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基本计算

sinθcscθ=1cosθsecθ=1tanθcotθ=1sin2θ+cos2θ=11+tan2θ=sec2θ1+cot2θ=csc2θ\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

sin(arcsinx)=x1+x2sin(arccosx)=1x2sin(arccot x)=11+x2sin(arccsc x)=1xsin(arcsec x)=11x2\LARGE \begin{array}{ll} \sin(\arcsin 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

cos(arctanx)=11+x2cos(arcsinx)=1x2cos(arccot x)=x1+x2cos(arccsc x)=11x2cos(arcsec x)=1x\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

tan(arcsinx)=x1x2tan(arccosx)=1x2xtan(arccsc x)=1x21tan(arcsec x)=x21tan(arccot x)=1x\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

cot(arcsinx)=1x2xcot(arccosx)=x1x2cot(arctanx)=1xcos(arccsc x)=x21cot(arcsec x)=1x21\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

sec(arcsinx)=11x2sec(arccosx)=1xsec(arctanx)=1+x2sec(arccot x)=1+1x2sec(arccsc x)=xx21\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

csc(arcsinx)=1xcsc(arccosx)=11x2csc(arctanx)=1+x2xcsc(arccot x)=x2+1csc(arcsec x)=xx21\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}

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