雙曲函數恆等式

sinh, cosh 和 tanh

csch, sech 和 coth

雙曲函數基本恒等式如下：

${\displaystyle \cosh ^{2}x-\sinh ^{2}x=1\,}$

${\displaystyle \tanh x\cdot \coth x\,=1}$

${\displaystyle 1\,-\tanh ^{2}x=\operatorname {sech} ^{2}x}$

${\displaystyle \coth ^{2}x-1\,=\operatorname {csch} ^{2}x}$

• ${\displaystyle \sinh x={{e^{x}-e^{-x}} \over 2}}$
• ${\displaystyle \cosh x={{e^{x}+e^{-x}} \over 2}}$
• ${\displaystyle \tanh x={{\sinh x} \over {\cosh x}}}$
• ${\displaystyle \coth x={1 \over {\tanh x}}}$
• ${\displaystyle {\mathop {\rm {sech}} }x={1 \over {\cosh x}}}$
• ${\displaystyle {\mathop {\rm {csch}} }x={1 \over {\sinh x}}}$

就如同三角函數，由上面的平方關係加上雙曲函數的基本定義，可以導出下面的表格，即每個雙曲函數都可以用其他五個表達。（严谨地说，所有根号前都应根据实际情况添加正负号）

函數	sinh	cosh	tanh	coth	sech	csch
${\displaystyle \sinh x}$	${\displaystyle \sinh x\ }$	${\displaystyle \operatorname {sgn} x{\sqrt {\cosh ^{2}x-1}}}$	${\displaystyle {\frac {\tanh x}{\sqrt {1-\tanh ^{2}x}}}}$	${\displaystyle {\frac {\operatorname {sgn} x}{\sqrt {\coth ^{2}x-1}}}}$	${\displaystyle \operatorname {sgn}(x){\frac {\sqrt {1-\operatorname {sech} ^{2}(x)}}{\operatorname {sech} (x)}}}$	${\displaystyle {\frac {1}{\operatorname {csch} (x)}}}$
${\displaystyle \cosh x}$	${\displaystyle {\sqrt {1+\sinh ^{2}x}}}$	${\displaystyle \cosh x\ }$	${\displaystyle {\frac {1}{\sqrt {1-\tanh ^{2}x}}}}$	${\displaystyle \,{\frac {\left|\coth(x)\right|}{\sqrt {\coth ^{2}(x)-1}}}}$	${\displaystyle \,{\frac {1}{\operatorname {sech} (x)}}}$	${\displaystyle \,{\frac {\sqrt {1+\operatorname {csch} ^{2}(x)}}{\left|\operatorname {csch} (x)\right|}}}$
${\displaystyle \tanh x}$	${\displaystyle {\frac {\sinh x}{\sqrt {1+\sinh ^{2}x}}}}$	${\displaystyle {\frac {\operatorname {sgn} x{\sqrt {\cosh ^{2}x-1}}}{\cosh x}}}$	${\displaystyle \tanh x\ }$	${\displaystyle {\frac {1}{\coth x}}}$	${\displaystyle \,\operatorname {sgn}(x){\sqrt {1-\operatorname {sech} ^{2}(x)}}}$	${\displaystyle \,{\frac {\operatorname {sgn}(x)}{\sqrt {1+\operatorname {csch} ^{2}(x)}}}}$
${\displaystyle \coth x}$	${\displaystyle {{\sqrt {1+\sinh ^{2}x}} \over \sinh x}}$	${\displaystyle {\cosh x \over \operatorname {sgn} x{\sqrt {\cosh ^{2}x-1}}}}$	${\displaystyle {1 \over \tanh x}}$	${\displaystyle \coth x\ }$	${\displaystyle \,{\frac {\operatorname {sgn}(x)}{\sqrt {1-\operatorname {sech} ^{2}(x)}}}}$	${\displaystyle \,\operatorname {sgn}(x){\sqrt {1+\operatorname {csch} ^{2}(x)}}}$
${\displaystyle \operatorname {sech} x}$	${\displaystyle {1 \over {\sqrt {1+\sinh ^{2}x}}}}$	${\displaystyle {1 \over \cosh \theta }}$	${\displaystyle {\sqrt {1-\tanh ^{2}x}}}$	${\displaystyle \,{\frac {\sqrt {\coth ^{2}(x)-1}}{\left|\coth(x)\right|}}}$	${\displaystyle \operatorname {sech} x\ }$	${\displaystyle \,{\frac {\left|\operatorname {csch} (x)\right|}{\sqrt {1+\operatorname {csch} ^{2}(x)}}}}$
${\displaystyle \operatorname {csch} x}$	${\displaystyle {1 \over \sinh x}}$	${\displaystyle {\frac {\operatorname {sgn} x}{\sqrt {\cosh ^{2}x-1}}}}$	${\displaystyle {\frac {\sqrt {1-\tanh ^{2}x}}{\tanh x}}}$	${\displaystyle \,\operatorname {sgn}(x){\sqrt {\coth ^{2}(x)-1}}}$	${\displaystyle \,\operatorname {sgn}(x){\frac {\operatorname {sech} (x)}{\sqrt {1-\operatorname {sech} ^{2}(x)}}}}$	${\displaystyle \operatorname {csch} x\ }$

