What values for theta(0<_theta<_2pi) satisfy the equation?

Question

3sin theta = sin theta-1

Answer 1

3sinθ = sinθ - 1
I suspect a typo, but as things stand,

2sinθ = -1
sinθ = -1/2

since sin π/6 = 1/2, and sinθ < 0 in QII,QIV, we have

θ = π+π/6 and 2π-π/6

Answer 2

If you are looking at solving
3sin(θ)=sin^-1(θ)
then you would be solving
f(x)=3sin(x)-asin(x)=0
in which case the trivial solution x=0 can be obtained by inspection.
If you're looking for
3sin(θ)=1/sin(θ)
then there are 4 solutions between 0 and 2&pi.
Transform the equation into a quadratic equation in s=sin(θ)
then the equation becomes
f(s)=3s²-1=0 where s≠0
so s=±sqrt(1/3), meaning 4 roots.

Answer 3

I guess Steve's interpretation of the question is correct. There was no indication of power!
Please go with Steve's solution.