Lista de identidades trigonométricas

En trigonometría, as identidades trigonométricas son igualdades que implican funcións trigonométricas e son verdadeiras para cada valor das variábeis que se producen para as que se definen ambos os dous lados da igualdade. Xeométricamente, estas son identidades que implican certas funcións dun ou máis ángulos. Son distintas das identidades de triángulos, que son identidades que poden implicar ángulos mais que tamén inclúen lonxitudes de lados ou outras lonxitudes dun triángulo.

Identidades pitagóricas

O triángulo sombreado en azul ilustra a identidade ${\displaystyle 1+\cot ^{2}\theta =\csc ^{2}\theta }$, e o triángulo vermello mostra que ${\displaystyle \tan ^{2}\theta +1=\sec ^{2}\theta }$ .

A relación básica entre o seno e o coseno vén dada pola identidade pitagórica:

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

onde ${\displaystyle \sin ^{2}\theta }$ significa ${\displaystyle (\sin \theta )^{2}}$ e ${\displaystyle \cos ^{2}\theta }$ significa ${\displaystyle (\cos \theta )^{2}.}$

Isto pódese ver como unha versión do teorema de Pitágoras, e dedúcese a partir da ecuación ${\displaystyle x^{2}+y^{2}=1}$ para a circunferencia unitaria. Esta ecuación pódese resolver tanto para o seno como para o coseno:

${\displaystyle {\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}}$

onde o signo depende do cuadrante de ${\displaystyle \theta .}$

Dividindo esta identidade por ${\displaystyle \sin ^{2}\theta }$, ${\displaystyle \cos ^{2}\theta }$, ou ambos os dous, proporcionan as seguintes identidades:

${\displaystyle {\begin{aligned}&1+\cot ^{2}\theta =\csc ^{2}\theta \\&1+\tan ^{2}\theta =\sec ^{2}\theta \\&\sec ^{2}\theta +\csc ^{2}\theta =\sec ^{2}\theta \csc ^{2}\theta \end{aligned}}}$

Usando estas identidades, é posíbel expresar calquera función trigonométrica en termos de calquera outra (ata un signo máis ou menos):

Cada función trigonométrica en función de cada unha das outras cinco.

en función de	${\displaystyle \sin \theta }$	${\displaystyle \csc \theta }$	${\displaystyle \cos \theta }$	${\displaystyle \sec \theta }$	${\displaystyle \tan \theta }$	${\displaystyle \cot \theta }$
${\displaystyle \sin \theta =}$	${\displaystyle \sin \theta }$	${\displaystyle {\frac {1}{\csc \theta }}}$	${\displaystyle \pm {\sqrt {1-\cos ^{2}\theta }}}$	${\displaystyle \pm {\frac {\sqrt {\sec ^{2}\theta -1}}{\sec \theta }}}$	${\displaystyle \pm {\frac {\tan \theta }{\sqrt {1+\tan ^{2}\theta }}}}$	${\displaystyle \pm {\frac {1}{\sqrt {1+\cot ^{2}\theta }}}}$
${\displaystyle \csc \theta =}$	${\displaystyle {\frac {1}{\sin \theta }}}$	${\displaystyle \csc \theta }$	${\displaystyle \pm {\frac {1}{\sqrt {1-\cos ^{2}\theta }}}}$	${\displaystyle \pm {\frac {\sec \theta }{\sqrt {\sec ^{2}\theta -1}}}}$	${\displaystyle \pm {\frac {\sqrt {1+\tan ^{2}\theta }}{\tan \theta }}}$	${\displaystyle \pm {\sqrt {1+\cot ^{2}\theta }}}$
${\displaystyle \cos \theta =}$	${\displaystyle \pm {\sqrt {1-\sin ^{2}\theta }}}$	${\displaystyle \pm {\frac {\sqrt {\csc ^{2}\theta -1}}{\csc \theta }}}$	${\displaystyle \cos \theta }$	${\displaystyle {\frac {1}{\sec \theta }}}$	${\displaystyle \pm {\frac {1}{\sqrt {1+\tan ^{2}\theta }}}}$	${\displaystyle \pm {\frac {\cot \theta }{\sqrt {1+\cot ^{2}\theta }}}}$
${\displaystyle \sec \theta =}$	${\displaystyle \pm {\frac {1}{\sqrt {1-\sin ^{2}\theta }}}}$	${\displaystyle \pm {\frac {\csc \theta }{\sqrt {\csc ^{2}\theta -1}}}}$	${\displaystyle {\frac {1}{\cos \theta }}}$	${\displaystyle \sec \theta }$	${\displaystyle \pm {\sqrt {1+\tan ^{2}\theta }}}$	${\displaystyle \pm {\frac {\sqrt {1+\cot ^{2}\theta }}{\cot \theta }}}$
${\displaystyle \tan \theta =}$	${\displaystyle \pm {\frac {\sin \theta }{\sqrt {1-\sin ^{2}\theta }}}}$	${\displaystyle \pm {\frac {1}{\sqrt {\csc ^{2}\theta -1}}}}$	${\displaystyle \pm {\frac {\sqrt {1-\cos ^{2}\theta }}{\cos \theta }}}$	${\displaystyle \pm {\sqrt {\sec ^{2}\theta -1}}}$	${\displaystyle \tan \theta }$	${\displaystyle {\frac {1}{\cot \theta }}}$
${\displaystyle \cot \theta =}$	${\displaystyle \pm {\frac {\sqrt {1-\sin ^{2}\theta }}{\sin \theta }}}$	${\displaystyle \pm {\sqrt {\csc ^{2}\theta -1}}}$	${\displaystyle \pm {\frac {\cos \theta }{\sqrt {1-\cos ^{2}\theta }}}}$	${\displaystyle \pm {\frac {1}{\sqrt {\sec ^{2}\theta -1}}}}$	${\displaystyle {\frac {1}{\tan \theta }}}$	${\displaystyle \cot \theta }$