其他函數的基本關係

三角函數還有正矢、餘矢、半正矢、半餘矢、外正割、外餘割等函數，利用他們的定義也可以導出雙曲函數。

名稱	函數	值
雙曲正矢, hyperbolic versine	${\displaystyle \operatorname {versinh} (x)}$ ${\displaystyle \operatorname {vsnh} (x)}$	${\displaystyle \cosh x-1}$
雙曲餘矢, hyperbolic coversine	${\displaystyle \operatorname {coversinh} (x)}$ ${\displaystyle \operatorname {cvsh} (x)}$	${\displaystyle \sinh x-1}$
雙曲半正矢 , hyperbolic haversine	${\displaystyle \operatorname {haversinh} (x)}$	${\displaystyle {\frac {\operatorname {versinh} (x)}{2}}}$
雙曲半餘矢 , hyperbolic hacoversine	${\displaystyle \operatorname {hacoversinh} (x)}$	${\displaystyle {\frac {\operatorname {cvsh} (x)}{2}}}$
雙曲外正割 , hyperbolic exsecant	${\displaystyle \operatorname {exsech} (x)}$	${\displaystyle 1-\operatorname {sech} (x)}$
雙曲外餘割 , hyperbolic excosecant	${\displaystyle \operatorname {excsch} (x)}$	${\displaystyle 1-\operatorname {csch} (x)}$

${\displaystyle \sinh(x+y)\ =\sinh x\cosh y+\cosh x\sinh y\,}$

${\displaystyle \sinh(x-y)\ =\sinh x\cosh y-\cosh x\sinh y\,}$

${\displaystyle \cosh(x+y)\ =\cosh x\cosh y+\sinh x\sinh y\,}$

${\displaystyle \cosh(x-y)\ =\cosh x\cosh y-\sinh x\sinh y\,}$

${\displaystyle \tanh(x+y)\ ={\frac {\tanh x+\tanh y}{1+\tanh x\tanh y}}\,}$

${\displaystyle \tanh(x-y)\ ={\frac {\tanh x-\tanh y}{1-\tanh x\tanh y}}\,}$

${\displaystyle \sinh x+\sinh y\ =2\sinh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\,}$

${\displaystyle \sinh x-\sinh y\ =2\cosh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\,}$

${\displaystyle \cosh x+\cosh y\ =2\cosh {\frac {x+y}{2}}\cosh {\frac {x-y}{2}}\,}$

${\displaystyle \cosh x-\cosh y\ =2\sinh {\frac {x+y}{2}}\sinh {\frac {x-y}{2}}\,}$

${\displaystyle \tanh x+\tanh y\ ={\frac {\sinh(x+y)}{\cosh x\cosh y}}\,}$

${\displaystyle \tanh x-\tanh y\ ={\frac {\sinh(x-y)}{\cosh x\cosh y}}\,}$

${\displaystyle \sinh x\sinh y\ ={\frac {\cosh(x+y)-\cosh(x-y)}{2}}\,}$

${\displaystyle \cosh x\cosh y\ ={\frac {\cosh(x+y)+\cosh(x-y)}{2}}\,}$

${\displaystyle \sinh x\cosh y\ ={\frac {\sinh(x+y)+\sinh(x-y)}{2}}\,}$

• 二倍角公式：

${\displaystyle \sinh 2x\ =2\sinh x\cosh x\,}$

${\displaystyle \cosh 2x\ =\cosh ^{2}x+\sinh ^{2}x=2\cosh ^{2}x-1=2\sinh ^{2}x+1\,}$

${\displaystyle \tanh 2x\ ={\frac {2\tanh x}{1+\tanh ^{2}x}}\,}$

• 三倍角公式：

${\displaystyle \sinh 3x\ =3\sinh x+4\sinh ^{3}x}$

${\displaystyle \cosh 3x\ =4\cosh ^{3}x-3\cosh x}$

${\displaystyle \sinh {\frac {x}{2}}\ =\operatorname {sgn} x{\sqrt {\frac {\cosh x-1}{2}}}}$

${\displaystyle \cosh {\frac {x}{2}}\ ={\sqrt {\frac {\cosh x+1}{2}}}}$

${\displaystyle \tanh {\frac {x}{2}}\ ={\frac {\cosh x-1}{\sinh x}}\ ={\frac {\sinh x}{1+\cosh x}}\,}$

