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# Inverse Trigonometric Function Formulas

## 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 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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