Reflexións, desprazamentos e periodicidade

Examinando a circunferencia unitaria, pódense estabelecer as seguintes propiedades das funcións trigonométricas.

Reflexións

Se unha liña (vector) con dirección ${\displaystyle \theta }$ reflíctese sobre unha liña con dirección ${\displaystyle \alpha ,}$ daquela o ángulo de dirección ${\displaystyle \theta ^{\prime }}$ desta liña reflectida (vector) ten o valor

${\displaystyle \theta ^{\prime }=2\alpha -\theta .}$

Os valores das funcións trigonométricas destes ángulos ${\displaystyle \theta ,\;\theta ^{\prime }}$ para ángulos específicos ${\displaystyle \alpha }$ satisfán identidades simples: ou son iguais, ou teñen signos opostos, ou empregan a función trigonométrica complementaria. Tamén se coñecen como fórmulas de redución (reduction formulae).^[1]

${\displaystyle \theta }$ reflectido en ${\displaystyle \alpha =0}$ identidades impares/pares	${\displaystyle \theta }$ reflectido en ${\displaystyle \alpha ={\frac {\pi }{4}}}$	${\displaystyle \theta }$ reflectido en ${\displaystyle \alpha ={\frac {\pi }{2}}}$	${\displaystyle \theta }$ reflectido en ${\displaystyle \alpha ={\frac {3\pi }{4}}}$	${\displaystyle \theta }$ reflectido en ${\displaystyle \alpha =\pi }$ comparar con ${\displaystyle \alpha =0}$
${\displaystyle \sin(-\theta )=-\sin \theta }$	${\displaystyle \sin \left({\tfrac {\pi }{2}}-\theta \right)=\cos \theta }$	${\displaystyle \sin(\pi -\theta )=+\sin \theta }$	${\displaystyle \sin \left({\tfrac {3\pi }{2}}-\theta \right)=-\cos \theta }$	${\displaystyle \sin(2\pi -\theta )=-\sin(\theta )=\sin(-\theta )}$
${\displaystyle \cos(-\theta )=+\cos \theta }$	${\displaystyle \cos \left({\tfrac {\pi }{2}}-\theta \right)=\sin \theta }$	${\displaystyle \cos(\pi -\theta )=-\cos \theta }$	${\displaystyle \cos \left({\tfrac {3\pi }{2}}-\theta \right)=-\sin \theta }$	${\displaystyle \cos(2\pi -\theta )=+\cos(\theta )=\cos(-\theta )}$
${\displaystyle \tan(-\theta )=-\tan \theta }$	${\displaystyle \tan \left({\tfrac {\pi }{2}}-\theta \right)=\cot \theta }$	${\displaystyle \tan(\pi -\theta )=-\tan \theta }$	${\displaystyle \tan \left({\tfrac {3\pi }{2}}-\theta \right)=+\cot \theta }$	${\displaystyle \tan(2\pi -\theta )=-\tan(\theta )=\tan(-\theta )}$
${\displaystyle \csc(-\theta )=-\csc \theta }$	${\displaystyle \csc \left({\tfrac {\pi }{2}}-\theta \right)=\sec \theta }$	${\displaystyle \csc(\pi -\theta )=+\csc \theta }$	${\displaystyle \csc \left({\tfrac {3\pi }{2}}-\theta \right)=-\sec \theta }$	${\displaystyle \csc(2\pi -\theta )=-\csc(\theta )=\csc(-\theta )}$
${\displaystyle \sec(-\theta )=+\sec \theta }$	${\displaystyle \sec \left({\tfrac {\pi }{2}}-\theta \right)=\csc \theta }$	${\displaystyle \sec(\pi -\theta )=-\sec \theta }$	${\displaystyle \sec \left({\tfrac {3\pi }{2}}-\theta \right)=-\csc \theta }$	${\displaystyle \sec(2\pi -\theta )=+\sec(\theta )=\sec(-\theta )}$
${\displaystyle \cot(-\theta )=-\cot \theta }$	${\displaystyle \cot \left({\tfrac {\pi }{2}}-\theta \right)=\tan \theta }$	${\displaystyle \cot(\pi -\theta )=-\cot \theta }$	${\displaystyle \cot \left({\tfrac {3\pi }{2}}-\theta \right)=+\tan \theta }$	${\displaystyle \cot(2\pi -\theta )=-\cot(\theta )=\cot(-\theta )}$

