Question

How do I use a half-angle identity to find the exact value of cos 165º ??

Answer 1

`cos 165 = - sqrt(2 + sqrt3)/2`

Explanation:

Use half angle identity:
`cos (a/2) = +- sqrt((1 + cos a)/2)`
In this case, `cos (a/2)` = cos 165, and cos a = cos 330.
Trig table gives -->
`cos 330 = cos (-30 + 360) = cos (-30) = cos 30 = sqrt3/2`
We then get:
`cos 165 = +- sqrt((1 + sqrt3/2)/2) = +- sqrt((2 + sqrt3)/4)`
Since cos 165 is negative (Quadrant 2), so,
`cos 165 = - sqrt(2 + sqrt3)/2`
Check by calculator.
`- sqrt(2 + sqrt3)/2 = - 1.93/2 = - 0.965`
Calculator gives -> cos 165 = - 0.965# . Proved.