Question

How do you find the value of `sin165` using the half angle identity?

Answer 1

(sqrt(2 - sqrt3)/2

Explanation:

Use trig identity
`1 - cos 2a = 2sin^2 a` (1)
First, find cos 2a = cos 330
`cos 330 = cos (-30 + 360) = cos (-30) = cos (30) = sqrt3/2`
From Identity (1):
`2sin^2 (165) = 1 - cos (330) = 1 - sqrt3/2 = (2 - sqrt3)/2`
`sin^2 (165) = (2 -sqrt3)/4`
`sin (165) = +- sqrt(2 - sqrt3)/2`
Since sin (165) is positive, take the positive answer.