Problem 80 - A Graphical Approach to Precalculus with Limits

Question

Write each expression as a product of trigonometric functions or values. $$\cos 5 t+\cos 8 t$$

Step-by-Step Solution

Answer

\(2 \cos \left( \frac{13t}{2} \right) \cos \left( \frac{3t}{2} \right)\)

Step 1: Identify the problem type

The given expression is the sum of two cosine functions with different angles, specifically \( \cos(5t) + \cos(8t) \). To rewrite this as a product, we will use trigonometric identities.

Step 2: Recall the Sum-to-Product Identities

The sum-to-product identities are used to rewrite sums or differences of trigonometric functions as products. For cosine functions, the identity is:\[\cos A + \cos B = 2 \cos \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right)\]

Step 3: Apply the Identity

Identify \( A \) and \( B \) in the expression. Here, \( A = 5t \) and \( B = 8t \). Apply the sum-to-product identity:\[\cos(5t) + \cos(8t) = 2 \cos \left( \frac{5t + 8t}{2} \right) \cos \left( \frac{5t - 8t}{2} \right)\]

Step 4: Simplify the Expression

Simplify the terms inside the cosine functions:1. \( \frac{5t + 8t}{2} = \frac{13t}{2} \).2. \( \frac{5t - 8t}{2} = \frac{-3t}{2} \). Thus, the expression becomes:\[2 \cos \left( \frac{13t}{2} \right) \cos \left( \frac{-3t}{2} \right)\]

Step 5: Simplify Further with Negative Angle Property

Use the property \( \cos(-x) = \cos(x) \) to simplify further, which gives:\[2 \cos \left( \frac{13t}{2} \right) \cos \left( \frac{3t}{2} \right)\]

Key Concepts

Trigonometric Identities

Trigonometric identities are crucial in transforming trigonometric expressions into different, often more usable forms. They serve as the foundation for simplifying complex equations, allowing us to convert sums into products among other operations. These identities might seem tedious at first, but they reveal the interconnectedness of trigonometric functions.
One essential set of identities is the sum-to-product identities. These identities enable the conversion of a sum into a product, which can be advantageous for further analysis or integration. For cosine, the pertinent identity is:

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

The application of these identities can simplify problems significantly, making them more tractable for calculation or conceptual understanding. When using this identity, it's important to correctly identify the terms \( A \) and \( B \) and then substitute them into the identity to transform the expression.

Cosine Function

The cosine function, \( \cos(x) \), is a fundamental trigonometric function that describes the horizontal coordinate of a point on the unit circle as it sweeps out an angle \( x \) from the positive x-axis. The cosine function is periodic with a period of \( 2\pi \), meaning it completes a full cycle over an interval of \( 2\pi \).
One of the notable features of the cosine function is its even nature. This means that it is symmetric about the y-axis, which is mathematically represented as \( \cos(-x) = \cos(x) \). This property greatly simplifies mathematical problems, especially when negative angles come into play.
Cosine is also involved in the sum-to-product identities, which help in simplifying expressions like \( \cos 5t + \cos 8t \). Understanding how the cosine function behaves and how it relates to angles is key to dissecting and solving trigonometric problems efficiently.

Negative Angle Property

The negative angle property of trigonometric functions is a pivotal concept that facilitates simplification of intricate expressions. For cosine, this property tells us that \( \cos(-x) = \cos(x) \). This evenness of cosine is a direct consequence of its symmetric graph about the y-axis.
When dealing with negative angles, this property helps reduce expressions involving \( \cos \left( \frac{-3t}{2} \right) \) to \( \cos \left( \frac{3t}{2} \right) \), as shown in the given solution. Instead of computing trigonometric values for negative angles separately, this property allows us to transform them into their positive counterparts, streamlining the calculation process.
This simplification not only makes solving equations easier, but it also supports better understanding of the cyclic, symmetric nature of the cosine function, nurturing a deeper comprehension of trigonometric functions.