Question

How do you use the half-angle identity to find the exact value of Cos(-22.5)?

Answer 1

`cos (22.5) = sqrt(2 + sqrt2)/2`

Explanation:

cos (-22.5) = cos (22.5)
Use half angle identity:
`cos (x/2) = +- sqrt((1 + cos x)/2)`
In this case,
`x/2 = 22.5`, and `x = 45` --> `cos x = cos 45 = sqrt2/2`
`cos (22.5) = +- sqrt((1 + sqrt2/2)/2) = +- sqrt(2 + sqrt2)/2`
Since cos (22.5) is positive, therefor,
`cos (22.5) = sqrt(2 + sqrt2)/2`