Desprazamentos e periodicidade

Desprazado por un período dun cuarto	Desprazado por un período dun medio	Desprazado por períodos completos	Período
${\displaystyle \sin(\theta \pm {\tfrac {\pi }{2}})=\pm \cos \theta }$	${\displaystyle \sin(\theta +\pi )=-\sin \theta }$	${\displaystyle \sin(\theta +k\cdot 2\pi )=+\sin \theta }$	${\displaystyle 2\pi }$
${\displaystyle \cos(\theta \pm {\tfrac {\pi }{2}})=\mp \sin \theta }$	${\displaystyle \cos(\theta +\pi )=-\cos \theta }$	${\displaystyle \cos(\theta +k\cdot 2\pi )=+\cos \theta }$	${\displaystyle 2\pi }$
${\displaystyle \csc(\theta \pm {\tfrac {\pi }{2}})=\pm \sec \theta }$	${\displaystyle \csc(\theta +\pi )=-\csc \theta }$	${\displaystyle \csc(\theta +k\cdot 2\pi )=+\csc \theta }$	${\displaystyle 2\pi }$
${\displaystyle \sec(\theta \pm {\tfrac {\pi }{2}})=\mp \csc \theta }$	${\displaystyle \sec(\theta +\pi )=-\sec \theta }$	${\displaystyle \sec(\theta +k\cdot 2\pi )=+\sec \theta }$	${\displaystyle 2\pi }$
${\displaystyle \tan(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\tan \theta \pm 1}{1\mp \tan \theta }}}$	${\displaystyle \tan(\theta +{\tfrac {\pi }{2}})=-\cot \theta }$	${\displaystyle \tan(\theta +k\cdot \pi )=+\tan \theta }$	${\displaystyle \pi }$
${\displaystyle \cot(\theta \pm {\tfrac {\pi }{4}})={\tfrac {\cot \theta \mp 1}{1\pm \cot \theta }}}$	${\displaystyle \cot(\theta +{\tfrac {\pi }{2}})=-\tan \theta }$	${\displaystyle \cot(\theta +k\cdot \pi )=+\cot \theta }$	${\displaystyle \pi }$

Signos

O signo das funcións trigonométricas depende do cuadrante do ángulo. Se ${\displaystyle {-\pi }<\theta \leq \pi }$ e sgn é a función signo,

${\displaystyle {\begin{aligned}\operatorname {sgn}(\sin \theta )=\operatorname {sgn}(\csc \theta )&={\begin{cases}+1&{\text{if}}\ \ 0<\theta <\pi \\-1&{\text{if}}\ \ {-\pi }<\theta <0\\0&{\text{if}}\ \ \theta \in \{0,\pi \}\end{cases}}\\[5mu]\operatorname {sgn}(\cos \theta )=\operatorname {sgn}(\sec \theta )&={\begin{cases}+1&{\text{if}}\ \ {-{\tfrac {1}{2}}\pi }<\theta <{\tfrac {1}{2}}\pi \\-1&{\text{if}}\ \ {-\pi }<\theta <-{\tfrac {1}{2}}\pi \ \ {\text{ou}}\ \ {\tfrac {1}{2}}\pi <\theta <\pi \\0&{\text{if}}\ \ \theta \in {\bigl \{}{-{\tfrac {1}{2}}\pi },{\tfrac {1}{2}}\pi {\bigr \}}\end{cases}}\\[5mu]\operatorname {sgn}(\tan \theta )=\operatorname {sgn}(\cot \theta )&={\begin{cases}+1&{\text{if}}\ \ {-\pi }<\theta <-{\tfrac {1}{2}}\pi \ \ {\text{ou}}\ \ 0<\theta <{\tfrac {1}{2}}\pi \\-1&{\text{if}}\ \ {-{\tfrac {1}{2}}\pi }<\theta <0\ \ {\text{ou}}\ \ {\tfrac {1}{2}}\pi <\theta <\pi \\0&{\text{if}}\ \ \theta \in {\bigl \{}{-{\tfrac {1}{2}}\pi },0,{\tfrac {1}{2}}\pi ,\pi {\bigr \}}\end{cases}}\end{aligned}}}$

