Solve: 2sin(Ø/2)=sinØ on the interval [0,π]

Question

Need help figuring out how to do these types of questions for a test at 4 est today. 

Answer 1

: 2sin(Ø/2)=sinØ on the interval [0,π]Need help figuring out how to do these types of questions for a test at 4 est today. 

 

Solution: let Ø/2 = u , then Ø = 2u substitute

 

2sin(u)=sin(2u), from double angle formulas sin(2u)=2 sinu cosu

 

substitute. 2sin(u)=2sin(u)cos(u), divide both sides by 2 sin(u) then

 

cos(u)=1, substitute u = Ø/2 then cos(Ø/2)=1

 

the only angle whose cosine is 1 is 0 , then Ø/2=0, and Ø=0

 

 

 

Answer 2

2sin(θ/2) = sinθ

 

2sin(θ/2) = sin[2(θ/2)]

 

2sin(θ/2) = 2sin(θ/2)cos(θ/2)

 

2sin(θ/2) - 2sin(θ/2)cos(θ/2) = 0

 

2sin(θ/2)[1 - cos(θ/2)] = 0

 

sin(θ/2) = 0 or cos(θ/2) = 1

 

If sin(θ/2) = 0, then θ/2 = kπ    θ = 2kπ     θ = 0

 

If cos(θ/2) = 1, then θ/2 = 2kπ     θ = 4kπ     θ = 0

 

The only solution in the interval [0,π] is 0.