Question

Find the exact value of cos 7x/12

Answer 1

I suppose x should be π.

Recall the double angle identity for cosine:

$\cos^2\dfrac x2=\dfrac{1+\cos x}2$

Then remember for $0$, we have $\cos x>0$.

Let $x=\frac{7\pi}6$. Plugging this into the equation above gives

$\cos^2\dfrac{7\pi}{12}=\dfrac{1+\cos\frac{7\pi}6}2$

Take the square root of both sides; this introduces two possible values, but we know $\cos\frac{7\pi}{12}$ should be positive, so

$\cos\dfrac{7\pi}{12}=\sqrt{\dfrac{1+\cos\frac{7\pi}6}2}=\dfrac{\sqrt{2-\sqrt3}}2$