As funcións trigonométricas son periódicas con período común ${\displaystyle 2\pi ,}$ polo que para valores de θ fóra do intervalo ${\displaystyle ({-\pi },\pi ],}$ toman valores repetitivos.

Identidades de suma e diferenza de ángulos

Ilustración de fórmulas de suma de ángulos para o seno e o coseno dos ángulos agudos. O segmento enfatizado é de lonxitude unidade.

Estes tamén se coñecen como fórmulas da suma de ángulos.

${\displaystyle {\begin{aligned}\sin(\alpha +\beta )&=\sin \alpha \cos \beta +\cos \alpha \sin \beta \\\sin(\alpha -\beta )&=\sin \alpha \cos \beta -\cos \alpha \sin \beta \\\cos(\alpha +\beta )&=\cos \alpha \cos \beta -\sin \alpha \sin \beta \\\cos(\alpha -\beta )&=\cos \alpha \cos \beta +\sin \alpha \sin \beta \end{aligned}}}$

Estas identidades resúmense nas dúas primeiras filas da seguinte táboa, que tamén inclúe identidades de suma e diferenza para as outras funcións trigonométricas.

Seno	${\displaystyle \sin(\alpha \pm \beta )}$			${\displaystyle =}$	${\displaystyle \sin \alpha \cos \beta \pm \cos \alpha \sin \beta }$[2]
Coseno	${\displaystyle \cos(\alpha \pm \beta )}$			${\displaystyle =}$	${\displaystyle \cos \alpha \cos \beta \mp \sin \alpha \sin \beta }$[2]
Tanxente	${\displaystyle \tan(\alpha \pm \beta )}$			${\displaystyle =}$	${\displaystyle {\frac {\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }}}$[2]
Cosecante	${\displaystyle \csc(\alpha \pm \beta )}$			${\displaystyle =}$	${\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\sec \alpha \csc \beta \pm \csc \alpha \sec \beta }}}$[3]
Secante	${\displaystyle \sec(\alpha \pm \beta )}$			${\displaystyle =}$	${\displaystyle {\frac {\sec \alpha \sec \beta \csc \alpha \csc \beta }{\csc \alpha \csc \beta \mp \sec \alpha \sec \beta }}}$[3]
Cotanxente	${\displaystyle \cot(\alpha \pm \beta )}$			${\displaystyle =}$	${\displaystyle {\frac {\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha }}}$[2]
Arcoseno	${\displaystyle \arcsin x\pm \arcsin y}$			${\displaystyle =}$	${\displaystyle \arcsin \left(x{\sqrt {1-y^{2}}}\pm y{\sqrt {1-x^{2}{\vphantom {y}}}}\right)}$
Arcocoseno	${\displaystyle \arccos x\pm \arccos y}$			${\displaystyle =}$	${\displaystyle \arccos \left(xy\mp {\sqrt {\left(1-x^{2}\right)\left(1-y^{2}\right)}}\right)}$
Arcotanxente	${\displaystyle \arctan x\pm \arctan y}$			${\displaystyle =}$	${\displaystyle \arctan \left({\frac {x\pm y}{1\mp xy}}\right)}$
Arcocotanxente	${\displaystyle \operatorname {arccot} x\pm \operatorname {arccot} y}$			${\displaystyle =}$	${\displaystyle \operatorname {arccot} \left({\frac {xy\mp 1}{y\pm x}}\right)}$

Fórmulas de ángulos múltiples e ángulos metade

Tn é o n-ésimo polinomio de Chebyshev	${\displaystyle \cos(n\theta )=T_{n}(\cos \theta )}$
Fórmula de De Moivre, i é a unidade imaxinaria	${\displaystyle \cos(n\theta )+i\sin(n\theta )=(\cos \theta +i\sin \theta )^{n}}$

Fórmulas de ángulos múltiples

Fórmulas de ángulo duplo

Fórmulas para dúas veces un ángulo. ^[4]

