Question

Calcule:
a) cos(2x) se cos(x) = 1/4
b) tan(2x) se tan(x) = 2

Answer 1

Explicação passo-a-passo:

a) Pela relação fundamental da trigonometria:

$\sf sen^2~x+cos^2~x=1$

$\sf sen^2~x+\Big(\dfrac{1}{4}\Big)^2=1$

$\sf sen^2~x+\dfrac{1}{16}=1$

$\sf sen^2~x=1-\dfrac{1}{16}$

$\sf sen^2~x=\dfrac{16-1}{16}$

$\sf sen^2~x=\dfrac{15}{16}$

Temos que:

$\sf cos~(2x)=cos^2~x-sen^2~x$

$\sf cos~(2x)=\Big(\dfrac{1}{4}\Big)^2-\dfrac{15}{16}$

$\sf cos~(2x)=\dfrac{1}{16}-\dfrac{15}{16}$

$\sf cos~(2x)=\dfrac{-14}{16}$

$\sf \red{cos~(2x)=\dfrac{-7}{8}}$

b)

$\sf tg~(2x)=\dfrac{2\cdot tg~x}{1-tg^2~x}$

$\sf tg~(2x)=\dfrac{2\cdot2}{1-2^2}$

$\sf tg~(2x)=\dfrac{4}{1-4}$

$\sf tg~(2x)=\dfrac{4}{-3}$

$\sf \red{tg~(2x)=\dfrac{-4}{3}}$