Identities

Basic Relationships

Pythagorean trigonometric identity	${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1\,}$
Ratio identity	${\displaystyle \tan \theta ={\frac {\sin \theta }{\cos \theta }}}$

Each trigonometric function in terms of the other five.

Function	${\displaystyle (\sin \theta )}$	${\displaystyle (\cos \theta )}$	${\displaystyle (\tan \theta )}$	${\displaystyle (\csc \theta )}$	${\displaystyle (\sec \theta )}$	${\displaystyle (\cot \theta )}$
${\displaystyle \sin \theta =}$	${\displaystyle \sin \theta \ }$	${\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}\ }$	${\displaystyle \pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}\ }$	${\displaystyle {\frac {1}{\csc \theta }}\ }$	${\displaystyle \pm {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}\ }$	${\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}\ }$
${\displaystyle \cos \theta =}$	${\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}\ }$	${\displaystyle \cos \theta \ }$	${\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}\ }$	${\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}\ }$	${\displaystyle {\frac {1}{\sec \theta }}\ }$	${\displaystyle \pm {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}\ }$
${\displaystyle \tan \theta =}$	${\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}\ }$	${\displaystyle \pm {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}\ }$	${\displaystyle \tan \theta \ }$	${\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}\ }$	${\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}\ }$	${\displaystyle {\frac {1}{\cot \theta }}\ }$
${\displaystyle \csc \theta =}$	${\displaystyle {\frac {1}{\sin \theta }}\ }$	${\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}\ }$	${\displaystyle \pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}\ }$	${\displaystyle \csc \theta \ }$	${\displaystyle \pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1}}}\ }$	${\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}\ }$
${\displaystyle \sec \theta =}$	${\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}\ }$	${\displaystyle {\frac {1}{\cos \theta }}\ }$	${\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}\ }$	${\displaystyle \pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}\ }$	${\displaystyle \sec \theta \ }$	${\displaystyle \pm {\frac {\sqrt {1+\cot ^{2}\theta }}{\cot \theta }}\ }$
${\displaystyle \cot \theta =}$	${\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}\ }$	${\displaystyle \pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta }}}\ }$	${\displaystyle {\frac {1}{\tan \theta }}\ }$	${\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}\ }$	${\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}\ }$	${\displaystyle \cot \theta \ }$

Historic Shorthands

Name(s)	Abbreviation(s)	Value
versed sine, versine	${\displaystyle {\textrm {versin}}\,\theta \ }$ ${\displaystyle {\textrm {vers}}\,\theta \ }$	${\displaystyle 1-\cos \theta \ }$
versed cosine, vercosine, coversed sine, coversine	${\displaystyle {\textrm {vercos}}\,\theta \ }$ ${\displaystyle {\textrm {coversin}}\,\theta \ }$ ${\displaystyle {\textrm {cvs}}\,\theta \ }$	${\displaystyle 1-\sin \theta \ }$
haversed sine, haversine	${\displaystyle {\textrm {haversin}}\,\theta \ }$ ${\displaystyle {\textrm {hav}}\,\theta \ }$	${\displaystyle {\tfrac {1}{2}}{\textrm {versin}}\ \theta \ }$
haversed cosine, havercosine, hacoversed sine, hacoversine, cohaversed sine, cohaversine	${\displaystyle {\textrm {havercos}}\,\theta \ }$ ${\displaystyle {\textrm {hacoversin}}\,\theta \ }$ ${\displaystyle {\textrm {cohav}}\,\theta \ }$	${\displaystyle {\tfrac {1}{2}}{\textrm {vercos}}\,\theta \ }$
exterior secant, exsecant	${\displaystyle {\textrm {exsec}}\,\theta \ }$	${\displaystyle \sec \theta -1\ }$
exterior cosecant, excosecant	${\displaystyle {\textrm {excsc}}\,\theta \ }$	${\displaystyle \csc \theta -1\ }$

Symmetries

Reflected in ${\displaystyle \theta =0}$	Reflected in ${\displaystyle \theta =\pi /2}$ (co-function identities)	Reflected in ${\displaystyle \theta =\pi }$
${\displaystyle {\begin{aligned}\sin(-\theta )&=-\sin \theta \\\cos(-\theta )&=+\cos \theta \\\tan(-\theta )&=-\tan \theta \\\csc(-\theta )&=-\csc \theta \\\sec(-\theta )&=+\sec \theta \\\cot(-\theta )&=-\cot \theta \end{aligned}}}$	${\displaystyle {\begin{aligned}\sin({\tfrac {\pi }{2}}-\theta )&=+\cos \theta \\\cos({\tfrac {\pi }{2}}-\theta )&=+\sin \theta \\\tan({\tfrac {\pi }{2}}-\theta )&=+\cot \theta \\\csc({\tfrac {\pi }{2}}-\theta )&=+\sec \theta \\\sec({\tfrac {\pi }{2}}-\theta )&=+\csc \theta \\\cot({\tfrac {\pi }{2}}-\theta )&=+\tan \theta \end{aligned}}}$	${\displaystyle {\begin{aligned}\sin(\pi -\theta )&=+\sin \theta \\\cos(\pi -\theta )&=-\cos \theta \\\tan(\pi -\theta )&=-\tan \theta \\\csc(\pi -\theta )&=+\csc \theta \\\sec(\pi -\theta )&=-\sec \theta \\\cot(\pi -\theta )&=-\cot \theta \\\end{aligned}}}$

Periodicity and Shifts

Shift by π/2	Shift by π Period for tan and cot	Shift by 2π Period for sin, cos, csc and sec
${\displaystyle {\begin{aligned}\sin(\theta +{\tfrac {\pi }{2}})&=+\cos \theta \\\cos(\theta +{\tfrac {\pi }{2}})&=-\sin \theta \\\tan(\theta +{\tfrac {\pi }{2}})&=-\cot \theta \\\csc(\theta +{\tfrac {\pi }{2}})&=+\sec \theta \\\sec(\theta +{\tfrac {\pi }{2}})&=-\csc \theta \\\cot(\theta +{\tfrac {\pi }{2}})&=-\tan \theta \end{aligned}}}$	${\displaystyle {\begin{aligned}\sin(\theta +\pi )&=-\sin \theta \\\cos(\theta +\pi )&=-\cos \theta \\\tan(\theta +\pi )&=+\tan \theta \\\csc(\theta +\pi )&=-\csc \theta \\\sec(\theta +\pi )&=-\sec \theta \\\cot(\theta +\pi )&=+\cot \theta \\\end{aligned}}}$	${\displaystyle {\begin{aligned}\sin(\theta +2\pi )&=+\sin \theta \\\cos(\theta +2\pi )&=+\cos \theta \\\tan(\theta +2\pi )&=+\tan \theta \\\csc(\theta +2\pi )&=+\csc \theta \\\sec(\theta +2\pi )&=+\sec \theta \\\cot(\theta +2\pi )&=+\cot \theta \end{aligned}}}$

Complex Numbers, De Moivre's Law and Argand Plane

Textbooks

Practice Questions

1  

${\displaystyle sin{\frac {\pi }{6}}=}$

2  

${\displaystyle cos{\frac {\pi }{3}}=}$

3  

${\displaystyle tan{\frac {\pi }{4}}=}$

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