Question

How do you use the half-angle identity to find the exact value of cos(67.5 degrees)?

Answer 1

Find cos (67.5)

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

Explanation:

Call cos (67.5) = cos t. Apply the trig identity: `cos 2t = 2cos^2 t - 1.`
`cos (135) = 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 = +- sqrt(2 - sqrt2)/2`.
Because the arc (67.5) is in Quadrant I, its cos is positive. Therefor,
`cos (67.5) = cos t = (2 - sqrt2)/2`
Check by calculator.
sin (67.5) = 0.38
sqrt(2 - sqrt2)/2 = 0.79/2 = 0.38. OK