Finite arithmetic sequence problem

Question

Choose such x that the following

$$\sin(3x+\pi/3) , \sin(2x+\pi/12), \sin(x-\pi/6)$$

forms finite arithmetical sequence.

I don't even know how to begin with such kind of problem. Thanks in advance for the answers.

Answer 1

Hint: Use that $$a_2=\frac{a_3+a_1}{2}$$ this means $$2\sin(2x+\frac{\pi}{12})=\sin(x-\frac{\pi}{6})+\sin(3x+\frac{\pi}{3})$$ Can you finish?

Answer 2

Use $$a_3-a_2=a_2-a_1$$

$$\sin(x-\pi/6)-\sin(2x+\pi/12)=\sin(2x+\pi/12)-\sin(3x+\pi/3)$$
$$\sin(x-\frac{\pi}{6})+\sin(3x+\frac{\pi}{3})=2\sin(2x+\frac{\pi}{12})$$