Question

find the exact value of cos(7\pi /12)

Answer 1

7π/12 lies in the second quadrant, so we expect cos(7π/12) to be negative.

Recall that

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

which tells us

$\cos\left(\dfrac{7\pi}{12}\right)=-\sqrt{\dfrac{1+\cos\left(\frac{7\pi}6\right)}2}$

Now,

$\cos\left(\dfrac{7\pi}6\right)=-\cos\left(\dfrac\pi6\right)=-\dfrac{\sqrt3}2$

and so

$\cos\left(\dfrac{7\pi}{12}\right)=-\sqrt{\dfrac{1-\frac{\sqrt3}2}2}=\boxed{-\dfrac{\sqrt{2-\sqrt3}}2}$