what is the exact value of sin(u/2) given the angle 112.5 degrees?

Answer 1

Is u/2 = 112.5 or u = 112.5? either way, you can find the value you want by using the half angle formula for sin:

sin(u/2) = sqrt(1/2-cos(u)/2) for 0< (u/2) <180

If u/2 = 112.5, then u=225 and cos(u) = -sqrt(2)/2

If u = 112.5, you can do this twice and find the cos(112.5) from pythagorean theorem (usual technique to find a trig function given any other trig function value for an angle) or from cos(u/2) = sqrt(1/2+cos(u)/2).

Good luck!

Answer 2

112.5/2 = 56.25 degrees

sin(56.25) = about 0.8315

sin(u/2) = sqr(1-cosu)/2) = sqr(1+ .383)/2) = sqr1.383/2 = sqr(0.691) = about 0.83

exact value of sin(u/2) = sqr((1-cos112.5)/2)

56.25 is almost 60. It's almost a 30-60-90 right triangle where sin60 = sqr3/2 = 1.732/2 = 0.866

with slightly less than 60 degrees, the sin56.25 is slightly less than 0.866 or about 0.8315