Solve the trigonometric equation: theta + 2 theta = 0 for all general solutions.

The equation given is:

We can factor out $\sin \theta$ from both terms:

This equation holds true if either factor is zero.
Case 1: $\sin \theta = 0$
Case 2: $1 + \sin \theta = 0 \implies \sin \theta = -1$

Case 1: $\sin \theta = 0$
The general solution for $\sin \theta = 0$ occurs at integer multiples of $\pi$.

where $n$ is an integer.

Case 2: $\sin \theta = -1$
The general solution for $\sin \theta = -1$ occurs at $\frac{3\pi}{2}$ plus any integer multiple of $2\pi$.

where $n$ is an integer.

Combining the solutions from both cases, the general solutions for $\theta$ are: