{"id":5632,"date":"2023-05-05T19:25:36","date_gmt":"2023-05-05T13:55:36","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=5632"},"modified":"2024-01-17T23:14:40","modified_gmt":"2024-01-17T17:44:40","slug":"inverse-trigonometric-function-formula","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/inverse-trigonometric-function-formula\/","title":{"rendered":"Inverse Trigonometric Function Formulas"},"content":{"rendered":"\n
\n\n\n\n
\"Inverse<\/figure>\n\n\n\n

Inverse Trigonometric Function Formula<\/strong><\/span>s<\/h2>\n\n\n\n
\n\n\n\n

Inverse Trigonometric Functions are important topic in Trigonometry. Here is detailed list of Inverse Trigonometric Function Formulas<\/p>\n\n\n\n

Domain and Range Of Inverse Trigonometric Functions<\/h3>\n\n\n\n
\"Inverse<\/a><\/figure>\n\n\n\n


The value of an inverse trigonometric functions which lies in its principal value branch is called the principal value of that inverse trigonometric functions<\/p>\n\n\n\n

Graph of Inverse Trigonometric Functions<\/h3>\n\n\n\n
\"sin<\/a><\/figure>\n\n\n\n

$sin^{-1} x$
<\/p>\n\n\n\n

\"cos<\/a><\/figure>\n\n\n\n

$cos^{-1} x$<\/p>\n\n\n\n

\"tan<\/a><\/figure>\n\n\n\n

$tan^{-1} x$<\/p>\n\n\n\n

\"cosec<\/a><\/figure>\n\n\n\n

$cosec^{-1} x$<\/p>\n\n\n\n

\"sec<\/a><\/figure>\n\n\n\n

$ sec^{-1} x$<\/p>\n\n\n\n

\"\"<\/a><\/figure>\n\n\n\n

$cot^{-1} x$<\/p>\n\n\n\n




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

More Formulas<\/h3>\n\n\n\n

$sin (sin^{-1} x) = x$ and $sin^{-1} (sin \\theta) = \\theta$, if $- \\frac {\\pi}{2} \\leq \\theta \\leq \\frac {\\pi}{2}$ and $- 1 \\leq x \\leq 1$.
$cos (cos^{-1} x) = x$ and $cos^{-1} (cos \\theta) = \\theta $, if $0 \\leq \\theta \\leq \\pi$ and $- 1 \\leq x \\leq 1$
$tan (tan^{-1} x) = x$ and $tan^{-1} (tan \\theta) = \\theta $ if $- \\frac {\\pi}{2} \\leq \\theta \\leq \\frac {\\pi}{2}$ and $ – \\infty < x < \\infty$.
$cosec (cosec^{-1} x) = x$ and $cosec^{-1} (cosec \\theta) = \\theta$ if $- \\frac {\\pi}{2} \\leq \\theta < 0$ , $ 0 < \\theta \\leq \\frac {\\pi}{2}$ and $- \\infty < x \\leq -1$ or $1 \\leq x < \\infty$ .
$sec (sec^{-1} x) = x$ and $sec^{-1} (sec \\theta) = \\theta$ if $0 \\leq \\theta < \\frac {\\pi}{2}$ or $\\frac {\\pi}{2} < \\theta \\leq \\pi $ and $- \\infty < x \\leq -1$ or $1 \\leq x < \\infty$.
$cot (cot^{-1} x) = x$ and $cot^{-1} (cot \\theta) = \\theta$, if $0 < \\theta < \\pi$ and $ – \\infty < x < \\infty$<\/p>\n\n\n\n

Inverse of Negative x<\/h3>\n\n\n\n

$sin^{-1} (-x) = -sin^{-1} (x)$
$cos^{-1} (-x) = \\pi – cos^{-1} (x)$
$tan^{-1} (-x) = -tan^{-1} (x)$
$sec^{-1} (-x) = \\pi – sec^{-1} (x)$
$cosec^{-1} (-x) = -cosec^{-1} (x)$
$cot^{-1} (-x) = \\pi – cot^{-1} (x)$<\/p>\n\n\n\n

\n

Other Integration Related Articles<\/p>\n\n\n\n

Integration of tan inverse x<\/a><\/td>Integration of cos inverse x<\/a><\/td><\/tr>
Integration of cos square x<\/a><\/td>Integration of fractional part of x<\/a><\/td><\/tr>
integration of uv formula<\/a><\/td>integration of log x<\/a><\/td><\/tr>
integration of sin inverse x<\/a><\/td>Integration of constant term<\/a><\/td><\/tr>
integration of hyperbolic functions<\/a><\/td>List of Integration Formulas <\/a><\/td><\/tr>
integration of log sinx<\/a><\/td>integration of sin x cos x dx<\/a><\/td><\/tr><\/tbody><\/table><\/figure>\n<\/div><\/div>\n\n\n\n

