Detailed explanation of inverse trigonometric functions

definition

name	symbol	definition	Domain	range
arcsine	$y=\arcsin x$	$x=\sin y$	$[-1,1]$	$[-\frac{}{2},\frac{}{2}]$
Arc cosine	$y=\arccos x$	$x=\cos y$	$[-1,1]$	$[0,p]$
arctangent	$y=\arctan x$	$x=\tan y$	$\mathbb{R}$	$(-\frac{}{2},\frac{}{2})$

arcsine

Function $f(x)=\sin x$在$x\in [-\frac{}{2},\frac{}{2}]$ part of the inverse function$f^{-1}\left(x\right)$is called the inverse sine function, denoted$\arcsin x$

Arc cosine

Function $f(x)=\cos x$ 在 $x\in [0,\pi ]$ part of the inverse function$f^{-1}\left(x\right)$is called the inverse cosine function, denoted$\arccos x$

arctangent

Function $f\left(x\right)=\tan x$在$x\in (-\frac{}{2},\frac{}{2})$ part of the inverse function$f^{-1}\left(x\right)$is called the arctangent function, denoted as$\arctan x$

image

arcsine

$$y=\arcsin x$$

Arc cosine

$$y=\arccos x$$

arctangent

$$y=\arctan x$$

identity

remaining angle

由$y=\arcsin x$和$y=\arccos The\; graph\; of\; x$ is easy to see
$$\arcsin x+\arccos x=\frac{}{2}$$

Negative parameters

• $\arcsin (-x)=-\arcsin x$
• $\arccos (-x)=Pi-\arccos x$
• $\arctan (-x)=-\arctan x$

reference

[1] Wikipedia Inverse Trigonometric Functions https://zh.wikipedia.org/wiki/Inverse Trigonometric Functions