Question

find the exact value of sin 112.5 in fraction form

Answer 1

112.5 = 5π/8 = (5π/2)/4
so, use your half-angle formula twice:

cos(x/2) = √((1+cos(x))/2)
cos(5π/4) = -√((1+0)/2) = -1/√2
since 5π/4 is in QIII

sin(x/4) = √((1-cos(x/2))/2)
sin(5π/8) = √((1-cos(5π/4)/2)
= √((1+1/√2)/2) = √((√2+1)/2√2)

Answer 2

112.5 = 90 + 22.5
ah ha, 22.5 is half of 45
we will be able to do this with half angle formulas
first:
sin 122.5 = sin(90+22.5) = sin 90 cos 22.5 + cos 90 sin 22.5 = cos 22.5 + 0
so we want cos (45/2)
cos 45/2 = sqrt[ (1+cos 45)/2]
= sqrt [ (1 +1/sqrt2)/2 }

= sqrt [ (1 + sqrt 2) /2 sqrt 2 ]

= sqrt [ (2 + sqrt 2) /4 ]

= (1/2)sqrt (2+sqrt 2)
about .923879 which checks

Answer 3

good insight, simpler solution.

Shows that with trig, there is almost always more than one path to solutions.