Question

How do you use the half-angle identity to find the exact value of cos [ - (3pi) / 8]?

Answer 1

Find `cos ((-3pi)/8)`

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

Explanation:

Call cos ((-3pi)/8) = cos t
`cos 2t = cos ((-6pi)/8) = cos ((6pi)/8) = cos ((3pi)/4) = -sqrt2/2`

Use the trig identity: `cos 2t = 2cos^2 t - 1`
`cos 2t = -sqrt2/2 = 2cos^2 t - 1`
`2cos^2 t = 1 - sqrt2/2 = (2 - sqrt2)/2`
`cos^2 t = (2 - sqrt2)/4`
`cos t = cos ((-3pi)/8) = +- sqrt(2 - sqrt2)/2`
Only the positive answer is accepted, because the arc `(-3pi)/8` is in Quadrant IV.
Check by calculator.
`sqrt(2 - sqrt2)/2` = 0.382
`cos ((3pi)/8) = cos 67.5` deg = 0.382.# Correct