Write each expression as a product of trigonometric functions. See Example 8.
cos 5x + cos 8x

Verified step by step guidance

1. Recognize that the expression is a sum of two cosine functions: \(\cos 5x + \cos 8x\).
2. Recall the sum-to-product identity for cosine: \(\cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)\).
3. Identify \(A = 5x\) and \(B = 8x\) in the given expression.
4. Apply the identity by substituting \(A\) and \(B\): \(\cos 5x + \cos 8x = 2 \cos \left( \frac{5x + 8x}{2} \right) \cos \left( \frac{5x - 8x}{2} \right)\).
5. Simplify the arguments inside the cosine functions to express the sum as a product of cosines.

Key Concepts

Sum-to-Product Identities

Sum-to-product identities transform sums or differences of trigonometric functions into products. For example, the sum of cosines can be expressed as a product using the formula: cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2). This simplifies expressions and aids in solving equations.

Trigonometric Function Properties

Understanding the basic properties and periodicity of trigonometric functions like cosine is essential. Recognizing how angles combine and how cosine behaves under addition helps in applying identities correctly and simplifying expressions.

Angle Manipulation and Substitution

Manipulating angles by adding, subtracting, or factoring is crucial when applying identities. Substituting expressions like (A+B)/2 and (A−B)/2 allows rewriting sums as products, making complex expressions more manageable.