precal question

Question

Use a Double- or Half-Angle Formula to solve the equation in the interval [0, 2𝜋). (Enter your answers as a comma-separated list.)

2 cos²(𝜃) = 2 − cos(2𝜃)

𝜃 = 

 

 

Answer 1

Using the Double-angle Formula we can rewrite the equation as

2cos²(θ) = 2 - (1 - 2sin²(θ))

2(cos²(θ) - sin²(θ)) = 1

Using the identity sin²(θ) = 1 - cos²(θ)

2(cos²(θ) - 1 + cos²(θ)) = 1

4cos²(θ) = 3

cos(θ) = +- sqrt(3)/2

In the interval [0,2π) there are these solutions

θ = π/6, 5π/6, 7π/6, 11π/6