It is known that

$\sin 15^{\circ}+\sin 45^{\circ}=\sin 75^{\circ}$

Find other solutions to the equation

$$
\sin x+\sin y=\sin z
$$

where $x, y, z$ are integers representing angles in degrees.
We advance that we are only looking for solutions where $x, y, z$ are positive acute angles.

(1) Proof of its correctness:

$$
\sin 15^{\circ}+\sin 45^{\circ}=2 \sin \frac{15^{\circ}+45^{\circ}}{2} \cos \frac{15^{\circ}-45^{\circ}}{2}=\cos 15^{\circ}=\sin 75^{\circ}
$$

This shows that when $x+y=60^{\circ}$, that is, when $x=30^{\circ}+u, y=30^{\circ}-u$, it always holds that

$$
\sin x+\sin y=\cos \frac{x-y}{2}=\cos u=\sin \left(90^{\circ}-u\right)
$$

Therefore, each of $u=1^{\circ}, 2^{\circ}, \ldots, 29^{\circ}$ provides a solution of the desired form.

Further integer values of $u$ do not yield new solutions; for example, when $u=31^{\circ}$, $\sin 61^{\circ}+\sin \left(-1^{\circ}\right)=\sin 59^{\circ}$, and we do not consider this different from $\sin 59^{\circ}+\sin 1^{\circ}=\sin 61^{\circ}$. When $u=0^{\circ}$, $z=90^{\circ}$.

Accordingly, the number of non-trivial, essentially different solutions of the form $x=30^{\circ}+u, y=30^{\circ}-u, z=90^{\circ}-u$ is 29.

Of course, the above does not rule out the existence of solutions of other forms. However, the problem did not require the production of all solutions.