Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference. cos 7x cos 3x

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1. Identify the appropriate product-to-sum formula for the expression \( \cos A \cos B \). The formula is: \( \cos A \cos B = \frac{1}{2} [\cos(A + B) + \cos(A - B)] \).
2. Substitute \( A = 7x \) and \( B = 3x \) into the formula.
3. Calculate \( A + B = 7x + 3x = 10x \) and \( A - B = 7x - 3x = 4x \).
4. Substitute these values back into the formula: \( \cos 7x \cos 3x = \frac{1}{2} [\cos(10x) + \cos(4x)] \).
5. Express the product \( \cos 7x \cos 3x \) as the sum \( \frac{1}{2} [\cos(10x) + \cos(4x)] \).

Key Concepts

Product-to-Sum Formulas

Product-to-sum formulas are trigonometric identities that allow the conversion of products of sine and cosine functions into sums or differences. For example, the formula for cos(A)cos(B) is given by (1/2)(cos(A+B) + cos(A-B)). These formulas simplify the process of integrating or differentiating trigonometric expressions and are essential for solving problems involving products of trigonometric functions.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved, provided they are within the domain of the functions. These identities, such as the Pythagorean identities, angle sum and difference identities, and product-to-sum identities, are fundamental tools in trigonometry. They help in simplifying expressions, solving equations, and proving other mathematical statements.

Angle Measurement

Angle measurement is crucial in trigonometry, as it determines the values of trigonometric functions. Angles can be measured in degrees or radians, with radians being the standard unit in higher mathematics. Understanding how to convert between these units and how angles relate to the unit circle is essential for applying trigonometric identities and solving problems involving angles in various contexts.