Question

The number of solutions of cos 2θ = sin θ in (0, 2π) is ______

Options

• 1

• 2

• 3

• 4

Answer 1

The number of solutions of cos 2θ = sin θ in (0, 2π) is 3.

Explanation:

cos 2θ = sin θ ⇒ 1 - 2 sin²θ = sin θ

⇒ 2sin²θ + sinθ - 1 = 0 

⇒ (2sin θ - 1)(sin θ + 1) = 0

⇒ sin θ = `1/2` or sin θ = -1

∴ sin θ = `1/2` = sin`pi/6 ⇒ theta = npi + (-1)^n pi/6`

and sin θ = -1 = sin`(3pi)/2`

⇒ `theta = mpi + (-1)^m (3pi)/2`

∴ `theta = pi/6, (5pi)/6, (3pi)/2`

∴ number of solutions = 3