Question
find the exact value of sin 112.5 in fraction form
Answers
Steve
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)
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)
Damon
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
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
Steve
good insight, simpler solution.
Shows that with trig, there is almost always more than one path to solutions.
Shows that with trig, there is almost always more than one path to solutions.