Finding solutions to the equation in the interval?

Question

Find all the solutions to the equation tan(t)=3/(tan(t)) in the interval 0 ≤ t ≤ 2pi. (given as fractions, not decimals)

Answer 1

The trigonometric equation tan(t) = 3/tan(t)

tan(t) tan(t) = 3

tan²(t) = 3

tan² (t) = (√3)²

tan² (t) = [tan(π/3)]²

tan² (t) = tan²(π/3)

The general solution of tan²(θ) = tan²(α),then θ = nπ ± α  where n is an integer.

θ = t, α  = π/3

For n = 0, t = 0 ± π/3 = π/3 , -π/3

For n = 1, t = π ± π/3 = 4π/3 , 2π/3

For n = 2, t = 2π ± π/3 = 7π/3 , 5π/3

The solutions in the interval 0 ≤ t ≤ 2pi are π/3, 2π/3, 4π/3 and 5π/3.