What are the solutions to sine left parenthesis 3 theta right parenthesis equals negative 1 in the interval 0 less than or equals theta less than 2 pi ​?

Question

What are the solutions to sine left parenthesis 3 theta right parenthesis equals negative 1

in the interval 0 less than or equals theta less than 2 pi
?

Answer 1

sin(3θ) = -1

 

3θ = 3π/2 + 2kπ, k = 0, ±1, ±2, ...

 

θ = π/2 + 2kπ/3

 

If k = 0, then θ = π/2

 

If k = 1, then θ = π/2 + 2π/3 = 7π/6

 

If k = 2, then θ = π/2 + 4π/3 = 11π/6

 

Other choices for k give values of θ that lie outside of the interval [0, 2π).

 

There are 3 solutions for θ in the interval [0, 2π):   π/2, 7π/6, and 11π/6.

 

Answer 2

sin(3θ) = -1

 

3θ = sin^-1(-1)

 

3θ = 3π/2

 

θ = π/2 for θ ∈ [0,2π)

 

 

Answer 3

sin 3θ = -1

3θ = π

θ = π/3, for 0 ≤ θ ≤ 2π