Question

How do you find the exact values of cos(67.5) using the half angle formula?

Answer 1

Find cos (67.5)

Ans: `(2 - sqrt2)/2`

Explanation:

Call cos (67.5) = cos t

`cos 2t = cos 135 = cos (-45 + 180) = -cos 45 = - sqrt2/2`
Use trig identity:`cos 2t = cos^2 t - 1`
`cos 2t = -sqrt2/2 = 2cos^2 t - 1`
`2cos^2 t = (2 - sqrt2)/2` -->`cos^2 t = (2 - sqrt2)/4`

`cos (67.5) = cos t = +- (2 - sqrt2)/2`
Since arc 67.5 is located in Quadrant I, therefor, only the positive answer is accepted.
`cos (67.5) = (2 - sqrt2)/2`