To evaluate the expression $\sin\left(\frac{\pi}{7}\right) \sin\left(\frac{2\pi}{7}\right) \sin\left(\frac{4\pi}{7}\right)$, we can use some properties of trigonometric functions and symmetry.

Step-by-step approach:

The values $\frac{\pi}{7}$, $\frac{2\pi}{7}$, and $\frac{4\pi}{7}$ are angles that come from dividing the full circle $2\pi$ into 7 parts, and the angles have a specific symmetry. There is a known identity that relates these products of sines for angles like these:

$\sin\left(\frac{\pi}{7}\right) \sin\left(\frac{2\pi}{7}\right) \sin\left(\frac{4\pi}{7}\right) = \frac{1}{8}$

This result is derived from advanced techniques in trigonometry, including the use of roots of unity and symmetry in the unit circle.

Final Answer:

$\sin\left(\frac{\pi}{7}\right) \sin\left(\frac{2\pi}{7}\right) \sin\left(\frac{4\pi}{7}\right) = \frac{1}{8}$

Would you like more details on how to derive this result, or do you have any other questions?

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Further Questions to Explore:

1. Can we use similar methods to evaluate products of sines with other angles like $\frac{\pi}{5}$ or $\frac{\pi}{3}$?
2. How do the symmetries of the unit circle help in simplifying trigonometric products?
3. Can this identity be extended to other similar products like $\sin\left(\frac{3\pi}{7}\right)$ or $\sin\left(\frac{5\pi}{7}\right)$?
4. What are some advanced trigonometric identities used in solving problems involving roots of unity?
5. How does the symmetry of regular polygons (like heptagons) relate to these trigonometric identities?

Tip:

For products of sine functions involving angles that divide the circle into equal parts (like $\frac{\pi}{n}$ where $n$ is an integer), there are often known identities or simplifications that can make calculations much easier!