Solve for −12cos^2θ+3=−3 and 0≤θ<2π. Enter exact answers in terms of π. If there are multiple solutions, enter them separated by commas.

Answer 1

-12 cos² θ + 3 = -3

-12 cos² θ = -9

cos² θ = 0.75

cos θ ≈ 0.8θ66

θ ≈ 0.524^r

Answer 2

Solutions:

−12 cos² θ + 3 = −3

−12 cos² θ = −6 (Subtract 3 from both sides)

cos² θ = 1 ⁄ 2 (Divide both sides by −12)

cos θ = ±1 ⁄ √2 (Taking the square root)

(1) For cos θ = 1 ⁄ √2, 0 ≤ θ < 2π,

θ = π ⁄ 4, 7π ⁄ 4.

(2) For cos θ = −1 ⁄ √2, 0 ≤ θ < 2π,

θ = 3π ⁄ 4, 5π ⁄ 4.

∴ All the solutions between 0 and 2π are:

θ = π ⁄ 4, 3π ⁄ 4, 5π ⁄ 4, 7π ⁄ 4.

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Solutions:

−12 cos² 2θ + 3 = −3

−12 cos² 2θ = −6 (Subtract 3 from both sides)

cos² 2θ = 1 ⁄ 2 (Divide both sides by −12)

cos 2θ = ±1 ⁄ √2 (Taking the square root)

Since 0 ≤ θ < 2π, 0 ≤ 2θ < 4π.

(1) For cos 2θ = 1 ⁄ √2,

2θ = π ⁄ 4, 7π ⁄ 4, 9π ⁄ 4, 15π ⁄ 4

θ = π ⁄ 8, 7π ⁄ 8, 9π ⁄ 8, 15π ⁄ 8

(2) For cos 2θ = −1 ⁄ √2,

2θ = 3π ⁄ 4, 5π ⁄ 4, 11π ⁄ 4, 13π ⁄ 4

θ = 3π ⁄ 8, 5π ⁄ 8, 11π ⁄ 8, 13π ⁄ 8

∴ All the solutions are

π ⁄ 8, 7π ⁄ 8, 9π ⁄ 8, 15π ⁄ 8,

3π ⁄ 8, 5π ⁄ 8, 11π ⁄ 8, 13π ⁄ 8.