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

Fórmulas de ángulos múltiples

${\displaystyle {\begin{aligned}\sin(n\theta )&=\sum _{k{\text{ impar}}}(-1)^{\frac {k-1}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sin \theta \sum _{i=0}^{(n+1)/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i+1}{i \choose j}\cos ^{n-2(i-j)-1}\theta \\{}&=\sin(\theta )\cdot \sum _{k=0}^{\left\lfloor {\frac {n-1}{2}}\right\rfloor }(-1)^{k}\cdot {(2\cdot \cos(\theta ))}^{n-2k-1}\cdot {n-k-1 \choose k}\\{}&=2^{(n-1)}\prod _{k=0}^{n-1}\sin(k\pi /n+\theta )\end{aligned}}}$

${\displaystyle {\begin{aligned}\cos(n\theta )&=\sum _{k{\text{ par}}}(-1)^{\frac {k}{2}}{n \choose k}\cos ^{n-k}\theta \sin ^{k}\theta =\sum _{i=0}^{n/2}\sum _{j=0}^{i}(-1)^{i-j}{n \choose 2i}{i \choose j}\cos ^{n-2(i-j)}\theta \\{}&=\sum _{k=0}^{\left\lfloor {\frac {n}{2}}\right\rfloor }(-1)^{k}\cdot {(2\cdot \cos(\theta ))}^{n-2k}\cdot {n-k \choose k}\cdot {\frac {n}{2n-2k}}\end{aligned}}}$

• ${\displaystyle \cos((2n+1)\theta )=(-1)^{n}2^{2n}\prod _{k=0}^{2n}\cos(k\pi /(2n+1)-\theta )}$
• ${\displaystyle \cos(2n\theta )=(-1)^{n}2^{2n-1}\prod _{k=0}^{2n-1}\cos((1+2k)\pi /(4n)-\theta )}$
• ${\displaystyle \tan(n\theta )={\frac {\sum _{k{\text{ impar}}}(-1)^{\frac {k-1}{2}}{n \choose k}\tan ^{k}\theta }{\sum _{k{\text{ par}}}(-1)^{\frac {k}{2}}{n \choose k}\tan ^{k}\theta }}}$ 

Método de Chebyshev

O método Chebyshev é un algoritmo recursivo para atopar a fórmula n-ésima de ángulos múltiples coñecendo os valores de multiplicidade ${\displaystyle (n-1)}$ e ${\displaystyle (n-2)}$.^[5]

Así ${\displaystyle \cos(nx)}$ pódese calcular a partir de ${\displaystyle \cos((n-1)x)}$, ${\displaystyle \cos((n-2)x)}$, e ${\displaystyle \cos(x)}$ con

${\displaystyle \cos(nx)=2\cos x\cos((n-1)x)-\cos((n-2)x).}$

Dedúcese por indución que ${\displaystyle \cos(nx)}$ é un polinomio de ${\displaystyle \cos x,}$ o chamado polinomio de Chebyshev do primeiro tipo, véxase Polinomios de Chebyshev.

Similarmente

${\displaystyle \sin(nx)=2\cos x\sin((n-1)x)-\sin((n-2)x).}$

Fórmulas de ángulo métade

${\displaystyle {\begin{aligned}\sin {\frac {\theta }{2}}&=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {1-\cos \theta }{2}}}\\[3pt]\cos {\frac {\theta }{2}}&=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {1+\cos \theta }{2}}}\\[3pt]\tan {\frac {\theta }{2}}&={\frac {1-\cos \theta }{\sin \theta }}={\frac {\sin \theta }{1+\cos \theta }}=\csc \theta -\cot \theta ={\frac {\tan \theta }{1+\sec {\theta }}}\\[6mu]&=\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1-\cos \theta }{1+\cos \theta }}}={\frac {-1+\operatorname {sgn}(\cos \theta ){\sqrt {1+\tan ^{2}\theta }}}{\tan \theta }}\\[3pt]\cot {\frac {\theta }{2}}&={\frac {1+\cos \theta }{\sin \theta }}={\frac {\sin \theta }{1-\cos \theta }}=\csc \theta +\cot \theta =\operatorname {sgn}(\sin \theta ){\sqrt {\frac {1+\cos \theta }{1-\cos \theta }}}\\\sec {\frac {\theta }{2}}&=\operatorname {sgn} \left(\cos {\frac {\theta }{2}}\right){\sqrt {\frac {2}{1+\cos \theta }}}\\\csc {\frac {\theta }{2}}&=\operatorname {sgn} \left(\sin {\frac {\theta }{2}}\right){\sqrt {\frac {2}{1-\cos \theta }}}\\\end{aligned}}}$

Fórmulas de redución de potencias

Obtido resolvendo a segunda e terceira versións da fórmula do ángulo duplo do coseno.

