Question

Solve the equation: sin θ + sin 5θ = sin 3θ, 0 < θ < π.

Answer 1

θ = π /12 is the required answer.

Given that

sin θ + sin 5θ = sin 3θ, 0 < θ < π

sin 5θ + sin θ = sin 3θ

2 sin[(5θ + θ)/2].cos[(5θ - θ)/2] = sin 3θ

2 sin 3θ. cos 2θ - sin 3θ = 0

sin 3θ [2 cos 2θ - 1] = 0

sin 3θ = 0          ,   2 cos 2θ - 1 = 0

For sin 3θ = 0

θ = 0 Not valid according to given condition

For 2 cos 2θ - 1 = 0

2 cos 2θ - 1 = 0

2 cos 2θ = 1

cos 2θ = 1 /2

 2θ = π /6

θ = π /12 is the required answer.