Conditions on $\alpha$ and $\beta$ integers between $1$ and $89$ for $\sin \alpha^{\circ}+\sin \beta^{\circ}$ to be irrational

Question

Under what conditions $$\sin \alpha^{\circ}+\sin \beta^{\circ} \not \in \mathbb Q$$ where $\alpha \in \mathbb N, \alpha<90,\beta \in \mathbb N, \beta<90.$

My work so far:

I used:

1) $\sin x+\sin y=2\sin\frac{x+y}2\cos \frac{x-y}2$

2) $\sin 10^{\circ} \not \in \mathbb Q$

3) $\sin 18^{\circ} \not \in \mathbb Q$

4) Lemma: Let $\alpha = r \pi, r \in \mathbb Q$ and $0

If $r\not=\frac16$ then $\sin \alpha \not \in \mathbb Q$

I don't know what to do next