equation on the interval

Question

Solve the equation on the interval

0 less than or equals theta less than 2 pi0≤θ<2π.

sine left parenthesis 3 theta right parenthesis equals negative 1sin(3θ)=−1

Question content area bottom

Part 1

What are the solutions to

sine left parenthesis 3 theta right parenthesis equals negative 1sin(3θ)=−1

in the interval

0 less than or equals theta less than 2 pi0≤θ<2π​?

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

Answer 1

sin3T = -1

3T = arcsin-1= -90=270

T= 270/3= 90= pi/2 radians

also

3T = 270+360n = 630, 990, ...

T = 630/3, 990/3 = 210, 330 = 7pi/6, 11pi/6

T = pi/2, 7pi/6, 11pi/6

Answer 2

sin(3θ) = -1

3θ = 3π/2 + 2kπ

θ = π/2 + 2kπ/3, k = 0, ±1, ±2, ...

if k = 0, then θ = π/2

if k = 1, then θ = 7π/6

if k = 2, then θ = 11π/6

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