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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1. Recognize that the expression \( \sin x + \sin 2x \) is a sum of sines, which can be expressed as a product using the sum-to-product identities.
2. Recall the sum-to-product identity for sine: \( \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \).
3. Identify \( A = x \) and \( B = 2x \) in the given expression.
4. Substitute \( A \) and \( B \) into the identity: \( \sin x + \sin 2x = 2 \sin \left( \frac{x + 2x}{2} \right) \cos \left( \frac{x - 2x}{2} \right) \).
5. Simplify the expression: \( 2 \sin \left( \frac{3x}{2} \right) \cos \left( -\frac{x}{2} \right) \).

Key Concepts

Sum-to-Product Identities

Sum-to-product identities are trigonometric formulas that express sums or differences of sine and cosine functions as products. For example, the identity for sine states that sin A + sin B can be rewritten as 2 sin((A+B)/2) cos((A-B)/2). These identities simplify the process of solving trigonometric equations and are essential for transforming expressions like sin x + sin 2x into a more manageable form.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate angles to ratios of sides in right triangles. The sine function, for instance, is defined as the ratio of the length of the opposite side to the hypotenuse. Understanding these functions is crucial for manipulating and solving trigonometric expressions, as they form the basis of many identities and equations used in trigonometry.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the specific values of sine, cosine, and tangent at key angles, such as 0°, 30°, 45°, 60°, and 90°. Knowing these values allows for quick calculations and simplifications in trigonometric problems. When finding the exact value of a product derived from a sum or difference, recognizing these key angles can significantly aid in the solution process.