Question #f4579

1 Answer
Jan 25, 2017

#- 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#