Seno	Coseno	Outros
${\displaystyle \sin ^{2}\theta ={\frac {1-\cos(2\theta )}{2}}}$	${\displaystyle \cos ^{2}\theta ={\frac {1+\cos(2\theta )}{2}}}$	${\displaystyle \sin ^{2}\theta \cos ^{2}\theta ={\frac {1-\cos(4\theta )}{8}}}$
${\displaystyle \sin ^{3}\theta ={\frac {3\sin \theta -\sin(3\theta )}{4}}}$	${\displaystyle \cos ^{3}\theta ={\frac {3\cos \theta +\cos(3\theta )}{4}}}$	${\displaystyle \sin ^{3}\theta \cos ^{3}\theta ={\frac {3\sin(2\theta )-\sin(6\theta )}{32}}}$
${\displaystyle \sin ^{4}\theta ={\frac {3-4\cos(2\theta )+\cos(4\theta )}{8}}}$	${\displaystyle \cos ^{4}\theta ={\frac {3+4\cos(2\theta )+\cos(4\theta )}{8}}}$	${\displaystyle \sin ^{4}\theta \cos ^{4}\theta ={\frac {3-4\cos(4\theta )+\cos(8\theta )}{128}}}$
${\displaystyle \sin ^{5}\theta ={\frac {10\sin \theta -5\sin(3\theta )+\sin(5\theta )}{16}}}$	${\displaystyle \cos ^{5}\theta ={\frac {10\cos \theta +5\cos(3\theta )+\cos(5\theta )}{16}}}$	${\displaystyle \sin ^{5}\theta \cos ^{5}\theta ={\frac {10\sin(2\theta )-5\sin(6\theta )+\sin(10\theta )}{512}}}$

Identidades produto a suma e suma a produto

Identidades produto a suma

${\displaystyle {\begin{aligned}\cos \theta \,\cos \varphi &={\tfrac {1}{2}}{\bigl (}\!\!~\cos(\theta -\varphi )+\cos(\theta +\varphi ){\bigr )}\\[3mu]\sin \theta \,\sin \varphi &={\tfrac {1}{2}}{\bigl (}\!\!~\cos(\theta -\varphi )-\cos(\theta +\varphi ){\bigr )}\\[3mu]\sin \theta \,\cos \varphi &={\tfrac {1}{2}}{\bigl (}\!\!~\sin(\theta +\varphi )+\sin(\theta -\varphi ){\bigr )}\\[3mu]\cos \theta \,\sin \varphi &={\tfrac {1}{2}}{\bigl (}\!\!~\sin(\theta +\varphi )-\sin(\theta -\varphi ){\bigr )}\end{aligned}}}$

• ${\displaystyle \tan \theta \,\tan \varphi ={\frac {\cos(\theta -\varphi )-\cos(\theta +\varphi )}{\cos(\theta -\varphi )+\cos(\theta +\varphi )}}}$
• ${\displaystyle \tan \theta \,\cot \varphi ={\frac {\sin(\theta +\varphi )+\sin(\theta -\varphi )}{\sin(\theta +\varphi )-\sin(\theta -\varphi )}}}$
• ${\displaystyle {\begin{aligned}\prod _{k=1}^{n}\cos \theta _{k}&={\frac {1}{2^{n}}}\sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\\[6pt]&{\text{where }}e=(e_{1},\ldots ,e_{n})\in S=\{1,-1\}^{n}\end{aligned}}}$
• ${\displaystyle \prod _{k=1}^{n}\sin \theta _{k}={\frac {(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }}{2^{n}}}{\begin{cases}\displaystyle \sum _{e\in S}\cos(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\prod _{j=1}^{n}e_{j}\;{\text{if}}\;n\;{\text{is even}},\\\displaystyle \sum _{e\in S}\sin(e_{1}\theta _{1}+\cdots +e_{n}\theta _{n})\prod _{j=1}^{n}e_{j}\;{\text{if}}\;n\;{\text{is odd}}\end{cases}}}$

Identidades suma a produto

As identidades de suma a produto son as seguintes:

• ${\displaystyle \sin \theta \pm \sin \varphi =2\sin \left({\frac {\theta \pm \varphi }{2}}\right)\cos \left({\frac {\theta \mp \varphi }{2}}\right)}$
• ${\displaystyle \cos \theta +\cos \varphi =2\cos \left({\frac {\theta +\varphi }{2}}\right)\cos \left({\frac {\theta -\varphi }{2}}\right)}$
• ${\displaystyle \cos \theta -\cos \varphi =-2\sin \left({\frac {\theta +\varphi }{2}}\right)\sin \left({\frac {\theta -\varphi }{2}}\right)}$
• ${\displaystyle \tan \theta \pm \tan \varphi ={\frac {\sin(\theta \pm \varphi )}{\cos \theta \,\cos \varphi }}}$ 

