Question

How do you use half angle formula to find `sin 165`?

Answer 1

`sin 165 = sqrt( 2 - sqrt3)/2`

Explanation:

Use half angle identity:
`sin t = +- sqrt((1 - cos 2t)/2)`
`2sin^2 t = (1 - cos 2t)`
In this case:
`cos 2t = cos 330 = cos (- 30 + 360) = cos (-30) = cos 30 = sqrt3/2`.
Hence,
`2sin^2 165 = 1 - sqrt3/2 = (2 - sqrt3)/2`
`sin^2 165 = (2 - sqrt3)/4`
`sin 165 = +- sqrt(2 - sqrt3)/2`
Since sin 165 is positive, then,
`sin 165 = sqrt(2 - sqrt3)/2`