Without using calculator, find the exact value of
𝑃 = sin 20° sin 40° sin 80°

To find the exact value of \( P = \sin 20^\circ \sin 40^\circ \sin 80^\circ \), we can use a known trigonometric identity involving the product of sines of angles that are in arithmetic progression. The identity is:

\[
\sin x \sin(60^\circ - x) \sin(60^\circ + x) = \frac{1}{4} \sin 3x
\]

For \( x = 20^\circ \), the angles become:
- \( 60^\circ - 20^\circ = 40^\circ \)
- \( 60^\circ + 20^\circ = 80^\circ \)

Substituting \( x = 20^\circ \) into the identity:

\[
\sin 20^\circ \sin 40^\circ \sin 80^\circ = \frac{1}{4} \sin 60^\circ
\]

We know that \( \sin 60^\circ = \frac{\sqrt{3}}{2} \). Therefore,

\[
\sin 20^\circ \sin 40^\circ \sin 80^\circ = \frac{1}{4} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{8}
\]

Thus, the exact value of \( P \) is:

\[
P = \sin 20^\circ \sin 40^\circ \sin 80^\circ = \frac{\sqrt{3}}{8}
\]

Would you like more details or have any questions?

Here are 8 related questions:
1. What is the value of \(\sin 60^\circ\)?
2. How can trigonometric identities simplify calculations?
3. What is the exact value of \(\cos 30^\circ\)?
4. Can you prove the identity \(\sin x \sin(60^\circ - x) \sin(60^\circ + x) = \frac{1}{4} \sin 3x\)?
5. How would you find the exact value of \(\sin 45^\circ \sin 75^\circ \sin 105^\circ\)?
6. What are other trigonometric identities involving products of sines or cosines?
7. How can you find the exact values of other trigonometric functions for specific angles?
8. Why are exact values of trigonometric functions important in mathematics?

**Tip:** Remember that using trigonometric identities can simplify complex expressions and help find exact values without a calculator.

Learn how to find the exact value of \( P = \sin 20^\circ \sin 40^\circ \sin 80^\circ \) using trigonometric identities. This problem involves applying the product-to-sum identity for sine, simplifying calculations without a calculator. Suitable for high school students.

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