Question

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. cos 3x/2 + cos x/2

Accepted Answer

Recognize that the expression involves the sum of two cosine terms: $$\cos\left(\frac{3x}{2}\right) + \cos\left(\frac{x}{2}\right). $$This suggests using the cosine sum-to-product identity. Recall the cosine sum-to-product formula: $$\cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right). $$Identify $$A = \frac{3x}{2}$$ and $$B = \frac{x}{2}$$, then compute the average and difference inside the cosines: $$\frac{A+B}{2} = \frac{\frac{3x}{2} + \frac{x}{2}}{2}$$ and $$\frac{A-B}{2} = \frac{\frac{3x}{2} - \frac{x}{2}}{2}. $$Simplify these expressions to get the arguments for the product form: $$\cos\left(\frac{A+B}{2}\right)$$ and $$\cos\left(\frac{A-B}{2}\right). $$Write the original sum as the product $$2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)$$, which expresses the sum of cosines as a product.

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value. cos 3x/2 + cos x/2

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Key Concepts

Sum-to-Product Formulas

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Angle Manipulation and Simplification