Question

What is the exact value of `\sin \frac { 13\pi } { 8} |`?

Answer 1

`sqrt(2 - sqrt2)/2`

Explanation:

`sin ((13pi)/8) = sin ((-3pi)/8 + (16pi)/8) = sin ((-3pi)/8 + pi) =`
`= - sin ((-3pi)/8) = sin ((3pi)/8)`.
Find `sin ((3pi)/8)` by using the trig identity:
`2sin^2 a = 1 - cos 2a`.
In this case:
`2sin^2 ((3pi)/8) = 1 - cos ((3pi)/4) = 1 - sqrt2/2 = (2 - sqrt2)/2`
`sin^2 ((3pi)/8) = (2 - sqrt2)/4`
`sin ((3pi)/8) = +- sqrt(2 - sqrt2)/2`
Since the arc `(3pi)/8` is in Quadrant 1, its sin is positive. Take the positive value.