正切半角公式

正切半角公式又稱萬能公式，這一組公式有四個功能：

1. 將角統一為${\displaystyle {\frac {\alpha }{2}}}$^[1]；
2. 將函數名稱統一為${\displaystyle \tan }$；
3. 任意實數都可以${\displaystyle \tan {\frac {\alpha }{2}}}$的形式表達，可用正切函數換元。
4. 在某些積分中，可以將含有三角函數的積分變為有理分式的積分。

因此，這組公式被稱為以切表弦公式，簡稱以切表弦。它們是由二倍角公式求得的。

${\displaystyle \sin \alpha ={\frac {2\tan {\frac {\alpha }{2}}}{1+\tan ^{2}{\frac {\alpha }{2}}}}}$

${\displaystyle \cos \alpha ={\frac {1-\tan ^{2}{\frac {\alpha }{2}}}{1+\tan ^{2}{\frac {\alpha }{2}}}}}$

${\displaystyle \tan \alpha ={\frac {2\tan {\frac {\alpha }{2}}}{1-\tan ^{2}{\frac {\alpha }{2}}}}}$

${\displaystyle \cot \alpha ={\frac {1-\tan ^{2}{\frac {\alpha }{2}}}{2\tan {\frac {\alpha }{2}}}}}$

${\displaystyle \sec \alpha ={\frac {1+\tan ^{2}{\frac {\alpha }{2}}}{1-\tan ^{2}{\frac {\alpha }{2}}}}}$

${\displaystyle \csc \alpha ={\frac {1+\tan ^{2}{\frac {\alpha }{2}}}{2\tan {\frac {\alpha }{2}}}}}$

而被稱為萬能公式的原因是利用${\displaystyle \tan {\frac {\alpha }{2}}}$的代換可以解決一些有關三角函數的積分。參見三角換元法。

${\displaystyle {\begin{aligned}\tan \left({\frac {\eta }{2}}\pm {\frac {\theta }{2}}\right)&={\frac {\sin \eta \pm \sin \theta }{\cos \eta +\cos \theta }}=-{\frac {\cos \eta -\cos \theta }{\sin \eta \mp \sin \theta }},\\[10pt]\tan \left(\pm {\frac {\theta }{2}}\right)&={\frac {\pm \sin \theta }{1+\cos \theta }}={\frac {\pm \tan \theta }{\sec \theta +1}}={\frac {\pm 1}{\csc \theta +\cot \theta }},~~~~(\eta =0)\\[10pt]\tan \left(\pm {\frac {\theta }{2}}\right)&={\frac {1-\cos \theta }{\pm \sin \theta }}={\frac {\sec \theta -1}{\pm \tan \theta }}=\pm (\csc \theta -\cot \theta ),~~~~(\eta =0)\\[10pt]\tan \left({\frac {\pi }{4}}\pm {\frac {\theta }{2}}\right)&={\frac {1\pm \sin \theta }{\cos \theta }}=\sec \theta \pm \tan \theta ={\frac {\csc \theta \pm 1}{\cot \theta }},~~~~(\eta ={\frac {\pi }{2}})\\[10pt]\tan \left({\frac {\pi }{4}}\pm {\frac {\theta }{2}}\right)&={\frac {\cos \theta }{1\mp \sin \theta }}={\frac {1}{\sec \theta \mp \tan \theta }}={\frac {\cot \theta }{\csc \theta \mp 1}},~~~~(\eta ={\frac {\pi }{2}})\\[10pt]{\frac {1-\tan {\frac {\theta }{2}}}{1+\tan {\frac {\theta }{2}}}}&={\sqrt {\frac {1-\sin \theta }{1+\sin \theta }}}.\end{aligned}}}$

