(i)\u00a0 y = \\(sin^{-1}(sinx)\\), x \\(\\in\\) R, y \\(\\in\\) (-\\(\\pi\\over 2\\), \\(\\pi\\over 2\\)) periodic with period \\(2\\pi\\)
and it is an odd function.<\/p>\n

(ii)\u00a0 y = \\(cos^{-1}(cosx)\\), x \\(\\in\\) R, y \\(\\in\\) [0, \\(\\pi\\)], periodic with period \\(2\\pi\\) and it is an even function.<\/p>\n

(iii)\u00a0 y = \\(tan^{-1}(tanx)\\), x \\(\\in\\) R – { (2n-1)\\(\\pi\\over 2\\), n \\(\\in\\) I }, y \\(\\in\\) (-\\(\\pi\\over 2\\), \\(\\pi\\over 2\\)) periodic with period \\(\\pi\\) and it is an odd function.<\/p>\n

(iv)\u00a0 y = \\(cot^{-1}(cotx)\\), x \\(\\in\\) R – { n\\(\\pi\\), n \\(\\in\\) I }, y \\(\\in\\) (0, \\(\\pi\\)) periodic with period \\(\\pi\\) and neither even or odd function.<\/p>\n

(v)\u00a0 y = \\(cosec^{-1}(cosecx)\\), x \\(\\in\\) R – { n\\(\\pi\\), n \\(\\in\\) I }, y \\(\\in\\) [-\\(\\pi\\over 2\\), 0] \\(\\cup\\) (0, \\(\\pi\\over 2\\)] periodic with period \\(2\\pi\\) and it is an odd function.<\/p>\n

(vi)\u00a0 y = \\(sec^{-1}(secx)\\), x \\(\\in\\) R – { (2n-1)\\(\\pi\\over 2\\), n \\(\\in\\) I }, y \\(\\in\\) [0, \\(\\pi\\over 2\\)] \\(\\cup\\) (\\(\\pi\\over 2\\), \\(\\pi\\)], y is periodic with period \\(2\\pi\\) and it is an even function.<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Evaluate \\(sin^{-1}(sin10)\\)<\/p>\n

Solution : <\/span>We know that \\(sin^{-1}(sinx)\\) = x, if \\(-\\pi\\over 2\\) \\(\\le\\) x \\(\\le\\) \\(\\pi\\over 2\\)

\n Here, x = 10 radians which does not lie between -\\(\\pi\\over 2\\) and \\(\\pi\\over 2\\)

\n But, \\(3\\pi\\) – x i.e. \\(3\\pi\\) – 10 lie between -\\(\\pi\\over 2\\) and \\(\\pi\\over 2\\)

\n Also, sin(\\(3\\pi\\) – 10) = sin 10

\n \\(\\therefore\\)   \\(sin^{-1}(sin10)\\) = \\(sin^{-1}(sin(3\\pi – 10)\\) = (\\(3\\pi\\) – 10)

<\/p>\n\n\n

Property – 2<\/strong><\/p>\n

\n

(i)  \\(sin^{-1}x\\) + \\(cos^{-1}x\\) = \\(\\pi\\over 2\\)<\/p>\n

(ii)  \\(tan^{-1}x\\) + \\(cot^{-1}x\\) = \\(\\pi\\over 2\\)<\/p>\n

(iii)  \\(cosec^{-1}x\\) + \\(sec^{-1}x\\) = \\(\\pi\\over 2\\)<\/p>\n<\/blockquote>\n

Property – 3<\/strong><\/p>\n

\n

(i)   \\(sin^{-1}(-x)\\) = -\\(sin^{-1}x\\)<\/p>\n

(ii)  \\(cosec^{-1}(-x)\\) = -\\(cosec^{-1}x\\)<\/p>\n

(iii)  \\(tan^{-1}(-x)\\) = -\\(tan^{-1}x\\)<\/p>\n

(iv)  \\(cot^{-1}(-x)\\) = \\(\\pi\\) – \\(cot^{-1}x\\)<\/p>\n

(v)  \\(cos^{-1}(-x)\\) = \\(\\pi\\) – \\(cos^{-1}x\\)<\/p>\n

(vi) \\(sec^{-1}(-x)\\) = \\(\\pi\\) – \\(sec^{-1}x\\)<\/p>\n<\/blockquote>\n

Property – 4<\/strong><\/p>\n

\n

(i)  \\(cosec^{-1}x\\) = \\(sin^{-1}{1\\over x}\\)<\/p>\n

(ii)  \\(sec^{-1}x\\) = \\(cos^{-1}{1\\over x}\\)<\/p>\n

(iii)  \\(cot^{-1}x\\) = \\(\\begin{cases} tan^{-1}{1\\over x}, & \\text{if}\\ x > 0 \\\\
\\pi + tan^{-1}{1\\over x}, & \\text{if}\\ x < 0 \\end{cases}\\)<\/p>\n<\/blockquote>\n\n\n

Example : <\/span> Find the value of x if \\(cos^{-1}(-x)\\) + \\(tan^{-1}(-x)\\) – 2\\(sin^{-1}x\\) + \\(sec^{-1}({-1\\over x})\\) = \\(\\pi\\over 4\\) for |x| \\(\\le\\) 1.<\/p>\n

Solution : <\/span>\\(\\pi\\) – \\(cos^{-1}(x)\\) – \\(tan^{-1}(x)\\) – 2\\(sin^{-1}x\\) + \\(cos^{-1}(-x)\\) = \\(\\pi\\over 4\\)

