The inverse of the trigonometric functions with restricted domains are known as inverse trigonometric functions. The inverse trigonometric function is denoted with an exponent "-1". Say,the inverse sine is denoted as $sin^{-1}(x)$

This notation does not mean that its reciprocal.

$sin^{-1}(x) \neq$ $\frac{1}{sin (x)}$

Another notation for inverse trigonometric functions can be written as

$sin^{-1}(x) = arcsin(x)$  (It is pronounced as "arc sine".)

$sin^{-1}$ $(\frac{1}{2})$ can be read as " the angle whose sine is $\frac{1}{2}$".

Inverse trigonometric functions operates on number and produces angle.

if $sin 30^{\circ}$ = $\frac{1}{2}$, then $sin^{-1}$$(\frac{1}{2})$ = $30^{\circ}$

Inverse of Trigonometric Functions

1. arcsine:
If $x = sin y$, inverse sine function is defined as $y = arcsin x$ in the range $\frac{-\pi}{2}$ $\leq y \leq$   $\frac{\pi}{2}$ .

domain of $y = arcsin x$ is $-1 \leq x \leq 1$

2. arccosine:
If $x = cos y$, inverse cosine function is defined as $y = arccos x$ in the range  $0 \leq y \leq \pi$ .

domain of $y = arccos x$ is $-1 \leq x \leq 1$

3. arctangent:
If $x = tan y$, inverse tangent function is defined as $y = arctan x$ in the range $\frac{-\pi}{2}$ $< y <$  $\frac{\pi}{2}$.

domain of $y = \arctan x$ is $- \infty < \infty$

4. arccotangent:
If $x = cot y$, inverse cotangent function is defined as $y = arctan x$ in the range $0 < y < \pi$ .

domain of $y = arccot x$ is $- \infty < \infty$

5. arcsecant:
If $x = sec y$, inverse cotangent function is defined as $y = arcsec x$ in the range $0\leq y <$ $\frac{\pi}{2}$  or   $\frac{\pi}{2}$ $< y \leq \pi$ .

domain of $y = arcsec x$ is $x \leq -1$ or $1 \leq x$

6. arccosecant:
If $x = csc y$, inverse cosecant function is defined as $y = arccsc x$ in the range  $\frac{-\pi}{2}$ $\leq y < 0$ or $0 < y \leq$ $\frac{\pi}{2}$.

domain of $y = arccosec x$ is $x \leq -1$ or $1 \leq x$

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