萬能公式的證明

由二倍角公式，有：

${\displaystyle \sin \alpha =2\sin {\frac {\alpha }{2}}\cos {\frac {\alpha }{2}}={\frac {2\sin {\frac {\alpha }{2}}\cos {\frac {\alpha }{2}}}{\cos ^{2}{\frac {\alpha }{2}}+\sin ^{2}{\frac {\alpha }{2}}}}={\frac {2\sin {\frac {\alpha }{2}}\cos {\frac {\alpha }{2}}\div \cos ^{2}{\frac {\alpha }{2}}}{(\cos ^{2}{\frac {\alpha }{2}}+\sin ^{2}{\frac {\alpha }{2}})\div \cos ^{2}{\frac {\alpha }{2}}}}={\frac {2{\frac {\sin {\frac {\alpha }{2}}}{\cos {\frac {\alpha }{2}}}}}{1+{\frac {\sin ^{2}{\frac {\alpha }{2}}}{\cos ^{2}{\frac {\alpha }{2}}}}}}={\frac {2\tan {\frac {\alpha }{2}}}{1+\tan ^{2}{\frac {\alpha }{2}}}}}$

${\displaystyle \tan \alpha ={\frac {2\tan {\frac {\alpha }{2}}}{1-\tan ^{2}{\frac {\alpha }{2}}}}}$

再由同角三角函數間的關係，得出

${\displaystyle \cos \alpha ={\frac {\sin \alpha }{\tan \alpha }}={\frac {\frac {2\tan {\frac {\alpha }{2}}}{1+\tan ^{2}{\frac {\alpha }{2}}}}{\frac {2\tan {\frac {\alpha }{2}}}{1-\tan ^{2}{\frac {\alpha }{2}}}}}={\frac {1-\tan ^{2}{\frac {\alpha }{2}}}{1+\tan ^{2}{\frac {\alpha }{2}}}}}$

幾何證明

正切半角公式的幾何證明

在單位圓內，${\displaystyle t=\tan {\frac {\phi }{2}}}$。根據相似關係，${\displaystyle {\frac {t}{\sin \phi }}={\frac {1}{1+\cos \phi }}}$，可得出

${\displaystyle t={\frac {\sin \phi }{1+\cos \phi }}={\frac {\sin \phi (1-\cos \phi )}{(1+\cos \phi )(1-\cos \phi )}}={\frac {1-\cos \phi }{\sin \phi }}}$

。

顯然${\displaystyle \tan {\frac {a+b}{2}}={\frac {\sin {\frac {a+b}{2}}}{\cos {\frac {a+b}{2}}}}={\frac {\sin a+\sin b}{\cos a+\cos b}}}$。

雙曲函數

此公式亦可以對雙曲函數起到類似的作用，由雙曲線右支上的一點${\displaystyle (\cosh \theta ,\sinh \theta )}$給出。從${\displaystyle (-1,0)}$到y軸給出了如下等式：

${\displaystyle t=\tanh {\frac {1}{2}}\theta ={\frac {\sinh \theta }{\cosh \theta +1}}={\frac {\cosh \theta -1}{\sinh \theta }}}$

可以得到

${\displaystyle \cosh \theta ={\frac {1+t^{2}}{1-t^{2}}},}$		${\displaystyle \sinh \theta ={\frac {2t}{1-t^{2}}},}$
${\displaystyle \tanh \theta ={\frac {2t}{1+t^{2}}},}$		${\displaystyle \coth \theta ={\frac {1+t^{2}}{2t}},}$
${\displaystyle \mathrm {sech} \,\theta ={\frac {1-t^{2}}{1+t^{2}}},}$		${\displaystyle \mathrm {csch} \,\theta ={\frac {1-t^{2}}{2t}},}$

和

${\displaystyle e^{\theta }={\frac {1+t}{1-t}},}$

${\displaystyle e^{-\theta }={\frac {1-t}{1+t}}.}$

卡爾·維爾斯特拉斯引入這個式子來省去查找原函數的麻煩。

${\displaystyle \theta }$在${\displaystyle T}$而得出下面的雙曲反正切函數和自然對數之間的關係：

${\displaystyle \operatorname {artanh} t={\frac {1}{2}}\ln {\frac {1+t}{1-t}}.}$

參見

• 三角恆等式