Question

Find an exact value.

$cos(- \frac{ \77 \pi }{12})$

Answer 1

$\cos\left(-\dfrac{7\pi}{12}\right)=\cos\dfrac{7\pi}{12}$

Recall that

$\cos^2x=\dfrac{1+\cos2x}2$

Let $x=\dfrac{7\pi}{12}$, so that

$\cos^2\dfrac{7\pi}{12}=\dfrac{1+\cos\dfrac{7\pi}6}2=\dfrac{1-\dfrac{\sqrt3}2}2=\dfrac{2-\sqrt3}4$
$\cos\dfrac{7\pi}{12}=\pm\sqrt{\dfrac{2-\sqrt3}4}=\pm\dfrac{\sqrt{2-\sqrt3}}2$

Since the angle $\dfrac{7\pi}{12}$ lies in the second quadrant, you know that the cosine must be negative, so

$\cos\dfrac{7\pi}{12}=-\dfrac{\sqrt{2-\sqrt3}}2$