Question

Se sen(x) + cos(x) = 3/4, então um possível valor para o cos(2x) é:
A) raíz de 305/16
B) -raíz de 207/16
C) raíz de 207/ 256
D) - raíz de 305/256
E) 207/8

Answer 1

Alternativa B

$\displaystyle \sin{x}+\cos{x}=\frac{3}{4}~\Leftrightarrow~\left(\sin{x}+\cos{x}\right)^2=\frac{9}{16}\\\\\Leftrightarrow~ \underbrace{\sin^2{x}+\cos^2{x}}_{=~1}~+~\underbrace{2\sin{x}\cos{x}}_{=~\sin{2x}}=\frac{9}{16}\\\\\Leftrightarrow~1+\sin{2x}=\frac{9}{16}~\Leftrightarrow~\sin{2x}=-\frac{7}{16}~\Leftrightarrow~\sin^2{2x}=\frac{49}{256}\\\\\Leftrightarrow~\underbrace{\sin^2{2x}+\cos^2{2x}}_{=~1}=\frac{49}{256}+\cos^2{2x}~\Leftrightarrow~1=\frac{49}{256}+\cos^2{2x}$
$\displaystyle \Leftrightarrow~\cos^2{2x}=1-\frac{49}{256}~\Leftrightarrow~\cos^2{2x}=\frac{207}{256}~\Leftrightarrow~|\cos{2x}|=\frac{\sqrt{207}}{16}\\\\\Leftrightarrow~\cos{2x}=\pm\frac{\sqrt{207}}{16}$