Question

Solve the equation sin θ + sin 3θ + sin 5θ = 0

Answer 1

We have sin θ + sin 3θ + sin 5θ = 0

or (sin θ + sin 5θ) + sin 3θ = 0

or 2 sin 3θ cos 2θ + sin 3θ = 0   

or sin 3θ (2 cos 2θ + 1) = 0

or sin 3θ = 0 or cos 2θ = `- 1/2`

When sin 3θ = 0, then 3θ = nπ or θ = `("n"pi)/3`

When cos 2θ = `-1/2`

= `cos  (2pi)/3`

Then 2θ = `2"n"pi +- (2pi)/3` or θ = `"n"pi +- pi/3`

Which gives θ = `(3"n" + 1)  pi/3` or θ = `(3"n" - 1)  pi/3`

All these values of θ are contained in θ = `("n"pi)/3` , n ∈ Z.

Hence, the required solution set is given by `{θ : θ = ("n"pi)/3, "n" ∈ "Z"}`