Question

Se tgX=raiz de 3, então quanto vale sen elevado a 2 X

Answer 1

Sabendo que tg(x) é a razão entre sen(x) e cos(x), temos:

$tg(x)~=~\frac{sen(x)}{cos(x)}\\\\\\\left(~tg(x)~\right)^2~=~\left(\frac{sen(x)}{cos(x)}\right)^2\\\\\\tg^2x~=~\frac{sen^2x}{cos^2x}\\\\\\Utilizando~a~identidade~trigonometrica~\boxed{sen^2x+cos^2~=~1}:\\\\\\tg^2x~=~\frac{sen^2x}{1-sen^2x}\\\\\\(1-sen^2x)~.~tg^2(x)~=~sen^2x\\\\\\tg^2x-sen^2x~.~tg^2x~=~sen^2x\\\\\\sen^2x+sen^2x~.~tg^2x~=~tg^2x\\\\\\sen^2x~.~(1+tg^2x)=tg^2x\\\\\\\boxed{sen^2x~=~\frac{tg^2x}{1+tg^2x}}$

Substituindo o valor de tg(x):

$sen^2x~=~\frac{(\sqrt{3})^2}{1+(\sqrt{3})^2}\\\\\\sen^2x~=~\frac{3}{1+3}\\\\\\\boxed{sen^2x~=~\frac{3}{4}}$