Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.sin x cos 2x

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1. Identify the product-to-sum formula that relates to the expression \( \sin A \cos B \). The formula is: \( \sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] \).
2. Assign \( A = x \) and \( B = 2x \) in the formula.
3. Substitute \( A \) and \( B \) into the formula: \( \sin x \cos 2x = \frac{1}{2} [\sin(x + 2x) + \sin(x - 2x)] \).
4. Simplify the expressions inside the sine functions: \( \sin(x + 2x) = \sin(3x) \) and \( \sin(x - 2x) = \sin(-x) \).
5. Recall that \( \sin(-x) = -\sin(x) \), so the expression becomes: \( \frac{1}{2} [\sin(3x) - \sin(x)] \).

Key Concepts

Product-to-Sum Formulas

Product-to-sum formulas are trigonometric identities that allow the transformation of products of sine and cosine functions into sums or differences. For example, the formula for sin(x)cos(y) can be expressed as (1/2)[sin(x+y) + sin(x-y)]. These formulas simplify calculations and are essential for integrating or differentiating trigonometric expressions.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the variables involved, particularly angles. They include fundamental identities like the Pythagorean identity, reciprocal identities, and co-function identities. Understanding these identities is crucial for manipulating and simplifying trigonometric expressions in various problems.

Angle Addition and Subtraction

Angle addition and subtraction formulas express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of the individual angles. For instance, sin(a ± b) = sin(a)cos(b) ± cos(a)sin(b). These formulas are vital for solving problems involving the combination of angles, particularly in the context of the product-to-sum transformation.