Problem 15 - Algebra and Trigonometry

Question

\(13-18\) m Use an addition or subtraction formula to write the expression as a trigonometric function of one number, and then find its exact value. \(\cos \frac{3 \pi}{7} \cos \frac{2 \pi}{21}+\sin \frac{3 \pi}{7} \sin \frac{2 \pi}{21}\)

Step-by-Step Solution

Answer

The exact value is \( \frac{1}{2} \).

Step 1: Identify the Formula

The expression \( \cos \frac{3 \pi}{7} \cos \frac{2 \pi}{21} + \sin \frac{3 \pi}{7} \sin \frac{2 \pi}{21} \) can be matched with the cosine addition formula: \( \cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B) \). Identify \( A = \frac{3 \pi}{7} \) and \( B = \frac{2 \pi}{21} \).

Step 2: Apply the Formula

Using the cosine addition formula: \( \cos(A - B) \), replace \( A \) and \( B \) with \( \frac{3 \pi}{7} \) and \( \frac{2 \pi}{21} \), respectively. The expression becomes \( \cos \left( \frac{3 \pi}{7} - \frac{2 \pi}{21} \right) \).

Step 3: Simplify the Angle

Find the common denominator of the fractions: \( \frac{3 \pi}{7} - \frac{2 \pi}{21} \). The least common multiple of 7 and 21 is 21. Convert both fractions: \( \frac{9 \pi}{21} - \frac{2 \pi}{21} = \frac{7 \pi}{21} \).

Step 4: Calculate the Simplified Expression

Simplify the fraction: \( \frac{7 \pi}{21} = \frac{\pi}{3} \). Therefore, the expression becomes \( \cos \frac{\pi}{3} \).

Step 5: Find the Exact Value

The value of \( \cos \frac{\pi}{3} \) is \( \frac{1}{2} \). Thus, the expression evaluates to \( \frac{1}{2} \).

Key Concepts

Cosine Addition Formula

The cosine addition formula plays a critical role in simplifying trigonometric expressions. It allows us to express a sum or difference of angles in terms of cosine and sine of the individual angles. The formula is given as \( \cos(A - B) = \cos(A) \cos(B) + \sin(A) \sin(B) \). This formula helps transform complex trigonometric expressions into simpler forms by identifying components that match the product-to-sum identities.

In practical applications, recognizing when expressions can be rewritten using the cosine addition formula can dramatically reduce the complexity involved in calculations. In our exercise, we had the expression \( \cos \frac{3 \pi}{7} \cos \frac{2 \pi}{21} + \sin \frac{3 \pi}{7} \sin \frac{2 \pi}{21} \), which perfectly aligns with \( \cos(A - B) \). Identifying the right formula is crucial and serves as the first step in the simplification process.

Exact Trigonometric Values

Exact trigonometric values are specific values of trigonometric functions that can be defined precisely, typically without a calculator. These values usually correspond to well-known angles measured in radians or degrees.

For instance, some commonly referenced angles with exact trigonometric values include:

• \( \cos \frac{\pi}{3} = \frac{1}{2} \)
• \( \sin \frac{\pi}{6} = \frac{1}{2} \)
• \( \tan \frac{\pi}{4} = 1 \)

These values are fundamental in solving many trigonometric expressions without numeric approximation. In our scenario, after applying the cosine addition formula and simplifying the angle, we reach the expression \( \cos \frac{\pi}{3} \), which is a well-known angle. The exact value of this is \( \frac{1}{2} \), allowing us to conclude the problem efficiently.

Angle Simplification

Simplifying angles is an essential technique in trigonometry that often involves finding a common denominator to combine or convert angles in different fractions. Understanding how to handle angle simplification helps in reducing the complexity of an expression, effectively making it easier to evaluate.

In the given problem, we started with the angles \( \frac{3 \pi}{7} \) and \( \frac{2 \pi}{21} \). To subtract these fractions, we found their least common multiple (LCM), which was 21. This step transforms the subtraction into:

• Convert \( \frac{3 \pi}{7} \) to \( \frac{9 \pi}{21} \)
• Subtract \( \frac{2 \pi}{21} \) from \( \frac{9 \pi}{21} \) to get \( \frac{7 \pi}{21} \)
• Simplify \( \frac{7 \pi}{21} \) to \( \frac{\pi}{3} \)

This simplification to \( \frac{\pi}{3} \) is necessary to progress towards solving the expression effectively, using known trigonometric values.