1. Find all integers $n$ for which $0 \leqslant n \leqslant 90$ and

$$
\sin 80^{\circ}+\sin 50^{\circ}-\sin 20^{\circ}=\sqrt{2} \sin n^{\circ}
$$
## 1. Vaji

$$
\begin{aligned}
\sin 80^{\circ}-\sin 20^{\circ}+\sin 50^{\circ} & =2 \sin 30^{\circ} \cos 50^{\circ}+\sin 50^{\circ}=2 \cdot \frac{1}{2} \cdot \sin 40^{\circ}+\sin 50^{\circ} \\
& =2 \sin 45^{\circ} \cos 5^{\circ}=\sqrt{2} \sin 85^{\circ}
\end{aligned}
$$

so $n=85$ is the only solution to the problem.

![](https://cdn.mathpix.com/cropped/2024_06_04_adcebcbe7d41887b6eefg-34.jpg?height=288&width=391&top_left_y=745&top_left_x=225)

Op 2019 3B 2