Problem 52 - Algebra and Trigonometry

Question

47–52 Write the sum as a product. $$\sin 3 x+\sin 4 x$$

Step-by-Step Solution

Answer

\(2 \sin\left(\frac{7x}{2}\right) \cos\left(\frac{x}{2}\right)\)

Step 1: Understanding the Problem

The problem requires us to write the sum of two sine functions, \(\sin 3x + \sin 4x\), as a single product using a trigonometric identity.

Step 2: Using the Sum-to-Product Identity

We use the sum-to-product identity for sine: \(\sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right)\). In our case, \(A = 3x\) and \(B = 4x\).

Step 3: Calculate the Components of the Identity

First, compute the average of \(A\) and \(B\): \(\frac{3x + 4x}{2} = \frac{7x}{2}\). Then, compute the difference: \(\frac{4x - 3x}{2} = \frac{x}{2}\).

Step 4: Apply the Identity

Substitute the computed values into the identity: \(\sin 3x + \sin 4x = 2 \sin\left(\frac{7x}{2}\right) \cos\left(\frac{x}{2}\right)\).

Step 5: Write the Final Expression

The sum \(\sin 3x + \sin 4x\) is expressed as the product \(2 \sin\left(\frac{7x}{2}\right) \cos\left(\frac{x}{2}\right)\).

Key Concepts

Trigonometric Identities

Trigonometric identities are key tools in simplifying and transforming trigonometric expressions. They help us convert sums, differences, and products of trigonometric functions into more manageable forms for computation and analysis. These identities are derived from the inherent properties of trigonometric functions, largely due to their periodic nature and their roots in the geometry of the unit circle.

One such critical identity used in trigonometry is the **Sum-to-Product identity**. This identity helps us transform the sum of two sine functions into a product. By doing so, it aids in integrating, solving equations, or simplifying expressions that involve trigonometric sums.

For example, consider the sum \( \sin A + \sin B \), which can be rewritten as:

• \( 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \)

This identity takes advantage of the symmetrical relationship between the angles and the periodicity of the sine and cosine functions.

Sine Addition

When dealing with the sum of sine functions, like \( \sin 3x + \sin 4x \), it's important to understand how these sums can be transformed for easier manipulation.

The Sum-to-Product identity simplifies this addition into a product, making it straightforward to handle in calculus or algebraic contexts. In our example, we begin by identifying each sine term:

• First angle: \( A = 3x \)
• Second angle: \( B = 4x \)

By applying the identity, the angles are averaged and their difference calculated. This results in:

• Average: \( \frac{3x + 4x}{2} = \frac{7x}{2} \)
• Difference: \( \frac{4x - 3x}{2} = \frac{x}{2} \)

Thus, \( \sin 3x + \sin 4x \) transforms into the product:

• \( 2 \sin\left(\frac{7x}{2}\right) \cos\left(\frac{x}{2}\right) \)

This transformation makes solving and simplifying trigonometric equations more efficient.

Algebraic Manipulation

Algebraic manipulation is an essential skill in mathematics that involves rearranging expressions into simpler or more useful forms. When applying trigonometric identities, careful algebraic manipulation enables us to capture opportunities to simplify calculations or solutions.

In the context of applying the Sum-to-Product identity, the key manipulations involve averaging and subtracting the angles before substitution.

Let's take a closer look:

• **Average:** To find the average, add the angles \( A \) and \( B \), and then divide by 2. This step consolidates both terms into one part of the identity.
• **Difference:** Subtract the smaller angle from the larger one, and then divide by 2. This captures the "oscillating" behavior of the sines into the cosine function.
• **Substitution:** After calculating these components, substitute back into the Sum-to-Product identity formula to achieve the product form.

This technique exemplifies how algebraic manipulation is used to transition between different formats of mathematical expressions, highlighting the versatility and utility of mathematical identities.