Question

se sen x = 1/3 qual o valor de sen 2x?​

Answer 1

Vamos começar determinando o cos(x) utilizando a identidade trigonométrica:

$sen^2(x)~+~cos^2(x)~=~1\\\\\\\left(\frac{1}{3}\right)^2~+~cos^2(x)~=~1\\\\\\cos^2(x)~=~1-\frac{1}{9}\\\\\\cos^2(x)~=~\frac{8}{9}\\\\\\cos(x)~=~\sqrt{\frac{8}{9}}\\\\\\cos(x)~=~\frac{\sqrt{8}}{3}\\\\\\\boxed{cos(x)~=~\frac{2\sqrt{2}}{3}}$

Agora, utilizando a equação do seno da soma de dois arcos, temos:

$sen(x+y)~=~sen(x).cos(y)~+~sen(y).cos(x)\\\\\\sen(2x)~=~sen(x+x)~=~sen(x).cos(x)~+~sen(x).cos(x)\\\\\\sen(2x)~=~2.sen(x).cos(x)\\\\\\sen(2x)~=~2~.~\frac{1}{3}~.~\frac{2\sqrt{2}}{3}\\\\\\\boxed{sen(2x)~=~\frac{4\sqrt{2}}{9}}$