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.
$\cos \frac{3x}{2}\sin \frac{x}{2}$

Verified step by step guidance

1. Identify the product-to-sum formula that applies to the given expression. The expression is a product of cosine and sine functions with the same angle denominator, so we use the formula: \(\cos A \sin B = \frac{1}{2} [\sin(A + B) - \sin(A - B)]\).
2. Assign the angles in the expression to variables: let \(A = \frac{3x}{2}\) and \(B = \frac{x}{2}\).
3. Substitute \(A\) and \(B\) into the product-to-sum formula: \(\cos \left( \frac{3x}{2} \right) \sin \left( \frac{x}{2} \right) = \frac{1}{2} \left[ \sin \left( \frac{3x}{2} + \frac{x}{2} \right) - \sin \left( \frac{3x}{2} - \frac{x}{2} \right) \right]\).
4. Simplify the arguments inside the sine functions by combining the fractions: \(\frac{3x}{2} + \frac{x}{2} = \frac{4x}{2} = 2x\) and \(\frac{3x}{2} - \frac{x}{2} = \frac{2x}{2} = x\).
5. Rewrite the expression as \(\frac{1}{2} [\sin(2x) - \sin(x)]\), which expresses the original product as a difference of sines.

Key Concepts

Product-to-Sum Formulas

Product-to-sum formulas convert products of sine and cosine functions into sums or differences of trigonometric functions. These identities simplify expressions and solve integrals by rewriting products like cos A sin B as sums, e.g., cos A sin B = 1/2 [sin(A + B) - sin(A - B)].

Trigonometric Function Arguments and Angles

Understanding how to handle angles in trigonometric functions is essential. When applying formulas, correctly identify and manipulate the angles (like x/2) to ensure accurate substitution into identities. This includes recognizing how to add or subtract these angles within the formulas.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves applying identities and algebraic manipulation to rewrite expressions in simpler or more useful forms. This skill is crucial when converting products to sums, as it helps in reducing complex expressions to manageable terms for further analysis or computation.