Question

How do you use the half angle identity to find exact value of cos pi/8 degrees?

Answer 1

Find `cos ((pi)/8)`

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

Explanation:

Call `(pi/8) = x.`
`cos 2x = cos ((2pi)/8) = cos ((pi)/4) = sqrt2/2`
Apply the trig identity: `cos 2a = 2cos^2 a - 1.`
`cos ((pi)/4) = sqrt2/2 = 2cos^2 x - 1.`
`2cos^2 x = 1 + sqrt2 = (2 + sqrt2)/2`
`cos^2 x = (2 + sqrt2)/4`
`cos x = cos (pi/8) = +- (2 + sqrt2)/2.`
Since `(pi/8)` is located in Quadrant I, its cos is positive, then,
`sin (pi/8) = sqrt(2 + sqrt2)/2`

Check by calculator.
`sin (pi/8) = sin 22.5 deg = 0.92`
`sqrt(2 + sqrt2)/2 = (1.84)/2 = 0.92`. OK