In Exercises 45–46, express each sum or difference as a product. If possible, find this product's exact value.sin 2x - sin 4x

Verified step by step guidance

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

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 the difference of sines states that sin A - sin B = 2 cos((A + B)/2) sin((A - B)/2). These identities simplify the process of solving trigonometric equations and are essential for transforming expressions like sin 2x - sin 4x into a product form.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate angles to the ratios of sides in right triangles. Understanding these functions is crucial for manipulating and solving trigonometric expressions. In the context of the given problem, recognizing the specific angles (2x and 4x) allows for the application of identities to simplify the expression effectively.

Exact Values of Trigonometric Functions

Exact values of trigonometric functions refer to the precise values of sine, cosine, and tangent at specific angles, often expressed in terms of square roots or fractions. Knowing these values is important for evaluating the final product obtained from the sum-to-product identities. For instance, if the resulting product involves angles like 30°, 45°, or 60°, being familiar with their exact sine and cosine values aids in finding the solution.