Question

How do you use the half angle identity to find exact value of sin^2(pi/8)?

Answer 1

Find `sin^2 (pi/8)`

Ans: `sqrt(2 - sqrt2)/2`

Explanation:

`Call sin (pi/8) = sin t`
`cos 2t = cos (2pi)/8 = cos (pi)/4 = sqrt2/2`
Apply the trig identity: `cos 2t = 1 - 2sin^2 t`
`sqrt2/2 = 1 - 2sin^2 t`
`2sin^2 t = 1 - sqrt2/2 = (2 - sqrt2)/2`
`sin^2 t = (2 - sqrt2)/4`
`sin t = +- (2 - sqrt2)/2`
since arc `pi/8` is located in Quadrant I, only the positive answer is accepted.
`sin ((7pi)/8) = sin t = sqrt(2 - sqrt2)/2`