Question

If `(sin^2θ - 3sinθ + 2)/(cos^2θ) = 1`, then θ = ______.

Answer 1

If `(sin^2θ - 3sinθ + 2)/(cos^2θ) = 1`, then θ = 30°.

Explanation:

Let s = sin θ.

Substitute cos²θ = 1 – sin²θ (so cos²θ ≠ 0).

Then `(s^2 - 3s + 2)/(1 - s^2) = 1`

⇒ s² – 3s + 2 = 1 – s² 

⇒ 2s² – 3s + 1 = 0

⇒ (2s – 1)(s – 1) = 0

⇒ s = 1 or s = `1/2`

s = 1 must be rejected because it makes cos²θ = 0 (division by zero in the original equation).

So `sin θ = 1/2`.

Therefore `θ = π/6 + 2kπ` or `θ = (5π)/6 + 2kπ`, where k is any integer. (In degrees: 30° + 360°k or 150° + 360°k.)