In Exercises 43–44, express each product as a sum or difference. sin 7x cos 3x

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1. Recall the product-to-sum identity for sine and cosine: \(\sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)]\).
2. Identify the angles in the problem: here, \(A = 7x\) and \(B = 3x\).
3. Substitute \(A\) and \(B\) into the identity: \(\sin 7x \cos 3x = \frac{1}{2} [\sin(7x + 3x) + \sin(7x - 3x)]\).
4. Simplify the expressions inside the sine functions: \(\sin(7x + 3x) = \sin 10x\) and \(\sin(7x - 3x) = \sin 4x\).
5. Write the final expression as a sum: \(\sin 7x \cos 3x = \frac{1}{2} (\sin 10x + \sin 4x)\).

Key Concepts

Product-to-Sum Formulas

Product-to-sum formulas convert products of sine and cosine functions into sums or differences of trigonometric functions. For example, sin A cos B can be expressed as ½[sin(A + B) + sin(A - B)], simplifying integration or further manipulation.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. Understanding these identities, such as angle addition and subtraction formulas, is essential for transforming and simplifying expressions.

Angle Notation and Manipulation

Recognizing and manipulating angles in expressions like sin 7x and cos 3x requires understanding how to handle multiples of variables within trigonometric functions. This skill is crucial for correctly applying formulas and simplifying results.