Problem 26 - Precalculus: Functions and Graphs

Question

Express as a sum. $$(\cos a u)(\cos b u)$$

Step-by-Step Solution

Answer

\((\cos a u)(\cos b u) = \frac{1}{2}[\cos((a+b)u) + \cos((a-b)u)]\).

Step 1: Recognize the Trigonometric Identity

To express the product \((\cos a u)(\cos b u)\) as a sum, we will use the trigonometric identity for the product of cosines: \[\cos x \cos y = \frac{1}{2}(\cos(x+y) + \cos(x-y)).\]

Step 2: Apply the Identity

Apply the identity to \((\cos a u)(\cos b u)\) by letting \(x = a u\) and \(y = b u\). Use the identity from Step 1:\[\cos(a u) \cos(b u) = \frac{1}{2}[\cos((a+b)u) + \cos((a-b)u)].\]

Step 3: Simplify the Expression

Write down the resulting expression:\[\cos(a u) \cos(b u) = \frac{1}{2}(\cos((a+b)u) + \cos((a-b)u)).\] This expression is the sum form of the original product.

Key Concepts

Product-to-Sum Identities

Product-to-sum identities are very useful in trigonometry, simplifying complex expressions into more manageable forms. These identities allow us to convert a product of trigonometric functions into a sum or difference of trigonometric functions.

The product-to-sum identities come in handy when integrating or simplifying expressions. For the cosine function, the identity is:

• \(\cos x \cos y = \frac{1}{2}(\cos(x+y) + \cos(x-y))\)

This formula transforms the product of two cosine terms into the sum of two other cosine terms. It breaks down a potentially difficult multiplication into a simpler addition problem.

By applying this identity, we can efficiently handle trigonometric expressions in different contexts such as signal processing, engineering, or pure mathematics.

Cosine Function

Understanding the cosine function is crucial in trigonometry. It's one of the primary functions that describe the relationship between the angles and sides of a right triangle.

The cosine function, often represented as \(\cos \theta\), expresses the ratio of the length of the adjacent side to the hypotenuse.

Besides its geometrical representation, cosine has a periodic 'wave' pattern that repeats every \(2\pi\) radians (or 360 degrees). This is vital in analyzing periodic phenomena such as sound waves or tides.

In our scope of identities, manipulating the cosine function using product-to-sum identities greatly enhances our ability to simplify and interpret intricate expressions.

Simplifying Expressions

Simplifying expressions is one of the key tasks in solving mathematical problems, and it often involves applying known identities.

For trigonometric expressions, this includes using identities such as product-to-sum to transform and simplify terms into forms that are easier to work with. This simplification reduces complexity and aids in recalculating or integrating functions.

For instance, using our example

• \((\cos a u)(\cos b u)\)

by applying the product-to-sum identity, we converted it into

• \(\frac{1}{2}(\cos((a+b)u) + \cos((a-b)u))\)

This resulted in a simpler format, making further calculations and derivations more straightforward.

Learning to simplify expressions is a powerful skill that simplifies problem-solving across various scenarios in mathematics and applied sciences.