Problem 17 - Precalculus Mathematics for Calculus

Question

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 of the expression is \(\frac{1}{2}\).

Step 1: Identify the Formula to Use

The expression \[\cos \frac{3\pi}{7} \cos \frac{2\pi}{21} + \sin \frac{3\pi}{7} \sin \frac{2\pi}{21}\]can be written as a single trigonometric function using the cosine addition formula, which states that\[\cos A \cos B + \sin A \sin B = \cos(A - B).\] In this case, we identify \(A = \frac{3\pi}{7} \) and \(B = \frac{2\pi}{21} \).

Step 2: Apply the Cosine Addition Formula

Since we have identified that the cosine addition formula is applicable, substitute \(A\) and \(B\) into it:\[\cos \left(\frac{3\pi}{7} - \frac{2\pi}{21}\right).\]

Step 3: Simplify the Argument

To simplify \[\frac{3\pi}{7} - \frac{2\pi}{21},\]we first find a common denominator. The common denominator for 7 and 21 is 21. Rewrite the fractions as\[\frac{3\pi}{7} = \frac{9\pi}{21}\]and\[\frac{2\pi}{21} = \frac{2\pi}{21}.\]Now we perform the subtraction:\[\frac{9\pi}{21} - \frac{2\pi}{21} = \frac{7\pi}{21} = \frac{\pi}{3}.\]

Step 4: Calculate the Cosine Value

Now we need to find the exact value of \(\cos \left(\frac{\pi}{3}\right)\). From known trigonometric values, \(\cos \frac{\pi}{3} = \frac{1}{2}\).

Key Concepts

Cosine Addition Formula

The cosine addition formula is a handy tool for simplifying certain trigonometric expressions. In this case, it deals with expressions like \[ \cos A \cos B + \sin A \sin B, \]which can be rewritten as a single cosine term: \[ \cos(A - B). \]This transformation is useful because it allows us to express a relatively complex expression in a much more straightforward form that involves just one angle. In our example, \( A = \frac{3\pi}{7} \) and \( B = \frac{2\pi}{21} \), so applying the formula transforms the sum of products into \( \cos \left(\frac{3\pi}{7} - \frac{2\pi}{21}\right).\)Using this technique can simplify the process of finding exact values, especially when facing expressions with unusual angles or non-standard fractions.

Exact Trigonometric Values

The key to solving trigonometric problems often lies in knowing the exact values of common trigonometric angles. These are specific angles whose trigonometric values are memorized for their simple and rational outputs. Some of the most frequently used angles and their values include:

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

In the context of the problem at hand, after simplifying the angle using the cosine addition formula, we reach \( \cos \left(\frac{\pi}{3}\right) \), which is a common angle with the exact value \( \frac{1}{2} \). This highlights how exact values facilitate solving trigonometric problems, turning seemingly complex expressions into simple numerical results.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often requires combining various mathematical skills, such as understanding formulas, manipulating fractions, and recognizing patterns. In this task, we start with an expression:\[ \cos \frac{3\pi}{7} \cos \frac{2\pi}{21} + \sin \frac{3\pi}{7} \sin \frac{2\pi}{21}. \]Applying the cosine addition formula is an effective first step, as it consolidates terms and reduces complexity. Next, calculating the exact value involves simplifying fractions by finding a common denominator. In our example, converting both terms into fractions over 21 allows us to simplify: \( \frac{3\pi}{7} \) becomes \( \frac{9\pi}{21} \), while \( \frac{2\pi}{21} \) remains the same.Subtracting these fractions gives us the simplified angle \( \frac{\pi}{3} \), where an exact trigonometric value is readily available.Simplification is crucial in trigonometry to make expressions more manageable and to identify easier paths for finding their values. Recognizing and using these strategies can also enhance problem-solving skills across different mathematical topics.