Question

Dada a equação 2.tg²(x) + k.cos(x) – 4.sen²(x) = 0, determine o valor “k”, sabendo que x = 60⁰.
a) -5
b) -4
c) -3
d) -2
e) -6

Answer 1

60º equivale a $\frac{\pi}{3}$ radianos. O seno, o cosseno e a tangente de

$\sin{\frac{\pi}{3}}=\frac{\sqrt{3}}{2}\\\\ \cos{\frac{\pi}{3}}=\frac{1}{2}\\\\ \tan{\frac{\pi}{3}}=\sqrt{3}$

Assim, para $x=\frac{\pi}{3}$, o valor de k na expressão será:

$2\tan^2{x}+k\cos{x}-4\sin^2{x}=0\ \therefore\\\\ 2(\sqrt{3})^2+k\dfrac{1}{2}-4\bigg(\dfrac{\sqrt{3}}{2}\bigg)^2=0\ \therefore$

$6+\dfrac{k}{2}-4\dfrac{3}{4}=0\ \therefore\ \dfrac{k}{2}=3-6\ \therefore\ k=2(-3)\\\\ \boxed{k=-6}$

Letra E.