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Referência de identidade trigonométrica

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)=senθcosϕ+cosθsenϕsen(θϕ)=senθcosϕcosθsenϕcos(θ+ϕ)=cosθcosϕsenθsenϕcos(θϕ)=cosθcosϕ+senθsenϕ\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)=tgθ+tgϕ1tgθtgϕtg(θϕ)=tgθtgϕ1+tgθtgϕ\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
senθ2=±1cosθ2cosθ2=±1+cosθ2tgθ2=±1cosθ1+cosθ=1cosθsenθ=senθ1+cosθ\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

senθ=cos(π2θ)cosθ=sen(π2θ)tgθ=cotg(π2θ)cotgθ=tg(π2θ)secθ=cossec(π2θ)cossecθ=sec(π2θ)\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.

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