Question

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

Accepted Answer

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: $$\cos A \sin B = \frac{1}{2} [\sin(A + B) - \sin(A - B)]. $$Assign the angles in the expression to variables: let $$A = \frac{3x}{2}$$ and $$B = \frac{x}{2}. $$Substitute $$A$$ and $$B$$ into the product-to-sum formula: $$\cos \left( \frac{3x}{2} \right) \sin \left( \frac{x}{2} \right) = \frac{1}{2} \left[ \sin \left( \frac{3x}{2} + \frac{x}{2} \right) - \sin \left( \frac{3x}{2} - \frac{x}{2} \right) \right]. $$Simplify the arguments inside the sine functions by combining the fractions: $$\frac{3x}{2} + \frac{x}{2} = \frac{4x}{2} = 2x$$ and $$\frac{3x}{2} - \frac{x}{2} = \frac{2x}{2} = x. $$Rewrite the expression as $$\frac{1}{2} [\sin(2x) - \sin(x)]$$, which expresses the original product as a difference of sines.

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

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Key Concepts

Product-to-Sum Formulas

Trigonometric Function Arguments and Angles

Simplification of Trigonometric Expressions