Question

The values of θ satisfying sin7θ = sin4θ - sinθ and 0 < θ < `pi/2` are ______

Options

• `pi/9, pi/4`

• `pi/3, pi/9`

• `pi/6, pi/9`

• `pi/3, pi/4`

Answer 1

The values of θ satisfying sin 7θ = sin4θ - sinθ and 0 < θ < `pi/2` are `underline(pi/9, pi/4)`.

Explanation:

sin7θ = sin4θ - sinθ

⇒ sin7θ + sinθ - sin4θ = 0

⇒ 2sin4θ cos3θ - sin4θ = 0

⇒ sin4θ(2 cos3θ - 1) = 0

⇒ sin4θ = 0 or cos3θ = `1/2`

Now, sin4θ = 0 ⇒ 4θ = π ⇒ θ = `pi/4`

and cos3θ = `1/2` ⇒ 3θ = `pi/3` ⇒ θ = `pi/9`