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. 3x x cos -------- sin ------- 2 2

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1. Identify the trigonometric product to be expressed as a sum or difference: \( \cos\left(\frac{3x}{2}\right) \sin\left(\frac{x}{2}\right) \).
2. Recall the product-to-sum identities: \( \cos A \sin B = \frac{1}{2} [\sin(A + B) - \sin(A - B)] \).
3. Assign \( A = \frac{3x}{2} \) and \( B = \frac{x}{2} \) to the identity.
4. Substitute \( A \) and \( B \) into the identity: \( \cos\left(\frac{3x}{2}\right) \sin\left(\frac{x}{2}\right) = \frac{1}{2} [\sin(\frac{3x}{2} + \frac{x}{2}) - \sin(\frac{3x}{2} - \frac{x}{2})] \).
5. Simplify the expressions inside the sine functions: \( \sin(\frac{4x}{2}) \) and \( \sin(\frac{2x}{2}) \).

Key Concepts

Product-to-Sum Formulas

Product-to-sum formulas are trigonometric identities that allow the transformation of products of sine and cosine functions into sums or differences. These formulas simplify calculations and are particularly useful in integration and solving trigonometric equations. For example, the product of sine and cosine can be expressed as a sum using the identity: sin(a)cos(b) = 1/2[sin(a+b) + sin(a-b)].

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 essential tools in simplifying trigonometric expressions and solving equations. Familiarity with these identities is crucial for effectively manipulating trigonometric functions.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions involves rewriting them in a more manageable form, often using identities to combine or reduce terms. This process is essential for solving trigonometric equations or evaluating expressions. Techniques include factoring, using identities, and converting between different forms (like sine and cosine). Mastery of simplification techniques is vital for success in trigonometry.