Question

If cos x= √2/2 and x is a fourth quadrant angle, evaluate cos 2x

Answer 1

Start by making a graph of the problem, taking into account that x is in the fourth quadrant and the triangle formed has a hypotenuse of 2 and an adjacent side of √2

use the Pythagorean theorem to find the missing side

$\begin{gathered} a^2+b^2=c^2 \\ a=\sqrt[]{c^2-b^2} \\ a=\sqrt[]{(2)^2-(\sqrt[]{2})^2} \\ a=\sqrt[]{4-2} \\ a=\sqrt[]{2} \end{gathered}$

then,

find the value of six(X)

$\begin{gathered} \sin x=\frac{op}{hy} \\ \sin x=-\frac{\sqrt[]{2}}{2} \end{gathered}$

use the double angle identities for cos

$\begin{gathered} \cos 2x=\cos ^2x-\sin ^2x \\ \cos 2x=(\frac{\sqrt[]{2}}{2})^2-(-\frac{\sqrt[]{2}}{2})^2 \\ \cos 2x=\frac{2}{4}-\frac{2}{4} \\ \cos 2x=\frac{1}{2}-\frac{1}{2} \\ \cos 2x=0 \end{gathered}$