Problem 55 - Algebra and Trigonometry

Question

\(53-56\) a Solve the equation by first using a Sum-to-Product Formula. \(\cos 4 \theta+\cos 2 \theta=\cos \theta\)

Step-by-Step Solution

Answer

\( \theta = \frac{\pi}{9} + \frac{2n\pi}{3} \) and \( \theta = -\frac{\pi}{9} + \frac{2n\pi}{3} \).

Step 1: Apply Sum-to-Product Formula

The sum-to-product formula for cosine is given by \( \cos X + \cos Y = 2 \cos \left( \frac{X+Y}{2} \right) \cos \left( \frac{X-Y}{2} \right) \). Using this formula, let's rewrite \( \cos 4\theta + \cos 2\theta \). Here, \( X = 4\theta \) and \( Y = 2\theta \). Substitute these values:\[ \cos 4\theta + \cos 2\theta = 2 \cos \left( \frac{4\theta + 2\theta}{2} \right) \cos \left( \frac{4\theta - 2\theta}{2} \right) \]Simplify the terms:\[ = 2 \cos (3\theta) \cos (\theta) \]

Step 2: Substitute and Rearrange Equation

Replace \( \cos 4\theta + \cos 2\theta \) in the original equation \( \cos 4\theta + \cos 2\theta = \cos \theta \) using the expression from Step 1:\[ 2 \cos (3\theta) \cos (\theta) = \cos \theta \]Divide both sides by \( \cos \theta \), assuming \( \cos \theta eq 0 \):\[ 2 \cos (3\theta) = 1 \]

Step 3: Solve for \( \cos 3\theta \)

From the equation \( 2 \cos (3\theta) = 1 \), solve for \( \cos 3\theta \):\[ \cos (3\theta) = \frac{1}{2} \]

Step 4: Determine General Solutions for \( \theta \)

Recall that \( \cos (3\theta) = \frac{1}{2} \) at angles where the cosine value is \( \frac{1}{2} \). These angles are \( 3\theta = \frac{\pi}{3} + 2n\pi \) and \( 3\theta = -\frac{\pi}{3} + 2n\pi \) for any integer \( n \). Solve for \( \theta \):1. \[ 3\theta = \frac{\pi}{3} + 2n\pi \rightarrow \theta = \frac{\pi}{9} + \frac{2n\pi}{3} \]2. \[ 3\theta = -\frac{\pi}{3} + 2n\pi \rightarrow \theta = -\frac{\pi}{9} + \frac{2n\pi}{3} \]

Step 5: General Solution

The general solutions for \( \theta \) combining both cases are:\[ \theta = \frac{\pi}{9} + \frac{2n\pi}{3}, \quad \theta = -\frac{\pi}{9} + \frac{2n\pi}{3}, \quad n \in \mathbb{Z} \]

Key Concepts

Sum-to-Product Formulas

The sum-to-product formulas are powerful tools in trigonometry. They help simplify expressions where two trigonometric functions are added or subtracted. In our problem, we have the sum of two cosine functions: \( \cos 4\theta + \cos 2\theta \). Using the sum-to-product formula, we simplify this sum into a product of cosines. This transformation leverages the formula:

• \( \cos X + \cos Y = 2 \cos \left( \frac{X+Y}{2} \right) \cos \left( \frac{X-Y}{2} \right) \)

Here, \( X = 4\theta \) and \( Y = 2\theta \), and applying the formula yields:

• \( \cos 4\theta + \cos 2\theta = 2 \cos (3\theta) \cos (\theta) \)

This simplification allows for easier manipulation of the equation, which is a crucial step in solving trigonometric identities. By converting sums into products, we can more readily exploit algebraic techniques to solve the equation.

Solving Trigonometric Equations

Solving trigonometric equations requires transforming the equation into a simpler form that can easily be solved for the relevant angles. In our example, this involved using an identity to express the sum of cosines as a product. The equation initially stated was:
\( 2 \cos (3\theta) \cos (\theta) = \cos \theta \).
By dividing both sides by \( \cos \theta \) (excluding where \( \cos \theta = 0 \), to avoid division by zero), we isolated our terms:

• \( 2 \cos (3\theta) = 1 \)

This equation simplifies to finding where \( \cos (3\theta) = \frac{1}{2} \). By determining the solutions for this equation, we take into account the periodic nature of the cosine function on the unit circle. Learning to manipulate these equations by isolating the trigonometric function, using identities or algebraic methods, forms the core part of solving such problems effectively.

General Solutions of Trigonometric Equations

Once you've simplified a trigonometric equation, you'll often need to find its general solutions, especially for problems involving periodic functions like sine and cosine. General solutions account for the periodicity of the trigonometric functions.
In our case, we have \( \cos (3\theta) = \frac{1}{2} \), which has known solutions based on the unit circle:

• \( 3\theta = \frac{\pi}{3} + 2n\pi \)
• \( 3\theta = -\frac{\pi}{3} + 2n\pi \)

We solve for \( \theta \) by dividing each solution:

• \( \theta = \frac{\pi}{9} + \frac{2n\pi}{3} \)
• \( \theta = -\frac{\pi}{9} + \frac{2n\pi}{3} \)

Each solution includes every integer \( n \), representing the infinite number of angles where the original equation holds true. This understanding of general solutions is essential, as it emphasizes how trigonometric functions repeat over their fundamental periods, impacting everything from mathematical theory to engineering applications.