Question

Question #f4579

Answer 1

`- sqrt(2 - sqrt2)/2`

Explanation:

`csc ((9pi)/8) = 1/(sin ((9pi)/8))`
First, find `sin ((9pi)/8)`.
Trig unit circle gives:
`sin ((9pi)/8) = sin (pi/8 + (8pi)/8) = sin (pi/8 + pi) = - sin (pi/8)`
To evaluate `sin (pi/8)`, use trig identity:
2sin^2 a = 1 - cos 2a.
In this case, cos 2a --> `cos pi/4 = sqrt2/2` (trig table)
`2sin^2 (pi/8) = 1 - sqrt2/2 = (2 - sqrt2)/2`
`sin^2 (pi/8) = (2 - sqrt2)/4`
`sin (pi/8) = sqrt(2 - sqrt2)/2`.
Take the positive answer because `sin (pi/8) > 0`.
Finally,
`csc ((9pi)/8) = - sin (pi/8) = - sqrt(2 - sqrt2)/2`