Transform the equation

cos2θ + 3cosθ = 1 → 2cos^{2}θ + 3cosθ – 1 = 1 → 2cos^{2}θ + 3cosθ – 2 = 0

Factor the equation (2cosθ – 1)(cosθ + 2) = 0

Then either 2cosθ – 1 = 0; cosθ = 1/2; θ = +– cos^{–1}(1/2) + 2πk;

θ = +– π/3 + 2πk, where k is any integer (k = 0, 1, 2, 3, …, –1, –2, –3, ...)

or cosθ + 2 = 0; cosθ = – 2, however, cosine values are between –1 and 1, so this equation does not have a solution.

Answer: θ = +– π/3 + 2πk, where k is any integer (k = 0, 1, 2, 3, …, –1, –2, –3, …).

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