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 7x ﹣ sin 3x

Verified step by step guidance

1. Recognize that the expression \( \sin 7x - \sin 3x \) is a difference of sines.
2. Use the identity for the difference of sines: \( \sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \).
3. Identify \( A = 7x \) and \( B = 3x \).
4. Substitute \( A \) and \( B \) into the identity: \( 2 \cos\left(\frac{7x + 3x}{2}\right) \sin\left(\frac{7x - 3x}{2}\right) \).
5. Simplify the expression: \( 2 \cos(5x) \sin(2x) \).

Key Concepts

Sum and Difference Formulas

Sum and difference formulas in trigonometry allow us to express the sine and cosine of sums or differences of angles in terms of products of sines and cosines. For example, the formula for the sine of a difference states that sin(a - b) = sin(a)cos(b) - cos(a)sin(b). These formulas are essential for simplifying expressions involving trigonometric functions.

Product-to-Sum Formulas

Product-to-sum formulas convert products of sine and cosine functions into sums or differences. This is particularly useful when simplifying expressions like sin(A)sin(B) or cos(A)cos(B). Understanding these formulas helps in transforming complex trigonometric expressions into more manageable forms, facilitating easier calculations.

Exact Values of Trigonometric Functions

Finding the exact values of trigonometric functions involves using known angles and their corresponding sine, cosine, and tangent values. For instance, angles like 0°, 30°, 45°, 60°, and 90° have specific sine and cosine values that can be used to evaluate expressions. Mastery of these values is crucial for solving trigonometric equations and simplifying expressions accurately.