Problem 20 - 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. $$ \cos \frac{13 \pi}{15} \cos \left(-\frac{\pi}{5}\right)-\sin \frac{13 \pi}{15} \sin \left(-\frac{\pi}{5}\right) $$

Step-by-Step Solution

Answer

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

Step 1: Identify the Trigonometric Formula

The expression \( \cos \frac{13 \pi}{15} \cos \left(-\frac{\pi}{5}\right)-\sin \frac{13 \pi}{15} \sin \left(-\frac{\pi}{5}\right) \) follows the form of the cosine addition formula: \( \cos(a)\cos(b) - \sin(a)\sin(b) = \cos(a + b) \).

Step 2: Apply the Cosine Addition Formula

By recognizing the expression as a variation of \( \cos(a + b) \), we set \( a = \frac{13 \pi}{15} \) and \( b = -\frac{\pi}{5} \). Substituting into the formula, we find that the expression simplifies to \( \cos \left( \frac{13 \pi}{15} + \left(-\frac{\pi}{5}\right) \right) \).

Step 3: Simplify the Angle Expression

Calculate \( \frac{13 \pi}{15} + \left(-\frac{\pi}{5}\right) \) by finding a common denominator. Convert \( -\frac{\pi}{5} \) to \( -\frac{3 \pi}{15} \). Thus, the expression becomes \( \frac{13\pi}{15} - \frac{3\pi}{15} = \frac{10\pi}{15} \). Further simplify \( \frac{10\pi}{15} \) to \( \frac{2\pi}{3} \).

Step 4: Calculate the Exact Trigonometric Value

The expression is now \( \cos \left( \frac{2\pi}{3} \right) \). Knowing the unit circle, \( \cos \left( \frac{2\pi}{3} \right) = -\frac{1}{2} \). This is the exact value of the trigonometric function.

Key Concepts

Cosine Addition Formula

The cosine addition formula is a powerful tool in trigonometry. This formula allows us to simplify expressions that involve the cosine and sine functions into a single cosine function. The general form of the cosine addition formula is:

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

This formula is particularly useful when dealing with angles that are sums or differences, such as the expression given. By recognizing parts of the formula in the problem, we can transform complex expressions into simpler ones. This conversion is often the first step in solving trigonometric problems, as it sets the stage for further simplification and finding exact values.

Simplifying Angle Expressions

In our example, we initially deal with the angles \( \frac{13 \pi}{15} \) and \( -\frac{\pi}{5} \). To simplify expressions involving angles, it's essential to bring them to a common denominator. This ensures that the arithmetic operations are straightforward and precise.

• Convert \( -\frac{\pi}{5} \) to \( -\frac{3\pi}{15} \)
• Perform the addition: \( \frac{13\pi}{15} - \frac{3\pi}{15} \)
• Obtain \( \frac{10\pi}{15} \), which simplifies to \( \frac{2\pi}{3} \)

By getting the expression to \( \frac{2\pi}{3} \), we are closer to finding exact trigonometric values. Simplifying angles aids in making complex trigonometric problems more manageable, allowing further application of trigonometric identities and formulas.

Exact Trigonometric Values

Exact trigonometric values are essential when evaluating expressions, especially when working with special angles on the unit circle. After simplifying the initial expression using the cosine addition formula and simplifying the angle, we need to find the exact value for \( \cos \left( \frac{2\pi}{3} \right) \).

• Refer to the unit circle: \( \frac{2\pi}{3} \) lies in the second quadrant.
• In the second quadrant, cosine values are negative.
• The reference angle for \( \frac{2\pi}{3} \) is \( \pi/3 \) with cosine value of \( \frac{1}{2} \).
• Thus, \( \cos \left( \frac{2\pi}{3} \right) = -\frac{1}{2} \).

Being familiar with the unit circle can greatly simplify the process of evaluating these functions. Knowing which quadrants the angles lie in, and their corresponding trigonometric values, allows for quick and accurate calculations.