Question

How do you use the half angle identity to find exact value of Cos(-5pi/12)?

Answer 1

Find `cos ((-5pi)/12)`

Ans: `+ sqrt(sqrt3 - 2)/2`

Explanation:

Call `cos ((-5pi)/12) = cos t`
`cos 2t = cos ((-10pi)/12) = cos ((5pi)/6) = -cos (pi/6) = -sqrt3/2`
Use trig identity: `cos 2t = 2cos^2 t - 1`

`cos 2t = sqrt3/2 = 2cos^2t - 1.`
`2cos^2 t = sqrt3/2 - 1 = (sqrt3 - 2)/2`
`cos^2t = (sqrt3 -2)/4`

`cos t = cos ((-5pi)/12) = +-sqrt(sqrt3 - 2)/2`
Since `(-5pi)/4` is in quadrant I, then only the positive answer is accepted -->`sqrt(sqrt3 - 2)/2`