SOLUTION: cot(x/2)=1/3 find csc(x/2)

Question 847607: cot(x/2)=1/3 find csc(x/2)
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
for[0,π/2]
cot(x/2)=1/3 find csc(x/2)
tan(x/2)=3
Identity: tan(x/2)=sinx/(1+cosx)
sinx/(1+cosx)=3
sinx=3+3cosx
3cosx=sinx-3
3√(1-sin^2x)=sinx-3
square both sides:
9(1-sin^2x)=sin^2x-6sinx+9
9-9sin^2x=sin^2x-6sinx+9
10sin^2x-6sinx=0
5sin^2x-3sinx=0
sinx(5sinx-3)=0
sinx=0 (reject,sinx>0)
or
sinx=3/5(in Q2)
cosx=-4/5(in Q2)
Identity:sin(x/2)=√((1-cosx)/2)=√((1-(-4/5))/2)=√((9/5)/2)=√(9/10)=3/√10
csc(x/2)=1/sin(x/2)=√10/3
calculator check:
tan(x/2)=3
x/2≈71.56˚
sin(x/2)=3/√10
x/2≈71.56˚
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