trig
posted by Sierra on .
How do you find the exact value of sin 5pi/8?

cos2θ = 1  2sin^2 θ
since 2θ = 5pi/4, cos 2θ = 1/√2
1/√2 = 1  2sin^2 5π/8
2sin^2 5π/8 = 1 + 1/√2
sin 5π/8 = √(1 + 1/√2)/2
= √(2+√2) / 2 
You are talking about 112.5 degrees
that is 90 + 22.5
22.5 is half of 45
so draw this in the third quadrant
I know functions of 45 degrees
sin 45 = +1/sqrt 2
cos 45 = +1/sqrt 2
so what are the functions of 22.5 degrees?
sin(45/2) = +/ sqrt[(1cos 45)/2]
cos (45/2) = +/sqrt[(1+cos 45)/2]
in this case we want the negative cos for the sin (look at sketch) because in quadrant 3
so sin 112.5 = sqrt [ (1+1/sqrt 2)/2]
= sqrt [ (1 + sqrt 2)/2 sqrt 2
= sqrt [ (2+sqrt2)/4 ]
=  .923
check sin 112.5 on calculator = .923 
sin is positive in QII

whoops, sorry