3sin(2A) = sin(A)

There are a number problems like this in my textbook and I am able to solve them, but I always end up with a polynomial in some trig function. I wanted to know if there were any less time consuming approaches to this question?

Here's what I did for the above example:

3sin(2A) = sin(A)

6sin(A)cos(A) = sin(A)

6(1-cos^2(A))(cos(A)) = 1 - cos^2(A)

6cos^3(A) - cos^2(A) - 6cos(A) + 1 = 0

and then solved to find cos(A).

I have a feeling that I am making the question more time-consuming than neccessary - is there a simpler approach?