قوانين التوابع القطعية

قوانين التوابع القطعية
(E: Hyperbolic Functions Formulas)
(T: Hiperbolik fonksiyonlar Formülleri)

\[\sinh (-x)=-\sinh x\]
\[\cosh (-x)=\cosh x\]
\[\tanh (-x)=-\tanh x\]
\[\coth (-x)=-\coth x\]
\[ sech (-x)=sech x\]
\[csch (-x)=-csch x\]
\[\cosh^2 x-\sinh^2 x=1\]
\[sech^2 x=1-\tanh^2 x\]
\[csch^2 x=\coth^2 x-1\]
\[\sinh(x+y)=\sinh x\cosh y+\cosh x\sinh y\]
\[\sinh(x-y)=\sinh x\cosh y-\cosh x\sinh y\]
\[\cosh(x+y)=\cosh x\cosh y+\sinh x\sinh y\]
\[\cosh(x-y)=\cosh x\cosh y-\sinh x\sinh y\]
\[\tanh (x+y)=\frac{\tanh x+\tanh y}{1+\tanh x \tanh y}\]
\[\tanh (x-y)=\frac{\tanh x-\tanh y}{1-\tanh x \tanh y}\]
\[\coth (x+y)=\frac{\coth x \coth y+1}{\coth x+\coth y}\]
\[\coth (x-y)=\frac{\coth x \coth y-1}{\coth x-\coth y}\]
\[\sinh 2x=2 \sinh x\cosh x\]
\[\cosh 2x=\cosh^2 x+\sinh^2 x\]
\[~~~~~~=2 \cosh^2 x-1\]
\[~~~~~~=1+2\sinh^2 x\]
\[\tanh 2x=\frac{2\tanh x}{1+\tanh^2 x}\]
\[\sinh 3x=3 \sinh x+4 \sinh^3 x\]
\[\cosh 3x=4\cosh^3 x-3\cosh x\]
\[\tanh 3x=\frac{3\tanh x+\tanh^3 x}{1+3\tanh^2 x}\]
\[\sinh \frac x 2=\pm \sqrt{\frac{\cosh x-1}{2}}\]
\[\cosh \frac x 2= \sqrt{\frac{\cos x+1}{2}}\]
\[\tanh \frac x 2=\pm \sqrt{\frac{\cosh x-1}{\cosh x+1}}\]
\[\sinh^2 x=\frac{\cosh 2x-1}{2}\]
\[\cosh^2 x=\frac{\cosh 2x+1}{2}\]
\[\tanh^2 x=\frac{\cosh 2x-1}{\cosh 2x+1}\]
\[\coth^2 x=\frac{\cosh 2x+1}{\cosh 2x-1}\]
\[\sinh x+\sinh y=2\sinh \frac{x+y}{2} \cosh \frac{x-y}{2}\]
\[\sinh x-\sinh y=2\cosh \frac{x+y}{2} \sinh \frac{x-y}{2}\]
\[\cosh x+\cosh y=2\cosh \frac{x+y}{2} \cosh \frac{x-y}{2}\]
\[\cosh x-\cosh y=-2\sinh \frac{x+y}{2} \sinh \frac{x-y}{2}\]
\[\sinh x \cosh y=\frac 12(\sinh (x+y)+\sinh(x-y))\]
\[\cosh x \cosh y=\frac 12(\cosh (x+y)+\cosh (x-y))\]
\[\sinh x \sinh y=\frac 12(\cosh (x+y)-\cosh (x-y))\]