Question

How do you find the value of `cos ((3pi)/8)` using the double or half angle formula?

Answer 1

`(2 - sqrt2)/2`

Explanation:

Trig table, unit circle, and property of complementary arcs -->
`cos ((3pi)/8) = cos (-pi/8 + (4pi/8)) = cos (-pi/8 + pi/2) =`
`= sin (pi/8).`
Find sin (pi/8) by using trig identity:
`cos 2a = 1 - 2sin^2 a`
`cos ((2pi)/8) = cos (pi/4) = sqrt2/2 = 1 - 2sin^2 (pi/8)`
`2sin^2 (pi/8) = 1 - sqrt2/2 = (2 - sqrt2)/2`
`sin^2 (pi/8) = (2 - sqrt2)/4`
`sin (pi/8) = sqrt(2 - sqrt2)/2` (because `sin (pi/8)` is positive.
Finally,
`cos ((3pi)/8) = sin (pi/8) = sqrt(2 - sqrt2)/2`