Question

If sin θ + sin 2θ + sin 3θ = sin α and cos θ + cos 2θ + cos 3θ = cos α, then θ is equal to ______.

Options

• `α/2`

• α

• 2α

• `α/6`

Answer 1

If sin θ + sin 2θ + sin 3θ = sin α and cos θ + cos 2θ + cos 3θ = cos α, then θ is equal to `underlinebb(α/2)`.

Explanation:

sin θ + sin 2θ + sin 3θ = sin α

`\implies` 2 sin 2θ cos θ + sin 2θ = sin α

`\implies` sin 2θ(2 cos θ + 1) = sin α  ...(i)

Now, cos θ + cos 3θ + cos 2θ = cos α

2 cos 2θ cos θ + cos 2θ = cos α

cos 2θ(2 cos θ + 1) = cos α  ...(ii)

From equations (i) and (ii),

tan 2θ = tan α

`\implies` 2θ = α

`\implies` θ = `α/2`