Precalculus: Solve the equation on the interval 0 ≤ 𝜃 < 2𝜋. sin(2theta) + sin theta = 0

Question

Solve the equation on the interval 0 ≤ 𝜃 < 2𝜋.

sin(2 theta) + sin theta = 0

Answer 1

Hi Katelyn A.,

sin(2θ) + sin(θ) = 0

2sin(θ)cos(θ) + sin(θ) = 0

sin(θ)[2cos(θ) +1] = 0

sin(θ) = 0, and, 2cos(θ) + 1 = 0 or cos(θ) = -1/2θ

For the interval [0, 2):

For sin(θ) = 0 we get θ = 0 and π

For cos(θ) = -1/2 we get = 2π/3 and 4π/3

Therefore the solution set is θ = {0, 2π/3, π, 4π/3}

Use Desmos and graph the equation. The solutions are the x-intercepts between the stated interval.

I hope this helps, Joe.

Answer 2

sin(2θ) = -sin(θ)

2sin(θ)cos(θ) = -sin(θ)

2cos(θ) = -1

cos(θ) = -1/2 Puts it into the 2nd and 3rd quadrants where cos(θ) < 0

θ = 2π/3, 4π/3