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[(1-cos 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
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