Problem 17 - Algebra and Trigonometry

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 Trigonometric Formula

The expression given is \( \cos \frac{3 \pi}{7} \cos \frac{2 \pi}{21} + \sin \frac{3 \pi}{7} \sin \frac{2 \pi}{21} \). This resembles the trigonometric addition formula for cosine: \( \cos(a - b) = \cos a \cos b + \sin a \sin b \).

Step 2: Apply the Addition Formula

Using the cosine addition formula, we recognize that this expression equals \( \cos\left(\frac{3\pi}{7} - \frac{2\pi}{21}\right) \).

Step 3: Simplify the Angle

Calculate the difference \( \frac{3\pi}{7} - \frac{2\pi}{21} \). Convert the fractions to a common denominator: \( \frac{3\pi}{7} = \frac{9\pi}{21} \). Thus, \( \frac{9\pi}{21} - \frac{2\pi}{21} = \frac{7\pi}{21} = \frac{\pi}{3} \).

Step 4: Evaluate the Cosine Function

Now we find \( \cos\left(\frac{\pi}{3}\right) \). From unit circle values, we know \( \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \).

Key Concepts

Addition Formulas

Addition formulas are essential tools in trigonometry. They help combine and simplify trigonometric functions into one expression. In particular, the addition formulas give us the relationship between angles in trigonometric expressions. The cosine addition formula is given by:

• \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
• \( \cos(a - b) = \cos a \cos b + \sin a \sin b \)

These formulas are used to simplify expressions and solve trigonometric equations. In the problem, we identify the expression resembles the cosine subtraction formula \( \cos(a - b) \), enabling simplification in further steps.

Subtraction Formulas

Subtraction formulas are just as important as addition formulas in trigonometry. They allow us to break down complex trigonometric expressions involving subtraction into simpler components. In the given problem, the subtraction formula \( \cos(a - b) = \cos a \cos b + \sin a \sin b \) is key.

• This formula effectively combines two trigonometric functions into one, based on the angles provided.
• It's straightforward: keep the same angle format and change the signs in subtraction vs. addition.

By using the subtraction formula, we can easily convert the given trigonometric expression into a single cosine function of a new angle, making it more manageable.

Cosine Function

The cosine function is a fundamental trigonometric function. It describes the ratio of the adjacent side over the hypotenuse in a right triangle. In the unit circle, cosine is the x-coordinate of a point where the terminal side of an angle intersects the circle.

• Common cosine values are derived from key angles like 0, \( \frac{\pi}{6} \), \( \frac{\pi}{4} \), \( \frac{\pi}{3} \), \( \frac{\pi}{2} \, \), and their respective values in negatives and over 2\(\pi\).

In this problem, by using identities and rational fraction simplifications, we end up with the need to find \( \cos\left(\frac{\pi}{3}\right) \), a standard value, which helps to find the exact solution.

Exact Trigonometric Values

Exact trigonometric values are the specific values of trigonometric functions at notable angles, such as \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \) and \( \frac{\pi}{2} \). These are often derived from geometry and the unit circle.

• The exact value of \( \cos\left(\frac{\pi}{3}\right) \) is \( \frac{1}{2} \).
• These values are essential for solving trigonometric equations without a calculator.
• Often used to verify identities and solve problems involving angles that are commonly found in geometric configurations.

In this task, recognizing \( \cos\left(\frac{\pi}{3}\right) \) as \( \frac{1}{2} \) revealed the exact value of the original trigonometric expression, simplifying what seemed like a more complex formula into a much simpler answer.