Find an exact value. cos(- ( \77 \pi )/(12))

Question

Find an exact value.


`cos(- ( \77 \pi )/(12))`

Answer 1

`\cos\left(-(7\pi)/(12)\right)=\cos(7\pi)/(12)`

Recall that


`\cos^2x=\frac{1+\cos2x}2`

Let
`x=(7\pi)/(12)`, so that


`\cos^2(7\pi)/(12)=\frac{1+\cos\frac{7\pi}6}2=\frac{1-\frac{\sqrt3}2}2=\frac{2-\sqrt3}4`

`\cos(7\pi)/(12)=\pm\sqrt{\frac{2-\sqrt3}4}=\pm\frac{√(2-\sqrt3)}2`

Since the angle
`(7\pi)/(12)` lies in the second quadrant, you know that the cosine must be negative, so


`\cos(7\pi)/(12)=-\frac{√(2-\sqrt3)}2`