Other Formulas<\/h3>\n\n\n\n

$sin^{-1} (\\frac {1}{x}) = cosec^{-1} (x)$
$cos^{-1} (\\frac {1}{x}) = sec^{-1} (x)$
$tan^{-1} (\\frac {1}{x}) = cot^{-1} (x)$
$sin^{-1} (x) + cos^ {-1} (x) = \\frac {\\pi}{2}$
$sec^{-1} (x) + cosec^ {-1} (x) = \\frac {\\pi}{2}$
$tan^{-1} (x) + cot^ {-1} (x) = \\frac {\\pi}{2}$
$sin^{-1} (x) + sin^ {-1} (y) = sin ^{-1} (x \\sqrt {1-y^2} + y \\sqrt {1-x^2})$ if $x,y \\geq 0 $, $x^2 + y^2 \\leq 1$
$sin^{-1} (x) + sin^ {-1} (y) = \\pi – sin ^{-1} (x \\sqrt {1-y^2} + y \\sqrt {1-x^2})$ if $x,y \\geq 0 $, $x^2 + y^2 > 1$
$sin^{-1} (x) – sin^ {-1} (y) = sin ^{-1} (x \\sqrt {1-y^2} – y \\sqrt {1-x^2)}$ if $x,y \\geq 0 $, $x^2 + y^2 \\leq 1$
$sin^{-1} (x) – sin^ {-1} (y) = \\pi – sin ^{-1} (x \\sqrt {1-y^2} – y \\sqrt {1-x^2})$ if $x,y \\geq 0 $, $x^2 + y^2 > 1$
$cos^{-1} (x) + cos^ {-1} (y) = cos ^{-1} (x y – \\sqrt {1-y^2} \\sqrt {1-x^2})$ if $x,y \\geq 0 $, $x^2 + y^2 \\leq 1$
$cos^{-1} (x) + cos^ {-1} (y) = \\pi – cos ^{-1} ((x y – \\sqrt {1-y^2} \\sqrt {1-x^2})$ if $x,y \\geq 0 $, $x^2 + y^2 > 1$
$cos^{-1} (x) – cos^ {-1} (y) = cos ^{-1} (x y + \\sqrt {1-y^2} \\sqrt {1-x^2})$ if $x,y \\geq 0 $, $x^2 + y^2 \\leq 1$
$cos^{-1} (x) – cos^ {-1} (y) = \\pi – cos ^{-1} (x y + \\sqrt {1-y^2} \\sqrt {1-x^2})$ if $x,y \\geq 0 $, $x^2 + y^2 > 1$
$tan^{-1} (x) + tan^ {-1} (y)= tan^{-1} (\\frac {x+y}{1-xy})$ , if $x,y > 0 $, $xy < 1$
$tan^{-1} (x) + tan^ {-1} (y)= \\pi + tan^{-1} (\\frac {x+y}{1-xy})$ , if $x,y > 0 $, $xy > 1$
$tan^{-1} (x) + tan^ {-1} (y)= tan^{-1} (\\frac {x+y}{1-xy}) – \\pi$ , if $x < 0, y > 0 $, $xy > 1$
$tan^{-1} (x) – tan^ {-1} (y)= tan^{-1} (\\frac {x-y}{1+xy}) – \\pi$ , if $xy > -1$
$tan^{-1} (x) + tan^ {-1} (y) + tan^ {-1} (z) = tan^{-1} (\\frac {x+y+z – xyz}{1-xy-yz-xz})$
$ 2 sin^{-1} (x) = sin^{-1} (2x \\sqrt {1-x^2})$ if $ -\\frac {1}{\\sqrt {2}} \\leq x \\frac {1}{\\sqrt {2}} $
$ 2 cos^{-1} (x) = cos^{-1} (2x^2 -1)$
$2 tan^{-1} (x) = tan^{-1} (\\frac {2x}{1-x^2})$ if $ -1 <x < 1$
$2 tan^{-1} (x) = sin^{-1} (\\frac {2x}{1+x^2})$ if $ |x| \\leq 1$
$2 tan^{-1} (x) = cos^{-1} (\\frac {1 -x^2}{1+x^2})$ if $ x \\geq 0$
$3 sin^{-1} (x) = sin^{-1} (3x -4x^3)$
$3 cos^{-1} (x) = cos^{-1} (4x^3 – 3x)$
$3 tan^{-1} (x) = tan^{-1} (\\frac {3x -x^3}{1-3x^2})$<\/p>\n\n\n\n

Inverse trigonometric functions are very useful in a wide range of applications. Understanding these functions is crucial for solving problems in mathematics, physics, and engineering. By mastering the properties and identities of inverse trigonometric functions, you can gain a deeper understanding of trigonometry and its applications.<\/p>\n\n\n\n

\n\n\n\n

Related Articles<\/p>\n\n\n\n

Trigonometry Formulas for class 11 (PDF download)<\/a>
sin cos tan table<\/a>
Integration Formulas<\/a>
sin 18 degrees<\/a>
tan 15 degrees<\/a>
Differentiation formulas<\/a>
https:\/\/en.wikipedia.org\/wiki\/Differentiation<\/a>
<\/p>\n\n\n\n

\n\n\n\n

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

Inverse Trigonometric Function Formulas Inverse Trigonometric Functions are important topic in Trigonometry. Here is detailed list of Inverse Trigonometric Function Formulas Domain and Range Of Inverse Trigonometric Functions The value of an inverse trigonometric functions which lies in its principal value branch is called the principal value of that inverse trigonometric functions Graph of Inverse […]<\/p>\n","protected":false},"author":8,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_uag_custom_page_level_css":"","site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","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":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[498],"tags":[],"class_list":["post-5632","post","type-post","status-publish","format-standard","hentry","category-maths"],"yoast_head":"\nInverse Trigonometric Function Formulas - 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Here is detailed list of Inverse Trigonometric Function Formulas Domain and Range Of Inverse Trigonometric Functions The value of an inverse trigonometric functions which lies in its principal value branch is called the principal value of that inverse trigonometric functions Graph of Inverse…","_links":{"self":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/5632","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/users\/8"}],"replies":[{"embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/comments?post=5632"}],"version-history":[{"count":7,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/5632\/revisions"}],"predecessor-version":[{"id":8796,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/posts\/5632\/revisions\/8796"}],"wp:attachment":[{"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/media?parent=5632"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/categories?post=5632"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/physicscatalyst.com\/article\/wp-json\/wp\/v2\/tags?post=5632"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}