${\displaystyle \sinh ^{2}x={\frac {\cosh 2x-1}{2}}\,}$

${\displaystyle \cosh ^{2}x={\frac {\cosh 2x+1}{2}}\,}$

${\displaystyle \tanh ^{2}x={\frac {\cosh 2x-1}{\cosh 2x+1}}\,}$

${\displaystyle \sinh x={\frac {2\tanh {\frac {x}{2}}}{1-\tanh ^{2}{\frac {x}{2}}}}}$

${\displaystyle \cosh x={\frac {1+\tanh ^{2}{\frac {x}{2}}}{1-\tanh ^{2}{\frac {x}{2}}}}}$

${\displaystyle \tanh x={\frac {2\tanh {\frac {x}{2}}}{1+\tanh ^{2}{\frac {x}{2}}}}}$

${\displaystyle \sinh x=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}}$

${\displaystyle \cosh x=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}}$

${\displaystyle \tanh x=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots =\sum _{n=1}^{\infty }{\frac {2^{2n}(2^{2n}-1)B_{2n}x^{2n-1}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}}$

${\displaystyle \coth x={\frac {1}{x}}+{\frac {x}{3}}-{\frac {x^{3}}{45}}+{\frac {2x^{5}}{945}}+\cdots ={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {2^{2n}B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi }$ (罗朗级数)

${\displaystyle \operatorname {sech} \,x=1-{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}-{\frac {61x^{6}}{720}}+\cdots =\sum _{n=0}^{\infty }{\frac {E_{2n}x^{2n}}{(2n)!}},\left|x\right|<{\frac {\pi }{2}}}$

${\displaystyle \operatorname {csch} \,x={\frac {1}{x}}-{\frac {x}{6}}+{\frac {7x^{3}}{360}}-{\frac {31x^{5}}{15120}}+\cdots ={\frac {1}{x}}+\sum _{n=1}^{\infty }{\frac {2(1-2^{2n-1})B_{2n}x^{2n-1}}{(2n)!}},0<\left|x\right|<\pi }$ (罗朗级数)

其中

${\displaystyle B_{n}\,}$ 是第n項 伯努利數

${\displaystyle E_{n}\,}$ 是第n項 欧拉數

利用三角恒等式的指數定義和雙曲函數的指數定義即可求出下列恆等式:

${\displaystyle e^{ix}=\cos x+i\;\sin x\qquad ,\;e^{-ix}=\cos x-i\;\sin x}$

${\displaystyle e^{x}=\cosh x+\sinh x\!\qquad ,\;e^{-x}=\cosh x-\sinh x\!}$

所以

${\displaystyle \cosh ix={\tfrac {1}{2}}(e^{ix}+e^{-ix})=\cos x}$

${\displaystyle \sinh ix={\tfrac {1}{2}}(e^{ix}-e^{-ix})=i\sin x}$

下表列出部分的三角函數與雙曲函數的恆等式:

三角函數	雙曲函數
${\displaystyle \sin \theta =-i\sinh {i\theta }\,}$	${\displaystyle \sinh {\theta }=i\sin {(-i\theta )}\,}$
${\displaystyle \cos {\theta }=\cosh {i\theta }\,}$	${\displaystyle \cosh {\theta }=\cos {(-i\theta )}\,}$
${\displaystyle \tan \theta ={\frac {\tanh {i\theta }}{i}}\,}$	${\displaystyle \tanh {\theta }=i\tan {(-i\theta )}\,}$
${\displaystyle \cot {\theta }=i\coth {i\theta }\,}$	${\displaystyle \coth \theta ={\frac {\cot {(-i\theta )}}{i}}\,}$
${\displaystyle \sec {\theta }=\operatorname {sech} {\,i\theta }\,}$	${\displaystyle \operatorname {sech} {\theta }=\sec {(-i\theta )}\,}$
${\displaystyle \csc {\theta }=i\;\operatorname {csch} {\,i\theta }\,}$	${\displaystyle \operatorname {csch} \theta ={\frac {\csc {(-i\theta )}}{i}}\,}$

• 其他恆等式:

${\displaystyle \cosh ix={\tfrac {1}{2}}(e^{ix}+e^{-ix})=\cos x}$

${\displaystyle \sinh ix={\tfrac {1}{2}}(e^{ix}-e^{-ix})=i\sin x}$

${\displaystyle \cosh(x+iy)=\cosh(x)\cos(y)+i\sinh(x)\sin(y)\,}$

${\displaystyle \sinh(x+iy)=\sinh(x)\cos(y)+i\cosh(x)\sin(y)\,}$

${\displaystyle \tanh ix=i\tan x\,}$

${\displaystyle \cosh x=\cos ix\,}$

${\displaystyle \sinh x=-i\sin ix\,}$

${\displaystyle \tanh x=-i\tan ix\,}$

• 三角函數恆等式
• 雙曲函數
• 雙曲線
• 三角函數
• 三角形

• 數學基本公式手冊 九章出版社 ISBN 957-603-010-2