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=\frac{1+\cos(2x)}2`

which tells us

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

Now,

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

and so

`\cos\left((7\pi)/(12)\right)=-\sqrt{\frac{1-\frac{\sqrt3}2}2}=\boxed{-\frac{√(2-\sqrt3)}2}`