Problem 55 - Precalculus Mathematics for Calculus

Question

Write the product as a sum. $$\sin 2 x \cos 3 x$$

Step-by-Step Solution

Answer

\( \sin 2x \cos 3x = \frac{1}{2} (\sin 5x - \sin x) \).

Step 1: Identify the Trigonometric Identity to Use

To express the product \( \sin 2x \cos 3x \) as a sum, we need to use the product-to-sum identities. The relevant identity for this form is \( \sin A \cos B = \frac{1}{2} [ \sin (A+B) + \sin (A-B) ] \).

Step 2: Substitute the Values into the Identity

Using the identity \( \sin A \cos B = \frac{1}{2} [ \sin (A+B) + \sin (A-B) ] \), substitute \( A = 2x \) and \( B = 3x \). This gives us: \( \sin 2x \cos 3x = \frac{1}{2} [ \sin (2x+3x) + \sin (2x-3x) ] \).

Step 3: Simplify the Expression

Simplify the terms inside the sine functions: \( 2x+3x = 5x \) and \( 2x-3x = -x \). Substitute back into the equation to get: \( \sin 2x \cos 3x = \frac{1}{2} [ \sin 5x + \sin (-x) ] \).

Step 4: Apply the Sine Odd Function Property

Recall that \( \sin(-x) = -\sin(x) \), an important trigonometric property. Applied to our expression, we substitute: \( \sin 2x \cos 3x = \frac{1}{2} [ \sin 5x - \sin x ] \).

Step 5: Rewrite as a Final Sum Expression

Now, the product \( \sin 2x \cos 3x \) is rewritten as a sum: \( \frac{1}{2} (\sin 5x - \sin x) \).

Key Concepts

Product-to-Sum Identities

The product-to-sum identities are incredibly useful in trigonometry for rewriting products of sine and cosine functions as sums or differences. This conversion helps simplify complex problems or integrate trigonometric functions. In our exercise, we started with the expression \( \sin 2x \cos 3x \), which is a product form. The product-to-sum identities provide a way to transform such products into sums.

• The general formula for converting \( \sin A \cos B \) into a sum is: \[ \sin A \cos B = \frac{1}{2} [ \sin (A+B) + \sin (A-B) ] \]
• This formula breaks down the original product into two sine terms.

By substituting the given angles, the identity simplifies the task of integration or solving equations where the sum expression might be easier to manage.

Sine and Cosine Functions

Sine and cosine are the foundational functions in trigonometry. These functions are periodic, meaning they repeat their values in regular intervals, known as cycles. Understanding their properties is essential when working with trigonometric identities, like turning a product into a sum.

• **Sine Function**: Represents the ratio of the opposite side to the hypotenuse in a right-angled triangle. The sine of an angle \( x \) can be positive or negative depending on its position across the unit circle.
  One key property is that \( \sin(-x) = -\sin(x) \), showing it is an odd function.
• **Cosine Function**: Represents the ratio of the adjacent side to the hypotenuse. Unlike sine, it is an even function, meaning \( \cos(-x) = \cos(x) \).

These functions, when combined, generate waves that can describe various physical phenomena, from sound to light energy. They play a crucial role in creating meaningful expressions via identities such as the product-to-sum identities.

Trigonometric Simplification

Simplifying trigonometric expressions is key to solving many mathematical problems more easily. This involves reducing expressions to simpler, equivalent forms using identities and functional properties.
In our solution, after applying the product-to-sum identity, we obtained:

• \( \sin 2x \cos 3x = \frac{1}{2} [ \sin 5x + \sin (-x) ] \)

The next simplification step leverages the odd property of the sine function: \( \sin(-x) = -\sin(x) \). This small adjustment results in a much cleaner expression:

• \( \frac{1}{2} [ \sin 5x - \sin x ] \)

The purpose is to find a simpler form that is easier to interpret and apply in further calculations, proving essential in calculus and physics.

Sum-to-Product Formulas

While the focus has been on converting products to sums, reversing the process using sum-to-product formulas can be equally beneficial in other contexts. These formulas help translate sums or differences back into a product, useful in solving certain integrals or equations.
Sum-to-product formulas are just as vital and are often expressed in the form:

• For sine: \( \sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \)
• For cosine: \( \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \)

These transformations allow flexibility in mathematical manipulations, offering paths to simplification depending on what the problem requires. Whether it's converting from product-to-sum or sum-to-product, these identities expand the toolbox for tackling a variety of trigonometric problems.