Question

How do you find the exact values of cos 75 degrees using the half angle formula?

Answer 1

Find cos (75 deg)
Ans: `+- (sqrt(2 - sqrt3))/2`

Explanation:

Call cos 75 = cos t --> cos 2t = cos 150
`cos 150 = cos (-30 + 180) = - cos 30 = -(sqrt3)/2`
Use trig identity: `cos 2t = 2cos^2 t - 1`
`cos 2t = -(sqrt3)/2 = 2co^2 t - 1`
`2cos^2 t = 1 - (sqrt3)/2 = (2 - sqrt3)/2`
`cos^2 t = (2 - sqrt3)/4`
`cos t = cos 75 = +- (sqrt(2 - sqrt3))/2`