Relación coa función exponencial complexa

A fórmula de Euler indica que, para calquera número real x:

${\displaystyle e^{ix}=\cos x+i\sin x,}$onde i é a unidade imaxinaria. Substituíndo −x por x dános:

${\displaystyle e^{-ix}=\cos(-x)+i\sin(-x)=\cos x-i\sin x.}$

Estas dúas ecuacións pódense usar para resolver o coseno e o seno en termos da función exponencial. En concreto,

${\displaystyle \cos x={\frac {e^{ix}+e^{-ix}}{2}}}$

${\displaystyle \sin x={\frac {e^{ix}-e^{-ix}}{2i}}}$

Estas fórmulas son útiles para demostrar moitas outras identidades trigonométricas. Por exemplo, que e^i(θ+φ) = e^iθ e^iφ significa que

cos(θ + φ) + i sin(θ + φ) = (cos θ + i sin θ) (cos φ + i sin φ) = (cos θ cos φ − sin θ sin φ) + i (cos θ sin φ + sin θ cos φ).

Que a parte real do lado esquerdo sexa igual á parte real do lado dereito é unha fórmula de suma de ángulos para o coseno. A igualdade das partes imaxinarias dá unha fórmula de suma de ángulos para o seno.

A seguinte táboa expresa as funcións trigonométricas e as súas inversas en función da función exponencial e do logaritmo complexo.

Función	Función inversa[6]
${\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}}$	${\displaystyle \arcsin x=-i\,\ln \left(ix+{\sqrt {1-x^{2}}}\right)}$
${\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}$	${\displaystyle \arccos x=-i\ln \left(x+{\sqrt {x^{2}-1}}\right)}$
${\displaystyle \tan \theta =-i\,{\frac {e^{i\theta }-e^{-i\theta }}{e^{i\theta }+e^{-i\theta }}}}$	${\displaystyle \arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)}$
${\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}}$	${\displaystyle \operatorname {arccsc} x=-i\,\ln \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}}\right)}$
${\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}}$	${\displaystyle \operatorname {arcsec} x=-i\,\ln \left({\frac {1}{x}}+i{\sqrt {1-{\frac {1}{x^{2}}}}}\right)}$
${\displaystyle \cot \theta =i\,{\frac {e^{i\theta }+e^{-i\theta }}{e^{i\theta }-e^{-i\theta }}}}$	${\displaystyle \operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)}$
${\displaystyle \operatorname {cis} \theta =e^{i\theta }}$	${\displaystyle \operatorname {arccis} x=-i\ln x}$

Nota: cis é unha notación que indica coseno e a parte imaxinaria (i) para o seno.

Relación con funcións hiperbólicas complexas

As funcións trigonométricas pódense deducir de funcións hiperbólicas con argumentos complexos. As fórmulas para as relacións móstranse a continuación^[7][8]

${\displaystyle {\begin{aligned}\sin x&=-i\sinh(ix)\\\cos x&=\cosh(ix)\\\tan x&=-i\tanh(ix)\\\cot x&=i\coth(ix)\\\sec x&=\operatorname {sech} (ix)\\\csc x&=i\operatorname {csch} (ix)\\\end{aligned}}}$

Expansión en series

Cando se utiliza unha expansión de serie de potencias para definir funcións trigonométricas, obtéñense as seguintes identidades:

${\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n+1}}{(2n+1)!}},}$

${\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}x^{2n}}{(2n)!}}.}$

Fórmulas infinitas de produtos

Para aplicacións con funcións especiais, son útiles as seguintes fórmulas de produtos infinitos para funcións trigonométricas.

${\displaystyle {\begin{aligned}&{\text{Funcións trigonométricas circulares:}}\\\sin x&=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cos x&=\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)\!{\vphantom {)}}^{2}}}\right).\\[10mu]&{\text{Funcións trigonométricas hiperbólicas:}}\\[10mu]\sinh x&=x\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),&\cosh x&=\prod _{n=1}^{\infty }\left(1+{\frac {x^{2}}{\pi ^{2}\left(n-{\frac {1}{2}}\right)\!{\vphantom {)}}^{2}}}\right).\end{aligned}}}$

Funcións trigonométricas inversas

Artigo principal: Funcións trigonométricas inversas.

