domain | range | domain | range | ||
arcsin | $\displaystyle [-1,1]$ | $\displaystyle [-\frac\pi2,\frac\pi2]$ | arccsc | $\displaystyle (-\infty,-1]\cup[1,\infty)$ | $\displaystyle [-\frac{\pi}2,0)\cup(0,\frac{\pi}2]$ (W) $\displaystyle \left(0,\frac{\pi}{2}\right]\cup\left(\pi,\frac32\pi\right]$ (S) |
arccos | $\displaystyle [-1,1]$ | $\displaystyle [0,\pi]$ | arcsec | $\displaystyle (-\infty,-1]\cup[1,\infty)$ | $\displaystyle [0,\frac{\pi}2)\cup(\frac{\pi}2,\pi]$ (W) $\displaystyle \left[0,\frac{\pi}{2}\right)\cup\left[\pi,\frac32\pi\right)$ (S) |
arctan | $\displaystyle \mathbb{R}$ | $\displaystyle (-\frac\pi2,\frac\pi2)$ | arccot | $\displaystyle \mathbb{R}$ | $\displaystyle (0,\pi)$ |
domain | range | domain | range | ||
arcsin | $\displaystyle -1\le x\le 1$ | $\displaystyle -\frac{\pi}{2}\le y\le\frac{\pi}{2}$ | arccsc | $\displaystyle x\le -1\textrm{ or }x\ge 1$ | $\displaystyle -\frac{\pi}{2}\le y\le\frac{\pi}{2},\,y\ne 0$ |
arccos | $\displaystyle -1\le x\le 1$ | $\displaystyle 0\le y\le \pi$ | arcsec | $\displaystyle x\le -1\textrm{ or }x\ge 1$ | $\displaystyle 0\le y\le\pi,\,y\ne\frac{\pi}{2}$ |
arctan | $\displaystyle -\infty < x < \infty$ | $\displaystyle -\frac{\pi}{2} < y < \frac{\pi}{2}$ | arccot | $\displaystyle -\infty < x < \infty$ | $\displaystyle 0 < y < \pi$ |
$\displaystyle y=\sin^{-1}x$ | ⇔ | $\displaystyle \sin y=x$ and $\displaystyle -\frac{\pi}2\le y\le \frac{\pi}2$ |
$\displaystyle y=\cos^{-1}x$ | ⇔ | $\displaystyle \cos y=x$ and $\displaystyle 0\le y\le \pi$ |
$\displaystyle y=\tan^{-1}x$ | ⇔ | $\displaystyle \tan y=x$ and $\displaystyle -\frac{\pi}2 < y < \frac{\pi}2$ |
$\displaystyle \sin^{-1}x$ | $\displaystyle \csc^{-1}x$ |
$\displaystyle \cos^{-1}x$ | $\displaystyle \sec^{-1}x$ |
$\displaystyle \tan^{-1}x$ | $\displaystyle \cot^{-1}x$ |
$\displaystyle \frac{1}{\sqrt{1-x^2}}$ | $\displaystyle \frac{1}{|x|\sqrt{x^2-1}}$ |
$\displaystyle \frac{-1}{\sqrt{1-x^2}}$ | $\displaystyle \frac{-1}{|x|\sqrt{x^2-1}}$ |
$\displaystyle \frac1{1+x^2}$ | $\displaystyle \frac{-1}{1+x^2}$ |
$\displaystyle \sin^{-1}u$ | $\displaystyle \csc^{-1}u$ |
$\displaystyle \cos^{-1}u$ | $\displaystyle \sec^{-1}u$ |
$\displaystyle \tan^{-1}u$ | $\displaystyle \cot^{-1}u$ |
$\displaystyle \frac{u'}{\sqrt{1-u^2}}$ | $\displaystyle gg$ |
$\displaystyle x$ | $\displaystyle x$ |
$\displaystyle x$ | $\displaystyle x$ |
식 | 정의역 | 식 | 정의역 |
$\displaystyle \frac{d}{dx}(\sin^{-1}x)=\frac1{\sqrt{1-x^2}}$ | $\displaystyle -1 | $\displaystyle \frac{d}{dx}(\csc^{-1}x)=-\frac1{|x|\sqrt{x^2-1}}$ | $\displaystyle \textrm{ for }|x|>1$ |
$\displaystyle \frac{d}{dx}(\cos^{-1}x)=-\frac1{\sqrt{1-x^2}}$ | $\displaystyle \textrm{ for }-1 | $\displaystyle \frac{d}{dx}(\sec^{-1}x)=\frac1{|x|\sqrt{x^2-1}}$ | $\displaystyle |x|>1$ |
$\displaystyle \frac{d}{dx}(\tan^{-1}x)=\frac1{1+x^2}$ | $\displaystyle -\infty | $\displaystyle \frac{d}{dx}(\cot^{-1}x)=-\frac1{1+x^2}$ | $\displaystyle \textrm{ for }-\infty |