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. sin x + sin 2x

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Recall the sum-to-product identity for sine functions: \(\sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right)\).

Identify the angles in the expression: here, \(A = x\) and \(B = 2x\).

Calculate the average of the angles: \(\frac{A+B}{2} = \frac{x + 2x}{2} = \frac{3x}{2}\).

Calculate half the difference of the angles: \(\frac{A-B}{2} = \frac{x - 2x}{2} = \frac{-x}{2}\).

Substitute these values into the sum-to-product formula to express \(\sin x + \sin 2x\) as a product: \(2 \sin \left( \frac{3x}{2} \right) \cos \left( \frac{-x}{2} \right)\).

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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 transform sums or differences of sine and cosine functions into products, simplifying expressions and solving equations. For example, sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2). This is essential for rewriting sin x + sin 2x as a product.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. Knowing identities like angle addition, double angle, and sum-to-product helps manipulate and simplify expressions such as sin x + sin 2x.

Exact Values of Trigonometric Functions

Exact values refer to precise trigonometric values for special angles (e.g., 0°, 30°, 45°, 60°, 90°) expressed in radicals or fractions. After expressing sin x + sin 2x as a product, evaluating the product's exact value requires familiarity with these standard angle values.