\n \\(\\pi\\) – \\(cos^{-1}(x)\\) – \\(tan^{-1}(x)\\) – 2\\(sin^{-1}x\\) + \\(\\pi\\) – \\(cos^{-1}(-x)\\) = \\(\\pi\\over 4\\)

\n 2\\(\\pi\\) – 2(\\(sin^{-1}x\\) + \\(cos^{-1}x\\)) – \\(\\pi\\over 4\\) = \\(tan^{-1}(x)\\)

\n 2\\(\\pi\\) – \\(\\pi\\) – \\(\\pi\\over 4\\) = \\(tan^{-1}(x)\\) \\(\\implies\\) \\(tan^{-1}(x)\\) = \\(3\\pi\\over 4\\)   Hence no solution.

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

Property – 5<\/strong><\/p>\n

\n

(i) <\/p>\n

(a)   \\(tan^{-1}x\\) + \\(tan^{-1}y\\) = \\(\\begin{cases} tan^{-1}{{x+y}\\over {1-xy}}, & \\text{where}\\ x > 0, y > 0 & xy < 1 \\\\ \\pi + tan^{-1}{{x+y}\\over {1-xy}}, & \\text{where}\\ x > 0, y > 0 & xy > 1 \\\\ {\\pi\\over 2} , & \\text{where}\\ x > 0, y > 0 & xy = 1 \\end{cases}\\)<\/p>\n

(b)  \\(tan^{-1}x\\) – \\(tan^{-1}y\\) = \\(tan^{-1}{{x-y}\\over {1+xy}}\\)<\/p>\n

(c)  \\(tan^{-1}x\\) + \\(tan^{-1}y\\) + \\(tan^{-1}z\\) = \\(tan^{-1}[{{x+y+z-xyz}\\over {1-xy-yz-zx}}]\\)<\/p>\n

(ii)<\/p>\n

(a)  \\(sin^{-1}x\\) + \\(sin^{-1}y\\) = \\(\\begin{cases} sin^{-1}[{x\\sqrt{1-y^2} + y{\\sqrt{1-x^2}}}], & \\text{where}\\ x > 0, y > 0 & (x^2 + y^2) \\le 1 \\\\ \\pi – sin^{-1}[{x\\sqrt{1-y^2} + y{\\sqrt{1-x^2}}}], & \\text{where}\\ x > 0, y > 0 & (x^2 + y^2) > 1 \\end{cases}\\)<\/p>\n

(b)  \\(sin^{-1}x\\) – \\(sin^{-1}y\\) = \\(sin^{-1}[{x\\sqrt{1-y^2} – y{\\sqrt{1-x^2}}}]\\), where x > 0, y > 0<\/p>\n

(iii)<\/p>\n

(a)  \\(cos^{-1}x\\) + \\(cos^{-1}y\\) = \\(cos^{-1}[xy – {\\sqrt{1-y^2}{\\sqrt{1-x^2}}}]\\), where x > 0, y > 0<\/p>\n

(b)  \\(cos^{-1}x\\) – \\(cos^{-1}y\\) = \\(\\begin{cases} cos^{-1}[xy + {\\sqrt{1-y^2}{\\sqrt{1-x^2}}}]; x < y,  \\ x, y > 0 \\\\ – cos^{-1}[xy + {\\sqrt{1-y^2}{\\sqrt{1-x^2}}}], x > y, \\  x, y > 0 \\end{cases}\\)<\/p>\n<\/blockquote>\n\n\n

Example : <\/span>Prove that : \\(tan^{-1}{1\\over 7}\\) + \\(tan^{-1}{1\\over 13}\\) = \\(tan^{-1}{2\\over 9}\\)<\/p>\n

Solution : <\/span>L.H.S = \\(tan^{-1}{1\\over 7}\\) + \\(tan^{-1}{1\\over 13}\\)

\n = \\(tan^{-1}[{{{1\\over 7}+{1\\over 13}}\\over {1 – {1\\over 7}\\times{1\\over 13}}}]\\)       { \\(\\because\\) \\(tan^{-1}x\\) + \\(tan^{-1}y\\) = \\(tan^{-1}{{x+y}\\over {1-xy}}\\); if xy < 1 }

\n = \\(tan^{-1}({20\\over 90})\\) = \\(tan^{-1}({2\\over 9})\\) = R.H.S.

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

\n
Next – Formulas for Inverse Trigonometric Functions<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Inverse Trigonometric Function Class 12 \u2013 Domain & Range<\/a><\/div>\n<\/div>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Here, you will learn all the properties of inverse trigonometric functions class 12 with examples. Let’s begin – Properties of Inverse Trigonometric functions Property – 1 (i)\u00a0 y = \\(sin^{-1}(sinx)\\), x \\(\\in\\) R, y \\(\\in\\) (-\\(\\pi\\over 2\\), \\(\\pi\\over 2\\)) periodic with period \\(2\\pi\\)and it is an odd function. (ii)\u00a0 y = \\(cos^{-1}(cosx)\\), x \\(\\in\\) R, …<\/p>\n

Properties of Inverse Trigonometric Functions with Example<\/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":[445,446],"yoast_head":"\nProperties of Inverse Trigonometric Functions with Example<\/title>\n<meta name=\"description\" content=\"In this post, you will learn all the properties of inverse trigonometric functions class 12 with 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