Question

use the exact value of cos 11pi/6 and the half-angle identity for sine to find the exact value of sin 11pi/12

Answer 1

Recall that


`\sin^2x=\frac{1-\cos2x}2`

So you have


`\sin^2(11\pi)/(12)=\frac{1-\cos\frac{11\pi}6}2=\frac{1-(\sqrt3)/(2)}2=\frac{2-\sqrt3}4`

This means


`\sin(11\pi)/(12)=\frac{√(2-\sqrt3)}2`

where you take the positive square root because
`\sin x` must be positive for angles
`x` in the interval
`0<x<\pi`.