三角函数的嵌套

基本计算

$$\LARGE \begin{array}{ll} \sin \theta \csc \theta =1 \\ \cos \theta \sec \theta =1 \\ \tan \theta \cot \theta =1 \\ \sin ^2 \theta + \cos ^2 \theta =1 \\ 1+ \tan ^2 \theta = \sec ^2 \theta \\ 1+ \cot ^2 \theta = \csc ^2 \theta \end{array}$$

正弦嵌套

$$\LARGE \begin{array}{ll} \sin(\arctan x)=\frac{x}{\sqrt{1+x^2}} \\ \sin(\arccos x)=\sqrt{1-x^2} \\ \sin(\text{arccot }x)=\frac{1}{\sqrt{1+x^2}} \\ \sin(\text{arccsc }x)=\frac{1}{x} \\ \sin(\text{arcsec }x)=\sqrt{1-\frac{1}{x^2}} \end{array}$$

余弦嵌套

$$\LARGE \begin{array}{ll} \cos(\arctan x)=\frac{1}{\sqrt{1+x^2}} \\ \cos(\arcsin x)=\sqrt{1-x^2} \\ \cos(\text{arccot }x)=\frac{x}{\sqrt{1+x^2}} \\ \cos(\text{arccsc }x)=\sqrt{1-\frac{1}{x^2}} \\ \cos(\text{arcsec }x)=\frac{1}{x} \end{array}$$

正切嵌套

$$\LARGE \begin{array}{ll} \tan(\arcsin x)=\frac{x}{\sqrt{1-x^2}} \\ \tan(\arccos x)=\frac{\sqrt{1-x^2}}{x} \\ \tan(\text{arccsc }x)=\frac{1}{\sqrt{x^2-1}} \\ \tan(\text{arcsec }x)=\sqrt{x^2-1} \\ \tan(\text{arccot }x)=\frac{1}{x} \end{array}$$

余切嵌套

$$\LARGE \begin{array}{ll} \cot(\arcsin x)=\frac{\sqrt{1-x^2}}{x} \\ \cot(\arccos x)=\frac{x}{\sqrt{1-x^2}} \\ \cot(\arctan x)=\frac{1}{x} \\ \cos(\text{arccsc }x)=\sqrt{x^2-1} \\ \cot(\text{arcsec }x)=\frac{1}{\sqrt{x^2-1}} \end{array}$$

正割嵌套

$$\LARGE \begin{array}{ll} \sec(\arcsin x)=\frac{1}{\sqrt{1-x^2}} \\ \sec(\arccos x)=\frac{1}{x} \\ \sec(\arctan x)=\sqrt{1+x^2} \\ \sec(\text{arccot }x)=\sqrt{1+\frac{1}{x^2}} \\ \sec(\text{arccsc }x)=\frac{x}{\sqrt{x^2-1}} \end{array}$$

余割嵌套

$$\LARGE \begin{array}{ll} \csc(\arcsin x)=\frac{1}{x} \\ \csc(\arccos x)=\frac{1}{\sqrt{1-x^2}} \\ \csc(\arctan x)=\frac{\sqrt{1+x^2}}{x} \\ \csc(\text{arccot }x)=\sqrt{x^2+1} \\ \csc(\text{arcsec }x)=\frac{x}{\sqrt{x^2-1}} \end{array}$$