قوانين التوابع المثلثية

قوانين التوابع المثلثية
(E: Trigonometric Functions Formulas)
(T: Trigonometrik Fonksiyonlar Formülleri)

\[\sin (-x)=-\sin x\]
\[\cos (-x)=\cos x\]
\[\tan (-x)=-\tan x\]
\[\cot (-x)=-\cot x\]
\[\sec(-x)=\sec x\]
\[\csc (-x)=-\csc x\]
\[\sin (\frac \pi 2-x)=\cos x\]
\[\cos (\frac \pi 2-x)=\sin x\]
\[\tan(\frac \pi 2-x)=\cot x\]
\[\cot(\frac \pi 2-x)=\tan x\]
\[\sin (\frac \pi 2+x)=\cos x\]
\[\cos (\frac \pi 2+x)=-\sin x\]
\[\tan(\frac \pi 2+x)=-\cot x\]
\[\cot(\frac \pi 2+x)=-\tan x\]
\[\sin ( \pi -x)=\sin x\]
\[\cos ( \pi -x)=-\cos x\]
\[\tan( \pi -x)=-\tan x\]
\[\cot( \pi -x)=-\cot x\]
\[\sin ( \pi +x)=-\sin x\]
\[\cos ( \pi +x)=-\cos x\]
\[\tan( \pi +x)=\tan x\]
\[\cot( \pi +x)=\cot x\]
\[\sin (\frac {3\pi}{ 2}-x)=-\cos x\]
\[\cos (\frac {3\pi}{ 2}-x)=-\sin x\]
\[\tan(\frac {3\pi}{ 2}-x)=\cot x\]
\[\cot(\frac {3\pi}{ 2}-x)=\sin x\]
\[\sin (\frac {3\pi}{ 2}+x)=-\cos x\]
\[\cos (\frac {3\pi}{ 2}+x)=\sin x\]
\[\tan(\frac {3\pi}{ 2}+x)=-\cot x\]
\[\cot(\frac {3\pi}{ 2}+x)=-\tan x\]
\[\sin^2 x+\cos^2 x=1\]
\[\sec^2 x=1+\tan^2 x\]
\[\csc^2 x=1+\cot^2 x\]
\[\sin(x+y)=\sin x\cos y+\cos x\sin y\]
\[\sin(x-y)=\sin x\cos y-\cos x\sin y\]
\[\cos(x+y)=\cos x\cos y-\sin x\sin y\]
\[\cos(x-y)=\cos x\cos y+\sin x\sin y\]
\[\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y}\]
\[\tan (x-y)=\frac{\tan x-\tan y}{1+\tan x \tan y}\]
\[\cot (x+y)=\frac{\cot x \cot y-1}{\cot x+\cot y}\]
\[\cot (x-y)=\frac{\cot x \cot y+1}{\cot x-\cot y}\]
\[\sin 2x=2 \sin x\cos x\]
\[~~~~~=\frac{2\tan x}{1+\tan^2 x}\]
\[\cos 2x=\cos^2 x-\sin^2 x\]
\[~~~~~~=2 \cos^2 x-1\]
\[~~~~~~=1-2\sin^2 x\]
\[~~~~~~=\frac{1-\tan ^2 x}{1+\tan ^2 x}\]
\[\tan 2x=\frac{2\tan x}{1-\tan^2 x}\]
\[\cot 2x=\frac{\cot^2 x-1}{2\cot x}\]
\[\sec 2x=\frac{\sec^2 x}{2-\sec^2 x}\]
\[\csc 2x=\frac{\sec x \csc x}{2}\]
\[\sin 3x=3 \sin x-4 \sin^3 x\]
\[\cos 3x=4\cos^3 x-3\cos x\]
\[\tan 3x=\frac{3\tan x-\tan^3 x}{1-3\tan^2 x}\]
\[\sin \frac x 2=\pm \sqrt{\frac{1-\cos x}{2}}\]
\[\cos \frac x 2=\pm \sqrt{\frac{1+\cos x}{2}}\]
\[\tan \frac x 2=\pm \sqrt{\frac{1-\cos x}{1+\cos x}}\]
\[~~~~~~~~~~=\frac{1-\cos x}{\sin x}\]
\[\cot \frac x 2=\pm \sqrt{\frac{1+\cos x}{1-\cos x}}\]
\[\sin^2 x=\frac{1-\cos 2x}{2}\]
\[\cos^2 x=\frac{1+\cos 2x}{2}\]
\[\tan^2 x=\frac{1-\cos 2x}{1+\cos 2x}\]
\[\cot^2 x=\frac{1+\cos 2x}{1-\cos 2x}\]
\[\sin x+\sin y=2\sin \frac{x+y}{2} \cos \frac{x-y}{2}\]
\[\sin x-\sin y=2\cos \frac{x+y}{2} \sin \frac{x-y}{2}\]
\[\cos x+\cos y=2\cos \frac{x+y}{2} \cos \frac{x-y}{2}\]
\[\cos x-\cos y=-2\sin \frac{x+y}{2} \sin \frac{x-y}{2}\]
\[\sin x \cos y=\frac 12(\sin (x+y)+\sin(x-y))\]
\[\cos x \cos y=\frac 12(\cos (x+y)+\cos(x-y))\]
\[\sin x \sin y=-\frac 12(\cos (x+y)+\cos (x-y))\]