Definition:Inverse Trigonometric Function

As there are six basic trigonometric functions, so each of these has its inverse functions:

Arcsine

Arcsine Function

From Shape of Sine Function, we have that $\sin x$ is continuous and strictly increasing on the interval $\left[{-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right]$.

From Sine of Multiple of Pi Plus Half, $\sin \left({-\dfrac {\pi} 2}\right) = -1$ and $\sin \dfrac {\pi} 2 = 1$.

Therefore, let $g: \left[{-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right] \to \left[{-1 \,.\,.\, 1}\right]$ be the restriction of $\sin x$ to $\left[{-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right]$.

Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ admits an inverse function, which will be continuous and strictly increasing on $\left[{-1 \,.\,.\, 1}\right]$.

This function is called arcsine of $x$ and is written $\arcsin x$.

Thus:

• The domain of $\arcsin x$ is $\left[{-1 \,.\,.\, 1}\right]$
• The image of $\arcsin x$ is $\left[{-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right]$.

Arccosine

Arccosine Function

From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\left[{1 \,.\,.\, \pi}\right]$.

From Cosine of Multiple of Pi, $\cos \pi = -1$ and $\cos 0 = 1$.

Therefore, let $g: \left[{0 \,.\,.\, \pi}\right] \to \left[{-1 \,.\,.\, 1}\right]$ be the restriction of $\cos x$ to $\left[{0 \,.\,.\, \pi}\right]$.

Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ admits an inverse function, which will be continuous and strictly decreasing on $\left[{-1 \,.\,.\, 1}\right]$.

This function is called arccosine of $x$ and is written $\arccos x$.

Thus:

• The domain of $\arccos x$ is $\left[{-1 \,.\,.\, 1}\right]$
• The image of $\arccos x$ is $\left[{0 \,.\,.\, \pi}\right]$.

Arctangent

Arctangent Function

From Shape of Tangent Function, we have that $\tan x$ is continuous and strictly increasing on the interval $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.

From the same source, we also have that:

• $\tan x \to + \infty$ as $x \to \dfrac \pi 2 ^-$
• $\tan x \to - \infty$ as $x \to -\dfrac \pi 2 ^+$

Let $g: \left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right) \to \R$ be the restriction of $\tan x$ to $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.

Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ admits an inverse function, which will be continuous and strictly increasing on $\R$.

This function is called arctangent of $x$ and is written $\arctan x$.

Thus:

• The domain of $\arctan x$ is $\R$
• The image of $\arctan x$ is $\left({-\dfrac \pi 2 \,.\,.\, \dfrac \pi 2}\right)$.

Arccotangent

From Shape of Cotangent Function, we have that $\cot x$ is continuous and strictly decreasing on the interval $\left({0 \,.\,.\, \pi}\right)$.

From the same source, we also have that:

• $\cot x \to + \infty$ as $x \to 0^+$
• $\cot x \to - \infty$ as $x \to \pi^-$

Let $g: \left({0 \,.\,.\, \pi}\right) \to \R$ be the restriction of $\cot x$ to $\left({0 \,.\,.\, \pi}\right)$.

Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ admits an inverse function, which will be continuous and strictly decreasing on $\R$.

This function is called arccotangent of $x$ and is written $\operatorname{arccot} x$.

Thus:

• The domain of $\operatorname{arccot} x$ is $\R$
• The image of $\operatorname{arccot} x$ is $\left({0 \,.\,.\, \pi}\right)$.