In Exercises 43–44, express each product as a sum or difference.sin 6x sin 4x

Verified step by step guidance

1. Recognize that the expression \( \sin A \sin B \) can be rewritten using the product-to-sum identities.
2. Recall the product-to-sum identity: \( \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \).
3. Identify \( A = 6x \) and \( B = 4x \) in the given expression \( \sin 6x \sin 4x \).
4. Substitute \( A = 6x \) and \( B = 4x \) into the identity: \( \sin 6x \sin 4x = \frac{1}{2} [\cos(6x - 4x) - \cos(6x + 4x)] \).
5. Simplify the expression: \( \sin 6x \sin 4x = \frac{1}{2} [\cos(2x) - \cos(10x)] \).

Key Concepts

Product-to-Sum Formulas

Product-to-sum formulas are trigonometric identities that allow the conversion of products of sine and cosine functions into sums or differences. For example, the product of two sine functions can be expressed as a sum of cosine functions. This transformation simplifies the process of integration and solving trigonometric equations.

Sine Function Properties

The sine function is a periodic function defined for all real numbers, with a range of [-1, 1]. It is essential to understand its properties, such as its periodicity (period of 2π) and symmetry (odd function), to manipulate and simplify expressions involving sine. Recognizing these properties aids in applying the correct identities when transforming products into sums.

Angle Addition and Subtraction Identities

Angle addition and subtraction identities express the sine and cosine of the sum or difference of two angles in terms of the sines and cosines of those angles. These identities are crucial when working with products of sine functions, as they provide a systematic way to rewrite expressions, facilitating easier calculations and interpretations in trigonometric problems.