Problem 56 - Precalculus Mathematics for Calculus

Question

Write the product as a sum. $$\sin x \sin 5 x$$

Step-by-Step Solution

Answer

\(\sin x \sin 5x = \frac{1}{2} [\cos(4x) - \cos(6x)]\).

Step 1: Introduction to Product-to-Sum Formulas

To write the product of two sine functions as a sum, we use the product-to-sum formulas. Specifically, for functions of the form \( ext{sin} A ext{sin} B\), the formula is:\[\sin A \sin B = \frac{1}{2} \left[\cos(A-B) - \cos(A+B)\right]\]Here, \(A\) and \(B\) are angles.

Step 2: Assign Angles

Identify the angles \(A\) and \(B\) in the expression \(\sin x \sin 5x\):\[ A = x \quad \text{and} \quad B = 5x \]

Step 3: Apply the Product-to-Sum Formula

Substitute \(A = x\) and \(B = 5x\) into the formula:\[ \sin x \sin 5x = \frac{1}{2} \left[ \cos(x - 5x) - \cos(x + 5x) \right] \]

Step 4: Simplify the Expression

Calculate the differences and sums inside the cosine functions:\[ x - 5x = -4x \]\[ x + 5x = 6x \]Substitute these results back into the expression:\[ \sin x \sin 5x = \frac{1}{2} \left[ \cos(-4x) - \cos(6x) \right] \]

Step 5: Use Cosine Symmetry Property

Recall that \(\cos(-\theta) = \cos(\theta)\). Thus, \(\cos(-4x) = \cos(4x)\):\[ \sin x \sin 5x = \frac{1}{2} \left[ \cos(4x) - \cos(6x) \right] \]

Key Concepts

Trigonometric Identities

Trigonometric identities are relationships between trigonometric functions that are true for every value of the occurring variables. They are fundamental tools in mathematics, particularly when working with equations and transformations involving angles. One common set of identities includes product-to-sum formulas. These formulas are extremely helpful when you need to convert the product of trigonometric functions, like sine and cosine, into a sum or difference.Understanding these identities allows us to simplify expressions, solve trigonometric equations, and even compute integrals. For example, for the product of two sine functions, the formula is:

• \( \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \)

By rewriting the expression \( \sin x \sin 5x \) using this formula, we can handle these products in a manner similar to linear terms. Using these identities often reveals more insights into the behavior of trigonometric functions, which can be beneficial in both theoretical and practical scenarios.

Sum of Angles

The concept of the sum of angles is integral when applying trigonometric identities. It represents the sum or difference of two given angles. In trigonometry, such concepts offer a way to express a single function or expression in terms of another, often simplifying calculations and solving equations.For example, in our expression \( \sin x \sin 5x \), where the sine functions are presented as a product, we used the sum of angles through the product-to-sum formula:

• The resulting expression involved \( \cos(x - 5x) \) and \( \cos(x + 5x) \).

These transformations make it possible to work with difference and sum forms directly, turning complex trigonometric equations into more manageable problems. Understanding how to manipulate these angles through their sum or difference is essential.

Sine and Cosine Functions

Sine and cosine functions are foundational elements in trigonometry. They describe the ratio of the lengths of sides in right triangles, but their role extends far beyond geometry, including wave motion, signal processing, and other scientific applications.The sine function is often expressed as the opposite side over the hypotenuse in a right triangle, while cosine is the adjacent side over the hypotenuse. In trigonometric identities, these functions frequently appear together, allowing for conversions depending on what is needed to solve a problem.In our scenario, using these functions:

• Converting a product like \( \sin x \sin 5x \) into a sum, we utilized cosine functions because they provide symmetry and are easier to manipulate.
• A property of the cosine function, \( \cos(-\theta) = \cos(\theta) \), simplifies expressions that involve negative angles.

Understanding the interplay between sine and cosine functions and their properties helps in effectively using trigonometric identities to solve problems.