Problem 55 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The solutions are \(\theta = \frac{\pi}{2} + k\pi\), \(\theta = \frac{\pi}{9} + \frac{2m\pi}{3}\), and \(\theta = \frac{5\pi}{9} + \frac{2m\pi}{3}\).

Step 1: Apply the Sum-to-Product Formula

The Sum-to-Product Formula for cosine states that \[\cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right).\]Let's apply it with \(A = 4\theta\) and \(B = 2\theta\). We have:\[\cos 4\theta + \cos 2\theta = 2 \cos \left( \frac{4\theta + 2\theta}{2} \right) \cos \left( \frac{4\theta - 2\theta}{2} \right).\]Simplify the terms:\[= 2 \cos (3\theta) \cos (\theta).\]

Step 2: Set Up the Equation

Now, replace \(\cos 4\theta + \cos 2\theta\) in the original equation with the result from Step 1:\[2 \cos (3\theta) \cos (\theta) = \cos (\theta).\]

Step 3: Factor the Equation

Subtract \(\cos (\theta)\) from both sides to set up for factoring:\[2 \cos (3\theta) \cos (\theta) - \cos (\theta) = 0.\]Factor out \(\cos (\theta)\):\[\cos (\theta) (2 \cos (3\theta) - 1) = 0.\]

Step 4: Solve for θ

Set each factor equal to zero and solve:1. \(\cos (\theta) = 0\) leads to: - \(\theta = \frac{\pi}{2} + k\pi\) for any integer \(k\). 2. \(2 \cos (3\theta) - 1 = 0\) simplifies to: - \(\cos (3\theta) = \frac{1}{2}\) - \(3\theta = \frac{\pi}{3} + 2m\pi\) or \(3\theta = \frac{5\pi}{3} + 2m\pi\), where \(m\) is any integer. - Solving for \(\theta\) gives: - \(\theta = \frac{\pi}{9} + \frac{2m\pi}{3}\) or \(\theta = \frac{5\pi}{9} + \frac{2m\pi}{3}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the variable within their domains. They simplify complex trigonometric expressions, making it easier to solve problems.
One useful identity in trigonometry is the Sum-to-Product Formula, which aids in transforming the sum of trigonometric functions into a product. This transformation is particularly helpful in solving equations or simplifying expressions.
When working with the sum of cosine functions, the Sum-to-Product Formula is given by:

• \(\cos A + \cos B = 2 \cos \left(\frac{A + B}{2}\right) \cos \left(\frac{A - B}{2}\right)\).

This formula comes in handy because products are often easier to manipulate and factor than sums. By using trigonometric identities, we are provided with numerous strategies to approach and solve trigonometric equations effectively.

Cosine Function

The cosine function is one of the fundamental trigonometric functions, commonly used to relate the angles and sides of a right-angled triangle. It is defined in the context of a unit circle as the x-coordinate of the point where the terminal side of the angle intersects the circle.
In equations, the cosine function can take various forms like \(\cos(\theta)\), \(\cos(2\theta)\), or \(\cos(3\theta)\), representing different angles, often resulting in the function oscillating between -1 and 1.
The cosine function is periodic, with a period of \(2\pi\), meaning its values repeat every \(2\pi\) radians. This periodicity is crucial when solving equations since it leads to multiple solutions, typically represented as a general solution involving an integer constant.
Understanding the behavior and properties of the cosine function is essential when manipulating and solving trigonometric equations, such as those requiring the application of the Sum-to-Product Formula.

Equation Solving

Solving trigonometric equations involves finding values of the variable that satisfy the given equation. Many times, these equations are in complex forms and require the use of specific formulas or identities to simplify them first.
In our example, the initial equation \(\cos 4\theta + \cos 2\theta = \cos \theta\) was made easier to handle with the Sum-to-Product Formula, converting the left side into \(2 \cos (3\theta) \cos (\theta)\).
Once simplified, equations often need to be rearranged, equating to zero where possible to make factorization simpler. Then, each factor can be solved separately, and solutions should include all possible angles due to the periodic nature of trigonometric functions, expressed in terms of one or more integer parameters.
Equation solving in trigonometry not only involves finding the immediate solutions but also understanding and applying periodic properties to find all solutions within given constraints.

Factorization

Factorization in trigonometry simplifies expressions or equations, making them easier to solve. Just like in algebra, where expressions are written as products of simpler expressions, trigonometric factorization involves breaking down complex trigonometric expressions into simpler factors.
In the problem at hand, after simplifying the trigonometric equation using Sum-to-Product Formula, we attempted to factorize the equation \(2 \cos (3\theta) \cos (\theta) - \cos (\theta) = 0\).
We factorized using the distributive property, \(\cos (\theta)(2 \cos (3\theta) - 1) = 0\). Hence, each factor can be solved separately:

• The first factor yields solutions where \(\cos (\theta) = 0\).
• The second factor gives solutions for \(2 \cos (3\theta) - 1 = 0\).

Through factorization, we reduce the problem into more manageable sub-problems, making it simpler to find solutions. This key step is critical in solving equations that involve complex trigonometric expressions.