csc ((9pi)/8) = 1/(sin ((9pi)/8)) csc(9π8)=1sin(9π8)
First, find sin ((9pi)/8)sin(9π8).
Trig unit circle gives:
sin ((9pi)/8) = sin (pi/8 + (8pi)/8) = sin (pi/8 + pi) = - sin (pi/8)sin(9π8)=sin(π8+8π8)=sin(π8+π)=−sin(π8)
To evaluate sin (pi/8)sin(π8), use trig identity:
2sin^2 a = 1 - cos 2a.
In this case, cos 2a --> cos pi/4 = sqrt2/2cosπ4=√22 (trig table)
2sin^2 (pi/8) = 1 - sqrt2/2 = (2 - sqrt2)/22sin2(π8)=1−√22=2−√22
sin^2 (pi/8) = (2 - sqrt2)/4sin2(π8)=2−√24
sin (pi/8) = sqrt(2 - sqrt2)/2sin(π8)=√2−√22.
Take the positive answer because sin (pi/8) > 0sin(π8)>0.
Finally,
csc ((9pi)/8) = - sin (pi/8) = - sqrt(2 - sqrt2)/2csc(9π8)=−sin(π8)=−√2−√22
