Trigonometry

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Definitions of Functions

$\sin {\theta }={\frac {y}{r}}={\frac {\text{opposite}}{\text{hypotenuse}}}$	$\csc {\theta }={\frac {r}{y}}={\frac {\text{hypotenuse}}{\text{opposite}}}$
$\cos {\theta }={\frac {x}{r}}={\frac {\text{adjacent}}{\text{hypotenuse}}}$	$\sec {\theta }={\frac {r}{x}}={\frac {\text{hypotenuse}}{\text{adjacent}}}$
$\tan {\theta }={\frac {y}{x}}={\frac {\text{opposite}}{\text{adjacent}}}$	$\cot {\theta }={\frac {x}{y}}={\frac {\text{adjacent}}{\text{opposite}}}$

$\sin {\theta }$ is an odd function: $\sin {(-\theta )}=-\sin {\theta }$
$\cos {\theta }$ is an even function: $\cos {(-\theta )}=\cos {\theta }$

Both the $\sin {\theta }$ and $\cos {\theta }$ functions are between -1 and 1.

Basic/Core Identities

$\tan {\theta }={\frac {\sin {\theta }}{\cos {\theta }}}$

$\csc {\theta }={\frac {1}{\sin {\theta }}}$

$\sec {\theta }={\frac {1}{\cos {\theta }}}$

$\cot {\theta }={\frac {1}{\tan {\theta }}}={\frac {\cos {\theta }}{\sin {\theta }}}$

$\sin ^{2}{\theta }+\cos ^{2}{\theta }=1\,\!$

$\tan ^{2}{\theta }+1=\sec ^{2}{\theta }\,\!$

$1+\cot ^{2}{\theta }=\csc ^{2}{\theta }\,\!$

Important Limits

$\displaystyle \lim _{x\to 0}{\frac {\sin {x}}{x}}=1$

$\displaystyle \lim _{x\to 0}{\frac {\cos {x}-1}{x}}=0$

Addition & Subtraction Formulas

Addition	Subtraction
$\sin {\left(x+y\right)}=\sin {x}\cos {y}+\cos {x}\sin {y}\,\!$	$\sin {\left(x-y\right)}=\sin {x}\cos {y}-\cos {x}\sin {y}\,\!$
$\cos {\left(x+y\right)}=\cos {x}\cos {y}-\sin {x}\sin {y}\,\!$	$\cos {\left(x-y\right)}=\cos {x}\cos {y}+\sin {x}\sin {y}\,\!$
$\tan {\left(x+y\right)}={\frac {\tan {x}+\tan {y}}{1-\tan {x}\tan {y}}}$	$\tan {\left(x-y\right)}={\frac {\tan {x}-\tan {y}}{1+\tan {x}\tan {y}}}$

Double-Angle Formulas

Derived by evaluating $\sin(x+x)$ and $\cos(x+x)$. The second and third identities for $\cos {2x}$ are derived by substituting $\sin ^{2}{x}=1-\cos ^{2}{x}$ and $\cos ^{2}{x}=1-\sin ^{2}{x}$ based on the basic Pythagorean identities.

$\sin {2x}=2\sin {x}\cos {x}\,\!$	$\cos {2x}=cos^{2}{x}-sin^{2}{x}\,\!$
	$\cos {2x}=2\cos ^{2}{x}-1\,\!$
	$\cos {2x}=1-2\sin ^{2}{x}\,\!$

Half-Angle Formulas

Derived by solving the second and third cosine double angle ($\cos {2x}$) formulas for $\sin ^{2}{x}$ and $\cos ^{2}{x}$.

$\sin ^{2}{x}={\frac {1-\cos {2x}}{2}}$

$\cos ^{2}{x}={\frac {1+\cos {2x}}{2}}$

Product Formulas

Formula for $\sin {x}\cos {y}$ derived by adding the sine addition and subtraction equations and solving for $\sin {x}\cos {x}$. Formulas for $\cos {x}\cos {y}$ and $\sin {x}\sin {y}$ derived by adding the cosine addition and subtraction equations and solving for $\cos {x}\cos {y}$ and $\sin {x}\sin {y}$, respectively.

$\sin {x}\sin {y}={\frac {1}{2}}\left[\cos \left(x-y\right)-\cos \left(x+y\right)\right]$	$\cos {x}\cos {y}={\frac {1}{2}}\left[\cos \left(x+y\right)+\cos \left(x-y\right)\right]$
$\sin {x}\cos {y}={\frac {1}{2}}\left[\sin \left(x+y\right)+\sin \left(x-y\right)\right]$

Derivatives of Inverse Functions

${\frac {d}{dx}}\left(\sin ^{-1}{x}\right)={\frac {1}{\sqrt {1-x^{2}}}}$	${\frac {d}{dx}}\left(\cos ^{-1}{x}\right)=-{\frac {1}{\sqrt {1-x^{2}}}}$
${\frac {d}{dx}}\left(\sec ^{-1}{x}\right)={\frac {1}{x{\sqrt {x^{2}-1}}}}$	${\frac {d}{dx}}\left(\csc ^{-1}{x}\right)=-{\frac {1}{x{\sqrt {x^{2}-1}}}}$
${\frac {d}{dx}}\left(\tan ^{-1}{x}\right)={\frac {1}{1+x^{2}}}$	${\frac {d}{dx}}\left(\cot ^{-1}{x}\right)=-{\frac {1}{1+x^{2}}}$

Integrals

The table below lists the integrals of the six basic trig functions and their squares.

$\int {\sin {x}\,dx}=-\cos {x}+C$	$\int {\sin ^{2}{x}\,dx}=\int {{\frac {1}{2}}\left(1-\cos \left(2x\right)\right)\,dx}={\frac {1}{2}}\left(x-{\frac {1}{2}}\sin {2x}\right)+C$
$\int {\cos {x}\,dx}=\sin {x}+C$	$\int {\cos ^{2}{x}\,dx}=\int {{\frac {1}{2}}\left(1+\cos \left(2x\right)\right)\,dx}={\frac {1}{2}}\left(x+{\frac {1}{2}}\sin {2x}\right)+C$
$\int {\tan {x}\,dx}=\ln {\left|\sec {x}\right|}+C$	$\int {\tan ^{2}{x}\,dx}=\int {\sec ^{2}{x}-1\,dx}=\tan {x}-x+C$
$\int {\cot {x}\,dx}=\ln {\left|\sin {x}\right|}+C$	$\int {\cot ^{2}{x}\,dx}=\int {\csc ^{2}{x}-1\,dx}=-\cot {x}-x+C$
$\int {\sec {x}\,dx}=\ln {\left|\sec {x}+\tan {x}\right|}+C$	$\int {\sec ^{2}{x}\,dx}=\tan {x}+C$
$\int {\csc {x}\,dx}=\ln {\left|\csc {x}-\cot {x}\right|}+C$	$\int {\csc ^{2}{x}\,dx}=-\cot {x}+C$

Derivatives

${\frac {d}{dx}}\sin {x}=\cos {x}$	${\frac {d}{dx}}\csc {x}=-\csc {x}\cot {x}$
${\frac {d}{dx}}\cos {x}=-\sin {x}$	${\frac {d}{dx}}\sec {x}=\sec {x}\tan {x}$
${\frac {d}{dx}}\tan {x}=\sec ^{2}{x}$	${\frac {d}{dx}}\cot {x}=-\csc ^{2}{x}$