查看“︁微积分学/三角函数表”︁的源代码

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== 定义 ==

* <math>\tan(x)=\frac{\sin(x)}{\cos(x)}</math>
* <math>\sec(x)=\frac{1}{\cos(x)}</math>
* <math>\cot(x)=\frac{\cos(x)}{\sin(x)}=\frac{1}{\tan(x)}</math>
* <math>\csc(x)=\frac{1}{\sin(x)}</math>

== 反三角函数 ==

* <math>\arcsin(x) = \int_0^x \frac{1}{\sqrt{1-t^2}}\mathrm{d}t= -i\log(ix + \sqrt{1-x^2})</math>
* <math>\arccos(x) = \frac{\pi}{2} - \arcsin(x) = \frac{\pi}{2} - \int_0^x \frac{1}{\sqrt{1-t^2}}\mathrm{d}t = \frac{\pi}{2} + i\log(ix + \sqrt{1-x^2})</math>
* <math>\arctan(x) = \int_0^x \frac{1}{1+t^2} \mathrm{d}t = \frac{i}{2}\log\left(\frac{1-ix}{1+ix}\right)</math>
* <math>\arccsc(x) = \arcsin\left(\frac{1}{x}\right) = -i\log\left(\frac{i}{x} + \sqrt{1 - \frac{1}{z^2}}\right)</math>
* <math>\arcsec(x) = \arccos\left(\frac{1}{x}\right) = \frac{\pi}{2} - \arcsin\left(\frac{1}{x}\right) = \frac{\pi}{2} + i\log\left(\frac{i}{x} + \sqrt{1 - \frac{1}{z^2}}\right)</math>
* <math>\arccot(x) = \arctan\left(\frac{1}{x}\right) = \frac{\pi}{2} - \arctan(x) = \frac{\pi}{2} + \frac{i}{2}\log\left(\frac{1+ix}{1-ix}\right)</math>
* <math>\arcsin(x) + \arcsin(y) = \arcsin\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right)</math>
* <math>\arccos(x) + \arccos(y) = \arccos\left(xy - \sqrt{(1-x^2)(1-y^2)}\right)</math>
* <math>\arctan(x) + \arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right) \pmod \pi</math>

== 畢氏定理 ==

* <math>\sin^2(x)+\cos^2(x)=1</math>
* <math>1+\tan^2(x)=\sec^2(x)</math>
* <math>1+\cot^2(x)=\csc^2(x)</math>

== 倍角公式 ==

* <math>\sin(2x)=2\sin(x)\cos(x)</math>
* <math>\cos(2x)=\cos^2(x)-\sin^2(x)</math>
* <math>\tan(2x)=\frac{2\tan(x)}{1-\tan^2(x)}</math>
* <math>\cos^2(x)=\frac{1+\cos(2x)}{2}</math>
* <math>\sin^2(x)=\frac{1-\cos(2x)}{2}</math>

== 和角公式 ==

:* <math>\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)</math>

:* <math>\sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)</math>

:* <math>\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)</math>

:* <math>\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)</math>

:* <math>\sin(x)+\sin(y)=2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)</math>

:* <math>\sin(x)-\sin(y)=2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)</math>

:* <math>\cos(x)+\cos(y)=2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)</math>

:* <math>\cos(x)-\cos(y)=-2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)</math>

:* <math>\tan(x)+\tan(y)=\frac{\sin(x+y)}{\cos(x)\cos(y)}</math>

:* <math>\tan(x)-\tan(y)=\frac{\sin(x-y)}{\cos(x)\cos(y)}</math>

:* <math>\cot(x)+\cot(y)=\frac{\sin(x+y)}{\sin(x)\sin(y)}</math>

:* <math>\cot(x)-\cot(y)=\frac{-\sin(x-y)}{\sin(x)\sin(y)}</math>

== 积化和差 ==

:* <math>\cos(x)\cos(y)= \frac{\cos(x+y)+\cos(x-y)}{2}</math>

:* <math>\sin(x)\sin(y)=\frac{\cos(x-y)-\cos(x+y)}{2}</math>

:* <math>\sin(x)\cos(y)=\frac{\sin(x+y)+\sin(x-y)}{2}</math>

:* <math>\cos(x)\sin(y)=\frac{\sin(x+y)-\sin(x-y)}{2}</math>

== 复指数形式 ==

:* <math>e^{i\theta} = \mathrm{cis}\theta = \cos\theta + i\sin\theta</math>
:* <math>\sin\theta = \mathrm{Im}(e^{i\theta}) = \frac{e^{i\theta}-e^{-i\theta}}{2i}</math>
:* <math>\cos\theta = \mathrm{Re}(e^{i\theta}) = \frac{e^{i\theta}+e^{-i\theta}}{2}</math>
:* <math>\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{e^{2i\theta}-1}{i(e^{2i\theta}+1)}</math>
:* <math>\csc\theta = \frac{1}{\sin\theta} = \frac{2i}{e^{i\theta}-e^{-i\theta}}</math>
:* <math>\sec\theta = \frac{1}{\cos\theta} = \frac{2}{e^{i\theta}+e^{-i\theta}}</math>
:* <math>\cot\theta = \frac{1}{\tan\theta} = \frac{i(e^{2i\theta}+1)}{e^{2i\theta}-1}</math>

== 双曲函数 ==

:* <math>e^x = \sinh x + \cosh x</math>
:* <math>\cosh^2 x - \sinh^2 x = 1</math>
:* <math>\mathrm{sech}^2 x = 1 - \tanh^2 x</math>
:* <math>\mathrm{csch}^2 x = \mathrm{coth}^2 x - 1</math>
:* <math>\sinh x = -i\sin ix = \frac{e^{x}-e^{-x}}{2}</math>
:* <math>\cosh x = \cos ix = \frac{e^{x}+e^{-x}}{2}</math>
:* <math>\tanh x = -i\tan ix = \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}</math>
:* <math>\mathrm{csch}  x = i\csc ix = \frac{2}{e^{x}-e^{-x}}</math>
:* <math>\mathrm{sech}  x = \sec ix = \frac{2}{e^{x}+e^{-x}}</math>
:* <math>\mathrm{coth}  x = i\cot ix = \frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}</math>

=== 反双曲函数 ===

:* <math>\mathrm{arsinh} x = \int_0^x \frac{1}{\sqrt{t^2 + 1}} \mathrm{d}t = \log\left(x + \sqrt{x^2 + 1}\right)</math>
:* <math>\mathrm{arcosh} x = \int_1^x \frac{1}{\sqrt{t^2 - 1}} \mathrm{d}t = \log\left(x + \sqrt{x^2 - 1}\right)</math>
:* <math>\mathrm{artanh} x = \int_0^x \frac{1}{1-t^2} \mathrm{d}t = \frac{1}{2}\log\left(\frac{1+x}{1-x}\right)</math>
:* <math>\mathrm{arccsh} x = \log\left(\frac{1 + \sqrt{1 + x^2}}{x}\right)</math>
:* <math>\mathrm{arsech} x = \log\left(\frac{1 + \sqrt{1 - x^2}}{x}\right)</math>
:* <math>\mathrm{arcoth} x = \frac{1}{2}\log\left(\frac{x+1}{x-1}\right)</math>
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