 find exact value of cos 3𝜋 7 cos 2𝜋 21 + sin 3𝜋 7 sin 2𝜋 21

We can solve this expression using the **cosine addition formula**, which states:

\[
\cos(A - B) = \cos(A) \cos(B) + \sin(A) \sin(B)
\]

In your case, the given expression is:

\[
\cos\left(\frac{3\pi}{7}\right) \cos\left(\frac{2\pi}{21}\right) + \sin\left(\frac{3\pi}{7}\right) \sin\left(\frac{2\pi}{21}\right)
\]

This matches the form of the cosine subtraction formula with \(A = \frac{3\pi}{7}\) and \(B = \frac{2\pi}{21}\), so we can rewrite it as:

\[
\cos\left(\frac{3\pi}{7} - \frac{2\pi}{21}\right)
\]

We need to subtract the angles inside the cosine:

\[
\frac{3\pi}{7} - \frac{2\pi}{21} = \frac{9\pi}{21} - \frac{2\pi}{21} = \frac{7\pi}{21} = \frac{\pi}{3}
\]

Thus, the expression becomes:

\[
\cos\left(\frac{\pi}{3}\right)
\]

The exact value of \(\cos\left(\frac{\pi}{3}\right)\) is:

\[
\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
\]

Therefore, the exact value of \(\cos\left(\frac{3\pi}{7}\right) \cos\left(\frac{2\pi}{21}\right) + \sin\left(\frac{3\pi}{7}\right) \sin\left(\frac{2\pi}{21}\right)\) is:

\[
\boxed{\frac{1}{2}}
\]

Would you like more details on this, or have any other questions?

Here are 5 related questions:
1. How do you derive the cosine subtraction formula?
2. What is the exact value of \(\cos\left(\frac{\pi}{6}\right)\)?
3. How can you apply the cosine formula to larger angles, like \(\frac{5\pi}{3}\)?
4. What is the sine subtraction formula, and when would you use it?
5. Can you apply the cosine formula to angles in different quadrants?

**Tip:** Understanding how to manipulate angles in radians is key to simplifying trigonometric expressions effectively.

Discover the exact value of cos(3π/7) cos(2π/21) + sin(3π/7) sin(2π/21) using trigonometric identities and the cosine subtraction formula. Learn step-by-step how to simplify this expression and find its solution. Suitable for high school students studying trigonometry.

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