Trigonometric Functions ,Inverse Trigonometric , Hyperbolic Function

Trigonometric Functions 

$\sin x,\cos x,\tan x,\cot x,\sec x,\csc x$ 

Some fundamental relationships among these functions are as follows .

$(a) \sin x=\cos \left(\cfrac{\pi}{2}-x\right) ,\cos x=\sin\left(\cfrac{\pi}{2}-x\right), \tan x=\cfrac{\sin x}{\cos x} , \cot x=\cfrac{\cos x}{\sin x}=\cfrac{1}{\tan x}, \sec x=\cfrac{1}{\cos x}, \csc =\cfrac{1}{\sin x}$

$(a) \sin^2 x+\cos^2 x=1, \sec^2x-\tan^2x=1 ,\csc^2x-\cot^2x=1$

$(c) \sin(-x)=-\sin x,\cos (-x)=\cos x,\tan (-x)=-\tan x$

$(d) \sin(x\pm y)=\sin x\cos y\pm \cos x\sin y,\cos (x\pm y)=\cos x\cos y \mp \sin x\sin y , \tan (x\pm y)=\cfrac{\tan x\pm \tan y}{1\mp \tan x\tan y}$

$(e) A\cos x+B\sin x=\sqrt{A^2+B^2}\sin (x+\alpha)$ where $\tan \alpha =A/B$

     The trigonometric functions are $periodic$ For example $\sin x$ and $\cos x$ shown in Fig 1-1 and 1-2 respectively ,have period $2\pi$

Inverse Trigonometric Function 

     $\sin^{-1} x,\cos^{-1}x,\tan^{-1}x,\cot^{-1}x ,\sec^{-1}x ,\csc^{-1}x$
There are $inverses$ of the trigonometric functions. For example if $\sin x=y$ then $x=\sin^{-1}y,$ or on interchanging $x$ and $y,$ $y=\sin^{-1}x$.

Hyperbolic Functions 

$(a) \mathrm{sinh}x=\cfrac{e^x -e^{-x}}{2}$   $\mathrm{cosh} x=\cfrac{e^x +e^{-x}}{2}$ ,

      $\mathrm{tanh}x=\cfrac{\mathrm{sinh}x}{\mathrm{cosh}x}=\cfrac{e^x-e^{-x}}{e^x+e^{-x}}$   $\mathrm{coth}x=\cfrac{\mathrm{cosh}x}{\mathrm{sinh}x}=\cfrac{1}{\mathrm{tanh}x}=\cfrac{e^x+e^{-x}}{e^x-e^{-x}}$,

$\mathrm{sech}x=\cfrac{1}{\mathrm{cosh}x}=\cfrac{2}{e^x+e^{-x}},$  $\mathrm{csch}x=\cfrac{1}{\mathrm{sinh}x} =\cfrac{2}{e^x-e^{-x}}$

$(b) \mathrm{cosh}^2x-\mathrm{sinh}^2x=1, \mathrm{sech}^2x+\mathrm{tanh}^2x=1, \mathrm{coth}^2x-\mathrm{csch}^2x=1$

$(c) \mathrm{sinh}(x\pm y)= \mathrm{sinh}x\mathrm{cosh}y\pm \mathrm{cosh}x\mathrm{sinh}y$

$\mathrm{cosh}(x\pm y)=\mathrm{cosh}x\mathrm{cosh}y\pm \mathrm{sinh}x\mathrm{sinh}y$

$\mathrm{tanh}(x\pm y)=\cfrac{\mathrm{tanh}x\pm \mathrm{tanh}y}{1\pm \mathrm{tanh}x\mathrm{tanh}y}$

     The inverse hyperbolic function ,given by $\mathrm{sinh}^{-1} x$, $\mathrm{cosh}^{-1} x,etc$ can be expressed in terms of logarithms

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