Problem 17 - A Graphical Approach to Precalculus with Limits

Question

Use identities to find the exact value of each expression. Do not use a calculator. $$\cos \frac{\pi}{3} \cos \frac{2 \pi}{3}-\sin \frac{\pi}{3} \sin \frac{2 \pi}{3}$$

Step-by-Step Solution

Answer

The exact value is \(-1\).

Step 1: Rewrite Using Cosine of Sum Formula

The given expression \( \cos \frac{\pi}{3} \cos \frac{2\pi}{3} - \sin \frac{\pi}{3} \sin \frac{2\pi}{3} \) can be rewritten using the identity for the cosine of a sum: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \). Here, let \( a = \frac{\pi}{3} \) and \( b = \frac{2\pi}{3} \).

Step 2: Simplify the Angles

Calculate \( a + b = \frac{\pi}{3} + \frac{2\pi}{3} = \pi \).

Step 3: Use the Identity

According to the cosine of a sum formula used in Step 1, we have \( \cos(\pi) = \cos\left(\frac{\pi}{3} + \frac{2\pi}{3}\right) = \cos \frac{\pi}{3} \cos \frac{2\pi}{3} - \sin \frac{\pi}{3} \sin \frac{2\pi}{3} \).

Step 4: Find Exact Value of Cosine

Recall that the cosine of \( \pi \) is \( \cos(\pi) = -1 \). Therefore, the expression simplifies to \(-1\).

Key Concepts

cosine of a sum formula

Understanding the cosine of a sum formula is crucial in trigonometry. This identity is used to express the cosine of the sum of two angles in terms of their individual trigonometric functions. The formula is the following:

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

This formula allows us to combine two angles into one single trigonometric expression.
As seen in the exercise, the expression \( \cos \frac{\pi}{3} \cos \frac{2\pi}{3} - \sin \frac{\pi}{3} \sin \frac{2\pi}{3} \) can be easily observed as a candidate for the cosine of a sum.
By letting \( a = \frac{\pi}{3} \) and \( b = \frac{2\pi}{3} \), we find that this expression simplifies to \( \cos(\pi) \), which can then be evaluated using known exact trigonometric values.

exact trigonometric values

Exact trigonometric values are specific trigonometric function values for certain standard angles. These values make complex trigonometric calculations manageable without needing a calculator.
For example, let's look into the angles \( \frac{\pi}{3} \), \( \frac{2\pi}{3} \), and \( \pi \):

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

These exact values simplify calculations significantly because you're working with well-defined numbers instead of approximations, leading to precise outcomes.
In our exercise, using these values allows us to confirm that \( \cos(\pi) = -1 \), the final result.

trigonometric identities

Trigonometric identities are equations that are true for all values of the variables they contain. They play a key role in simplifying trigonometric expressions or evaluating complex equations.
Different identities cater to different trigonometric functions and operations. In this exercise, we mainly used the identity for the cosine of a sum. There are a variety of other identities too:

• Pythagorean Identities, such as \( \sin^2 x + \cos^2 x = 1 \)
• Angle Sum and Difference Identities, such as the \( \sin(a \pm b) \) and \( \cos(a \pm b) \) formulas
• Double Angle Identities, e.g., \( \sin(2a) = 2\sin a \cos a \)
• Sum-to-Product and Product-to-Sum Identities

These identities help dissect a seemingly complex problem into parts that can be resolved using exact values and known formulas.
Understanding how to leverage these identities can simplify not just academic exercises but also real-world applications that deal with angles, waves, and cycles.