If you're seeing this message, it means we're having trouble loading external resources on our website.

Se você está atrás de um filtro da Web, certifique-se que os domínios *.kastatic.org e *.kasandbox.org estão desbloqueados.

Conteúdo principal

Pré-cálculo

Curso: Pré-cálculo > Unidade 2

Lição 10: Uso de identidades trigonométricas
Matemática
Pré-cálculo
Trigonometria
Uso de identidades trigonométricas
© 2023 Khan AcademyTermos de usoPolítica de privacidadeAviso de cookies

Referência de identidade trigonométrica

Google Sala de Aula
Procure E entenda todas as suas identidades trigonométricas favoritas

Identidades recíprocas e de quociente

\sec, left parenthesis, theta, right parenthesis, equals, start fraction, 1, divided by, cosine, left parenthesis, theta, right parenthesis, end fraction

c, o, s, s, e, c, left parenthesis, theta, right parenthesis, equals, start fraction, 1, divided by, s, e, n, left parenthesis, theta, right parenthesis, end fraction

c, o, t, g, left parenthesis, theta, right parenthesis, equals, start fraction, 1, divided by, t, g, left parenthesis, theta, right parenthesis, end fraction

t, g, left parenthesis, theta, right parenthesis, equals, start fraction, s, e, n, left parenthesis, theta, right parenthesis, divided by, cosine, left parenthesis, theta, right parenthesis, end fraction

c, o, t, g, left parenthesis, theta, right parenthesis, equals, start fraction, cosine, left parenthesis, theta, right parenthesis, divided by, s, e, n, left parenthesis, theta, right parenthesis, end fraction

Identidades pitagóricas

s, e, n, squared, left parenthesis, theta, right parenthesis, plus, cosine, squared, left parenthesis, theta, right parenthesis, equals, 1, squared
t, g, squared, left parenthesis, theta, right parenthesis, plus, 1, squared, equals, \sec, squared, left parenthesis, theta, right parenthesis
c, o, t, g, squared, left parenthesis, theta, right parenthesis, plus, 1, squared, equals, c, o, s, s, e, c, squared, left parenthesis, theta, right parenthesis

Identidades que vêm de somas, subtrações, multiplicações e frações de ângulos

Esses são todos parentes próximos, mas vamos examinar cada tipo.
Identidades de somas e subtrações de ângulos
sen⁡(θ+ϕ)=sen⁡θcos⁡ϕ+cos⁡θsen⁡ϕsen⁡(θ−ϕ)=sen⁡θcos⁡ϕ−cos⁡θsen⁡ϕcos⁡(θ+ϕ)=cos⁡θcos⁡ϕ−sen⁡θsen⁡ϕcos⁡(θ−ϕ)=cos⁡θcos⁡ϕ+sen⁡θsen⁡ϕ\begin{aligned} \operatorname{sen}(\theta+\phi)&=\operatorname{sen}\theta\cos\phi+\cos\theta\operatorname{sen}\phi\\\\ \operatorname{sen}(\theta-\phi)&=\operatorname{sen}\theta\cos\phi-\cos\theta\operatorname{sen}\phi\\\\ \cos(\theta+\phi)&=\cos\theta\cos\phi-\operatorname{sen}\theta\operatorname{sen}\phi\\\\ \cos(\theta-\phi)&=\cos\theta\cos\phi+\operatorname{sen}\theta\operatorname{sen}\phi \end{aligned}
tg⁡(θ+ϕ)=tg⁡θ+tg⁡ϕ1−tg⁡θtg⁡ϕtg⁡(θ−ϕ)=tg⁡θ−tg⁡ϕ1+tg⁡θtg⁡ϕ\begin{aligned} \operatorname{tg}(\theta+\phi)&=\dfrac{\operatorname{tg}\theta+\operatorname{tg}\phi}{1-\operatorname{tg}\theta\operatorname{tg}\phi}\\\\ \operatorname{tg}(\theta-\phi)&=\dfrac{\operatorname{tg}\theta-\operatorname{tg}\phi}{1+\operatorname{tg}\theta\operatorname{tg}\phi} \end{aligned}
Identidades de arco duplo
s, e, n, left parenthesis, 2, theta, right parenthesis, equals, 2, s, e, n, theta, cosine, theta
cosine, left parenthesis, 2, theta, right parenthesis, equals, 2, cosine, squared, theta, minus, 1
t, g, left parenthesis, 2, theta, right parenthesis, equals, start fraction, 2, t, g, theta, divided by, 1, minus, t, g, squared, theta, end fraction
Identidades do arco metade
sen⁡θ2=±1−cos⁡θ2cos⁡θ2=±1+cos⁡θ2tg⁡θ2=±1−cos⁡θ1+cos⁡θ=1−cos⁡θsen⁡θ=sen⁡θ1+cos⁡θ\begin{aligned} \operatorname{sen}\dfrac\theta2&=\pm\sqrt{\dfrac{1-\cos\theta}{2}}\\\\ \cos\dfrac\theta2&=\pm\sqrt{\dfrac{1+\cos\theta}{2}}\\\\ \operatorname{tg}\dfrac{\theta}{2}&=\pm\sqrt{\dfrac{1-\cos\theta}{1+\cos\theta}}\\ \\ &=\dfrac{1-\cos\theta}{\operatorname{sen}\theta}\\ \\ &=\dfrac{\operatorname{sen}\theta}{1+\cos\theta}\end{aligned}

Simetria e identidade de periodicidade

s, e, n, left parenthesis, minus, theta, right parenthesis, equals, minus, s, e, n, left parenthesis, theta, right parenthesis
cosine, left parenthesis, minus, theta, right parenthesis, equals, plus, cosine, left parenthesis, theta, right parenthesis
t, g, left parenthesis, minus, theta, right parenthesis, equals, minus, t, g, left parenthesis, theta, right parenthesis
sen⁡(θ+2π)=sen⁡(θ)cos⁡(θ+2π)=cos⁡(θ)tg⁡(θ+π)=tg⁡(θ)\begin{aligned} \operatorname{sen}(\theta+2\pi)&=\operatorname{sen}(\theta)\\\\ \cos(\theta+2\pi)&=\cos(\theta)\\\\ \operatorname{tg}(\theta+\pi)&=\operatorname{tg}(\theta) \end{aligned}

Identidades de cofunções

sen⁡θ=cos⁡(π2−θ)cos⁡θ=sen⁡(π2−θ)tg⁡θ=cotg⁡(π2−θ)cotg⁡θ=tg⁡(π2−θ)sec⁡θ=cossec⁡(π2−θ)cossec⁡θ=sec⁡(π2−θ)\begin{aligned} \operatorname{sen}\theta&= \cos\left(\dfrac{\pi}{2}-\theta\right)\\\\ \cos\theta&= \operatorname{sen}\left(\dfrac{\pi}{2}-\theta\right)\\\\ \operatorname{tg}\theta&= \operatorname{cotg}\left(\dfrac{\pi}{2}-\theta\right)\\\\ \operatorname{cotg}\theta&= \operatorname{tg}\left(\dfrac{\pi}{2}-\theta\right)\\\\ \sec\theta&= \operatorname{cossec}\left(\dfrac{\pi}{2}-\theta\right)\\\\ \operatorname{cossec}\theta&= \sec\left(\dfrac{\pi}{2}-\theta\right) \end{aligned}

Apêndice: todas as proporções trigonométricas no círculo unitário

Use o ponto móvel para ver como os comprimentos das proporções mudam de acordo com o ângulo.

Quer participar da conversa?

Entrar
Você entende inglês? Clique aqui para ver mais debates na versão em inglês do site da Khan Academy.