Solve on the interval [0,2pi) : 1-cos theta=1/2

Question

A. 2pi/3 , 4pi/3 B. pi/6 , 5pi/3 C. pi/3 , 5pi/3 D. 7pi/6 , 11pi/6

Answer 1

The trigonometric equation is 1 - cos θ = 1/2.

cos θ = 1 - 1/2

cos θ = 1/2

cos θ = cos (π/3).

The genaral solution of cos(θ) = cos(α) is θ = 2nπ ± α, where n is an integer.

⇒θ = 2nπ ± (π/3).

If n = 0, θ = 2(0)π + (π/3) and θ = 2(0)π - (π/3) = π/3 and - π/3,

If n = 1, θ = 2(1)π + (π/3) and θ = 2(1)π - (π/3) = 2π + π/3 and 2π - π/3 = 7π/3 and 5π/3.

Therefore, the solutions of the given equation are θ = π/3 and θ = 5π/3 in the interval [0, 2π).

Option C is the correct choice.