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Chapter 7: Problem 18

Use an addition or subtraction formula to write the expression as a trigonometric function of one number, and then find its exact value. $$\cos \frac{13 \pi}{15} \cos \left(-\frac{\pi}{5}\right)-\sin \frac{13 \pi}{15} \sin \left(-\frac{\pi}{5}\right)$$

Short Answer

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

Step by step solution

01

Identify the Trigonometric Identity

Recognize that the expression \(\cos \frac{13\pi}{15} \cos \left(-\frac{\pi}{5}\right) - \sin \frac{13\pi}{15} \sin \left(-\frac{\pi}{5}\right)\) matches the cosine angle addition formula: \(\cos(A)\cos(B) - \sin(A)\sin(B) = \cos(A + B)\). This will allow us to rewrite the expression in a simpler form.
02

Apply the Cosine Addition Formula

Using the identity \(\cos(A)\cos(B) - \sin(A)\sin(B) = \cos(A + B)\), set \(A = \frac{13\pi}{15}\) and \(B = -\frac{\pi}{5}\). Hence, we have \(\cos\left(\frac{13\pi}{15} + \left(-\frac{\pi}{5}\right)\right)\).
03

Simplify the Angle Addition

Perform the arithmetic to simplify the sum of angles: \(\frac{13\pi}{15} - \frac{\pi}{5}\). First, express \(\frac{\pi}{5}\) in terms of fifteenths: \(\frac{\pi}{5} = \frac{3\pi}{15}\). Then, \(\frac{13\pi}{15} - \frac{3\pi}{15} = \frac{10\pi}{15}\). Simplify this fraction to \(\frac{2\pi}{3}\).
04

Calculate the Cosine of the Simplified Angle

Find \(\cos\left(\frac{2\pi}{3}\right)\). Recall that \(\frac{2\pi}{3}\) is in the second quadrant, where cosine is negative. The reference angle is \(\pi/3\), for which \(\cos(\pi/3) = 1/2\). Therefore, \(\cos(\frac{2\pi}{3}) = -1/2\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Addition Formula
The cosine addition formula is a powerful tool in trigonometric calculations. It is expressed as \( \cos(A)\cos(B) - \sin(A)\sin(B) = \cos(A + B) \). This identity allows you to convert complex trigonometric expressions into simpler ones involving the cosine of a single angle.

When you encounter expressions like these, recognizing the structure is crucial. For example, recognizing that the terms match the pattern of the formula lets you simplify the expression into \( \cos(A + B) \). Through this, calculations often become more straightforward, allowing for easier solutions.
  • Simplifies expressions
  • Useful in angle addition problems
  • Transforms cosine and sine products into a single cosine function
Simplifying Angles
Simplifying angles is an essential skill when dealing with trigonometric identities. In our problem, we simplify \( \frac{13\pi}{15} - \frac{\pi}{5} \).

To simplify, you must manage the angle units properly. Start by finding a common denominator. In this case, convert \( \frac{\pi}{5} \) into fifteenths, which is \( \frac{3\pi}{15} \). Subtracting these angles gives us \( \frac{10\pi}{15} \), which simplifies to \( \frac{2\pi}{3} \).

This kind of simplification helps to make the next steps in solving the problem more direct and manageable.
  • Find a common denominator
  • Reduce fractions to their simplest form
  • Make complex calculations more manageable
Trigonometric Functions
Trigonometric functions are the foundation of understanding angles and sides of triangles. There are six primary functions: sine, cosine, tangent, cosecant, secant, and cotangent. Each one represents a unique aspect of a right-angled triangle based on a given angle.

In our problem, the focus is on cosine and sine. These functions help transform and simplify expressions using identities like the cosine addition formula. Their properties change based on the angle's location on the unit circle 鈥� with quadrant considerations affecting function sign and value.
  • Sine and Cosine are key trigonometric functions
  • Functions depend on angle and triangle side ratios
  • Useful in converting angle expressions
Exact Values of Trigonometric Functions
Finding the exact values of trigonometric functions involves understanding fundamental angles and their properties. For instance, having simplified our angle to \( \frac{2\pi}{3} \), we need to determine its cosine value.

Since \( \frac{2\pi}{3} \) is located in the second quadrant, we know that cosine values here are negative. Its reference angle \( \pi/3 \) gives us the familiar trigonometric ratio \( \cos(\pi/3) = 1/2 \). Thus, the exact value for \( \cos(\frac{2\pi}{3}) \) is \(-1/2\).
  • Memorize key angles and their trigonometric values
  • Know the sign of functions across different quadrants
  • Apply reference angles for determining precise values

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