# Task 4.

## Maximum 10 points.

Calculate using trigonometric transformations

$$
\sin \frac{\pi}{22} \cdot \sin \frac{3 \pi}{22} \cdot \sin \frac{5 \pi}{22} \cdot \sin \frac{7 \pi}{22} \cdot \sin \frac{9 \pi}{22}
$$

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# Solution:

Transform each factor using the formula $\sin \alpha=\cos \left(\frac{\pi}{2}-\alpha\right)$.

We get:

$$
\begin{gathered}
\sin \frac{\pi}{22} \cdot \sin \frac{3 \pi}{22} \cdot \sin \frac{5 \pi}{22} \cdot \sin \frac{7 \pi}{22} \cdot \sin \frac{9 \pi}{22}= \\
=\cos \left(\frac{\pi}{2}-\frac{\pi}{22}\right) \cdot \cos \left(\frac{\pi}{2}-\frac{3 \pi}{22}\right) \cdot \cos \left(\frac{\pi}{2}-\frac{5 \pi}{22}\right) \cdot \cos \left(\frac{\pi}{2}-\frac{7 \pi}{22}\right) \cdot \cos \left(\frac{\pi}{2}-\frac{9 \pi}{22}\right)= \\
=\cos \frac{5 \pi}{11} \cdot \cos \frac{4 \pi}{11} \cdot \cos \frac{3 \pi}{11} \cdot \cos \frac{2 \pi}{11} \cdot \cos \frac{\pi}{11}=
\end{gathered}
$$

Multiply and divide by $\sin \frac{\pi}{11} \neq 0$:

$$
=\frac{\cos \frac{5 \pi}{11} \cdot \cos \frac{4 \pi}{11} \cdot \cos \frac{3 \pi}{11} \cdot \cos \frac{2 \pi}{11} \cdot\left(\cos \frac{\pi}{11} \sin \frac{\pi}{11}\right)}{\sin \frac{\pi}{11}}=
$$

Apply the formula $\sin \alpha \cdot \cos \alpha=\frac{1}{2} \sin 2 \alpha$

$$
\begin{aligned}
& =\frac{\cos \frac{5 \pi}{11} \cdot \cos \frac{4 \pi}{11} \cdot \cos \frac{3 \pi}{11} \cdot \cos \frac{2 \pi}{11} \cdot \frac{1}{2} \sin \frac{2 \pi}{11}}{\sin \frac{\pi}{11}}= \\
& =\frac{\cos \frac{5 \pi}{11} \cdot \cos \frac{4 \pi}{11} \cdot \cos \frac{3 \pi}{11} \cdot\left(\cos \frac{2 \pi}{11} \cdot \sin \frac{2 \pi}{11}\right)}{2 \sin \frac{\pi}{11}}= \\
& =\frac{\cos \frac{5 \pi}{11} \cdot \cos \frac{4 \pi}{11} \cdot \cos \frac{3 \pi}{11} \cdot \sin \frac{4 \pi}{11}}{4 \sin \frac{\pi}{11}}= \\
& =\frac{\cos \frac{5 \pi}{11} \cdot \cos \frac{3 \pi}{11} \cdot\left(\cos \frac{4 \pi}{11} \cdot \sin \frac{4 \pi}{11}\right)}{4 \sin \frac{\pi}{11}}= \\
& =\frac{\cos \frac{5 \pi}{11} \cdot \cos \frac{3 \pi}{11} \cdot \sin \frac{8 \pi}{11}}{8 \sin \frac{\pi}{11}}=
\end{aligned}
$$

Notice that $\cos \frac{3 \pi}{11}=-\cos \left(\pi-\frac{3 \pi}{11}\right)=-\cos \frac{8 \pi}{11}$,

$\cos \frac{5 \pi}{11}=-\cos \left(\pi+\frac{5 \pi}{11}\right)=-\cos \frac{16 \pi}{11}$.

Considering this, we get:

$$
\begin{gathered}
=\frac{\cos \frac{16 \pi}{11} \cdot\left(\cos \frac{8 \pi}{11} \cdot \sin \frac{8 \pi}{11}\right)}{8 \sin \frac{\pi}{11}}= \\
=\frac{\cos \frac{16 \pi}{11} \cdot \sin \frac{16 \pi}{11}}{16 \sin \frac{\pi}{11}}=\frac{\sin \frac{32 \pi}{11}}{32 \sin \frac{\pi}{11}}=\frac{\sin \left(3 \pi-\frac{\pi}{11}\right)}{32 \sin \frac{\pi}{11}}= \\
=\frac{\sin \left(\pi-\frac{\pi}{11}\right)}{32 \sin \frac{\pi}{11}}==\frac{\sin \frac{\pi}{11}}{32 \sin \frac{\pi}{11}}=\frac{1}{32}
\end{gathered}
$$

Answer: $\frac{1}{32}$

## Criteria.

If calculated approximately on a calculator, then $0 \%$. - $\mathbf{0}$ points

If a constant factor (minus; 0.5; 0.25; ...) is lost in the solution, but the solution method is correct; trigonometric functions are absent in the answer, then $75 \%$ - 7 points

If the answer is correct, then with justification $100 \%$ - 10 points

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