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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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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sum-to-Product Formulas

Sum-to-product formulas convert sums or differences of trigonometric functions into products, simplifying expressions and solving equations. For example, cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2). These formulas are essential for rewriting sums or differences as products.

Evaluating Trigonometric Expressions

After rewriting expressions using sum-to-product identities, evaluating the exact value involves substituting known angle values and using unit circle values or special angle properties. This step ensures the expression is simplified to a precise numerical value.

Angle Manipulation and Simplification

Understanding how to manipulate angles, such as adding, subtracting, or halving angles, is crucial when applying sum-to-product formulas. Simplifying angles to standard positions or known reference angles helps in both rewriting and evaluating trigonometric expressions.