1467. Calculate $\sin 10^{\circ}$ with an accuracy of 0.0001.
Solution. We will use series (6) of § 4, expressing degrees in radians:

$$
\sin 10^{\circ}=\sin \frac{\pi}{18}=\frac{\pi}{18}-\left(\frac{\pi}{18}\right)^{3} \frac{1}{3!}+\left(\frac{\pi}{18}\right)^{5} \frac{1}{5!}-\ldots
$$

$\frac{\pi}{18}<0.2$, therefore, the third term of the expansion

$$
a_{3}=\left(\frac{\pi}{18}\right)^{5} \cdot \frac{1}{5!}<\frac{(0.2)^{5}}{5!}=\frac{2^{5}}{5!10^{5}}=\frac{32}{5!10^{5}}<0.00001
$$

Therefore, if we set $\sin 10^{\circ} \approx a_{1}+a_{2}$, the error, according to the Leibniz criterion, will not exceed the absolute value of the third term $a_{3}$, i.e., it will be less than the given value. Calculating with accuracy to the fourth decimal place, we find

$$
\begin{aligned}
& a_{1}=\frac{\pi}{18} \approx 0.1745, a_{2}=\left(\frac{\pi}{18}\right)^{3} \frac{1}{3!} \approx 0.0009 \\
& \quad \sin 10^{\circ} \approx 0.1745-0.0009=0.1736
\end{aligned}
$$

(compare the obtained result with the value of $\sin 10^{\circ}$ in the Bradis tables).