1. 反三角函数的定义值及值域
| 反三角函数 | 三角函数 | 定义域 | 值域 |
|---|---|---|---|
| y = arcsin ( x ) y = \arcsin(x) y=arcsin(x) | x = sin ( y ) x = \sin(y) x=sin(y) | − 1 ≤ x ≤ 1 -1 \leq x \leq 1 −1≤x≤1 | − π 2 ≤ y ≤ π 2 -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} −2π≤y≤2π |
| y = arccos ( x ) y = \arccos(x) y=arccos(x) | x = cos ( y ) x = \cos(y) x=cos(y) | − 1 ≤ x ≤ 1 -1 \leq x \leq 1 −1≤x≤1 | 0 ≤ y ≤ π 0 \leq y \leq \pi 0≤y≤π |
| y = arctan ( x ) y = \arctan(x) y=arctan(x) | x = tan ( y ) x = \tan(y) x=tan(y) | − ∞ ≤ x < + ∞ -\infty \leq x < +\infty −∞≤x<+∞ | − π 2 < y < π 2 -\frac{\pi}{2} < y < \frac{\pi}{2} −2π<y<2π |
| y = arccot ( x ) y = \text{arccot}(x) y=arccot(x) | x = cot ( y ) x = \cot(y) x=cot(y) | − ∞ ≤ x < + ∞ -\infty \leq x < +\infty −∞≤x<+∞ | 0 < y < π 0 < y < \pi 0<y<π |
| y = arcsec ( x ) y = \text{arcsec}(x) y=arcsec(x) | x = sec ( y ) x = \sec(y) x=sec(y) | x ≤ − 1 or 1 ≤ x x \leq -1 \text{ or } 1 \leq x x≤−1 or 1≤x | 0 ≤ y < π 2 or π 2 < y ≤ π 0 \leq y < \frac{\pi}{2} \text{ or } \frac{\pi}{2} < y \leq \pi 0≤y<2π or 2π<y≤π |
2.反三角函数的导数及其定义域
反函数的导 ( d a ˇ o ) (dǎo) (daˇo)数等于直接函数的导数的倒 ( d a ˋ o ) (dào) (daˋo)数。
d x d y = 1 d y d x \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} dydx=dxdy1
| 编号 | 导数 | 定义域 |
|---|---|---|
| 1 | ( arcsin x ) ′ = 1 1 − x 2 (\arcsin x)' = \dfrac{1}{\sqrt{1 - x^2}} (arcsinx)′=1−x2 1 | − 1 < x < 1 -1 < x < 1 −1<x<1 |
| 2 | ( arccos x ) ′ = − 1 1 − x 2 (\arccos x)' = -\dfrac{1}{\sqrt{1 - x^2}} (arccosx)′=−1−x2 1 | − 1 < x < 1 -1 < x < 1 −1<x<1 |
| 3 | ( arctan x ) ′ = 1 1 + x 2 (\arctan x)' = \dfrac{1}{1 + x^2} (arctanx)′=1+x21 | − ∞ < x < ∞ -\infty < x < \infty −∞<x<∞ |
| 4 | ( arccot x ) ′ = − 1 1 + x 2 (\text{arccot } x)' = -\dfrac{1}{1 + x^2} (arccot x)′=−1+x21 | − ∞ < x < ∞ -\infty < x < \infty −∞<x<∞ |
1\] [关于反三角函数及其导数](https://zhuanlan.zhihu.com/p/201720785)