As seguintes identidades dan o resultado de compoñer unha función trigonométrica cunha función trigonométrica inversa.^[9]

${\displaystyle {\begin{aligned}\sin(\arcsin x)&=x&\cos(\arcsin x)&={\sqrt {1-x^{2}}}&\tan(\arcsin x)&={\frac {x}{\sqrt {1-x^{2}}}}\\\sin(\arccos x)&={\sqrt {1-x^{2}}}&\cos(\arccos x)&=x&\tan(\arccos x)&={\frac {\sqrt {1-x^{2}}}{x}}\\\sin(\arctan x)&={\frac {x}{\sqrt {1+x^{2}}}}&\cos(\arctan x)&={\frac {1}{\sqrt {1+x^{2}}}}&\tan(\arctan x)&=x\\\sin(\operatorname {arccsc} x)&={\frac {1}{x}}&\cos(\operatorname {arccsc} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\tan(\operatorname {arccsc} x)&={\frac {1}{\sqrt {x^{2}-1}}}\\\sin(\operatorname {arcsec} x)&={\frac {\sqrt {x^{2}-1}}{x}}&\cos(\operatorname {arcsec} x)&={\frac {1}{x}}&\tan(\operatorname {arcsec} x)&={\sqrt {x^{2}-1}}\\\sin(\operatorname {arccot} x)&={\frac {1}{\sqrt {1+x^{2}}}}&\cos(\operatorname {arccot} x)&={\frac {x}{\sqrt {1+x^{2}}}}&\tan(\operatorname {arccot} x)&={\frac {1}{x}}\\\end{aligned}}}$

Tomando o inverso multiplicativo de ambos os dous lados de cada ecuación anterior resultan as ecuacións para ${\displaystyle \csc ={\frac {1}{\sin }},\;\sec ={\frac {1}{\cos }},{\text{ and }}\cot ={\frac {1}{\tan }}.}$

O lado dereito da fórmula anterior sempre se invertirá. Por exemplo, a ecuación para ${\displaystyle \cot(\arcsin x)}$ é:

${\displaystyle \cot(\arcsin x)={\frac {1}{\tan(\arcsin x)}}={\frac {1}{\frac {x}{\sqrt {1-x^{2}}}}}={\frac {\sqrt {1-x^{2}}}{x}}}$

mentres que as ecuacións para ${\displaystyle \csc(\arccos x)}$ e ${\displaystyle \sec(\arccos x)}$ son:

${\displaystyle \csc(\arccos x)={\frac {1}{\sin(\arccos x)}}={\frac {1}{\sqrt {1-x^{2}}}}.}$

${\displaystyle \sec(\arccos x)={\frac {1}{\cos(\arccos x)}}={\frac {1}{x}}.}$

As seguintes identidades están implicadas polas identidades de reflexión. Mantéñense sempre que ${\displaystyle x,r,s,-x,-r,{\text{ e }}-s}$ estean nos dominios das funcións relevantes.

${\displaystyle {\begin{alignedat}{9}{\frac {\pi }{2}}~&=~\arcsin(x)&&+\arccos(x)~&&=~\arctan(r)&&+\operatorname {arccot}(r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arccsc}(s)\\[0.4ex]\pi ~&=~\arccos(x)&&+\arccos(-x)~&&=~\operatorname {arccot}(r)&&+\operatorname {arccot}(-r)~&&=~\operatorname {arcsec}(s)&&+\operatorname {arcsec}(-s)\\[0.4ex]0~&=~\arcsin(x)&&+\arcsin(-x)~&&=~\arctan(r)&&+\arctan(-r)~&&=~\operatorname {arccsc}(s)&&+\operatorname {arccsc}(-s)\\[1.0ex]\end{alignedat}}}$

Tamén ,^[10]

${\displaystyle {\begin{aligned}\arctan x+\arctan {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&{\text{se }}x>0\\-{\frac {\pi }{2}},&{\text{se }}x<0\end{cases}}\\\operatorname {arccot} x+\operatorname {arccot} {\dfrac {1}{x}}&={\begin{cases}{\frac {\pi }{2}},&{\text{se }}x>0\\{\frac {3\pi }{2}},&{\text{se }}x<0\end{cases}}\\\end{aligned}}}$

${\displaystyle \arccos {\frac {1}{x}}=\operatorname {arcsec} x\qquad {\text{ e }}\qquad \operatorname {arcsec} {\frac {1}{x}}=\arccos x}$

${\displaystyle \arcsin {\frac {1}{x}}=\operatorname {arccsc} x\qquad {\text{ e }}\qquad \operatorname {arccsc} {\frac {1}{x}}=\arcsin x}$

A función arcotanxente pódese expandir como unha serie:^[11]

${\displaystyle \arctan(nx)=\sum _{m=1}^{n}\arctan {\frac {x}{1+(m-1)mx^{2}}}}$