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Question

Find all the values of θ satisfying the equation,sinθ+sin5θ=sin3θ such that 0θπ

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Solution

sinθ+sin5θ=sin3θ
2sin(θ+5θ2)cos(θ5θ2)=sin3θ
2sin(6θ2)cos(4θ2)=sin3θ
2sin3θcos(2θ)=sin3θ
2sin3θcos(2θ)=sin3θ
sin3θ(2cos2θ1)=0
sin3θ=0,2cos2θ=1
3θ=nπ,cos2θ=12
3θ=nπ,cos2θ=cosπ3
3θ=nπ,2θ=2nπ±π3
θ=nπ3,θ=nπ±π6
θ=0,π3,2π3,3π3(=π)
and θ=π6,ππ6(=5π6)
the solutions are {0,π6,π3,2π3,5π6,π}

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