Question

How do you evaluate `cos ((3pi)/8)`?

Answer 1

`sqrt(2 - sqrt2)/2`

Explanation:

Property of complementary arcs -->
`cos ((3pi)/8) = cos (-pi/8 + pi/2) = sin (pi/8).`
Evaluate `sin (pi/8)` by applying the trig identity:
`cos 2a = 1 - 2sin^2 a.`
Trig table -->
`cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)`.
`2sin^2 (pi/8) = 1 - sqrtt2/2 = (2 - sqrt2)/2`
`sin^2 (pi/8) = (2 - sqrt20/4`
`sin (pi/8) = sqrt(2 - sqrt2)/2`
Since `pi/8` is in Quadrant I, then we select the positive value.
`cos ((3pi)/8) = sin (pi/8) = sqrt(2 - sqrt2)/2`