Question

How do you find the value of `sin((7pi)/8)` using the double or half angle formula?

Answer 1

`sqrt(2 - sqrt2)/2`

Explanation:

Trig table of special arcs, and unit circle -->
`sin ((7pi/8) = sin (pi - pi/8) = sin (pi/8)`
Find `sin (pi/8)` by using trig identity:
`2sin^2 a = 1 - cos 2a`
`2sin^2 (pi/8) = 1 - cos (pi/4) = 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,
`sin ((7pi)/8) = sin (pi/8) = sqrt(2 -sqrt2)/2`