Inverse Trigonometric Function Formulas

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Inverse Trigonometric Function Formulas

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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 Trigonometric Functions

$sin^{-1} x$

$cos^{-1} x$

$tan^{-1} x$

$cosec^{-1} x$$ sec^{-1} x$

$cot^{-1} x$

More Formulas

$sin (sin^{-1} x) = x$ 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$

Inverse of Negative x

$sin^{-1} (-x) = -sin^{-1} (x)$
$cos^{-1} (-x) = \pi – cos^{-1} (x)$
$tan^{-1} (-x) = -tan^{-1} (x)$
$sec^{-1} (-x) = \pi – sec^{-1} (x)$
$cosec^{-1} (-x) = -cosec^{-1} (x)$
$cot^{-1} (-x) = \pi – cot^{-1} (x)$

Other Formulas

$sin^{-1} (\frac {1}{x}) = cosec^{-1} (x)$
$cos^{-1} (\frac {1}{x}) = sec^{-1} (x)$
$tan^{-1} (\frac {1}{x}) = cot^{-1} (x)$
$sin^{-1} (x) + cos^ {-1} (x) = \frac {\pi}{2}$
$sec^{-1} (x) + cosec^ {-1} (x) = \frac {\pi}{2}$
$tan^{-1} (x) + cot^ {-1} (x) = \frac {\pi}{2}$
$sin^{-1} (x) + sin^ {-1} (y) = sin ^{-1} (x \sqrt {1-y^2} + y \sqrt {1-x^2})$ 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$
$3 sin^{-1} (x) = sin^{-1} (3x -4x^3)$
$3 cos^{-1} (x) = cos^{-1} (4x^3 – 3x)$
$3 tan^{-1} (x) = tan^{-1} (\frac {3x -x^3}{1-3x^2})$

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