How do you solve 2 sin²x − 5 cos x − 4 = 0?

Question

Solve 2 sin²x − 5 cos x − 4 = 0 and choose the correct answer.

Answer choices

1. x = ±π/3 + 2πk, k ∈ ℤ.
2. x = ±2π/3 + 2πk, k ∈ ℤ.
3. x = −1/2; x = −2.
4. x = ±2π/3 + 2πk, k ∈ ℤ; x = ±arccos(−2) + 2πn, n ∈ ℤ.

Explanation

Use sin²x = 1 − cos²x:

2(1 − cos²x) − 5 cos x − 4 = 0.

Collecting terms gives:

2 cos²x + 5 cos x + 2 = 0,

(2 cos x + 1)(cos x + 2) = 0.

Thus cos x = −1/2 or cos x = −2. The second value is impossible for real x, since −1 ≤ cos x ≤ 1. Therefore x = ±2π/3 + 2πk, k ∈ ℤ.