Math Problem Statement

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

Solution

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

cos(AB)=cos(A)cos(B)+sin(A)sin(B)\cos(A - B) = \cos(A) \cos(B) + \sin(A) \sin(B)

In your case, the given expression is:

cos(3π7)cos(2π21)+sin(3π7)sin(2π21)\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=3π7A = \frac{3\pi}{7} and B=2π21B = \frac{2\pi}{21}, so we can rewrite it as:

cos(3π72π21)\cos\left(\frac{3\pi}{7} - \frac{2\pi}{21}\right)

We need to subtract the angles inside the cosine:

3π72π21=9π212π21=7π21=π3\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(π3)\cos\left(\frac{\pi}{3}\right)

The exact value of cos(π3)\cos\left(\frac{\pi}{3}\right) is:

cos(π3)=12\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}

Therefore, the exact value of cos(3π7)cos(2π21)+sin(3π7)sin(2π21)\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:

12\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(π6)\cos\left(\frac{\pi}{6}\right)?
  3. How can you apply the cosine formula to larger angles, like 5π3\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.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Angle subtraction formula
Cosine function

Formulas

Cosine addition formula
Cosine subtraction formula

Theorems

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Suitable Grade Level

High School