Problem 78 - A Graphical Approach to Precalculus with Limits

Question

Write each expression as a sum or difference of trigonometric functions or values. $$\sin 4 x \sin 5 x$$

Step-by-Step Solution

Answer

\( \sin 4x \sin 5x = \frac{1}{2} \cos(x) - \frac{1}{2} \cos(9x) \)

Step 1: Recall the Trigonometric Identity

To solve this problem, we need to use the product-to-sum formulas for trigonometric functions. Specifically, we use the identity: \[ \sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)] \] This identity allows us to convert the product of two sine functions into a sum or difference of cosine functions.

Step 2: Substitute Values into the Identity

In the given expression \( \sin 4x \sin 5x \), identify \( A = 4x \) and \( B = 5x \). Substitute these values into the product-to-sum formula: \[ \sin 4x \sin 5x = \frac{1}{2} [\cos(4x - 5x) - \cos(4x + 5x)] \]

Step 3: Simplify the Expression

Now, simplify the expression: \[ \cos(4x - 5x) = \cos(-x) \] and since \( \cos \) is an even function, we have \( \cos(-x) = \cos(x) \). So the expression becomes: \[ \frac{1}{2} [\cos(x) - \cos(9x)] \]

Step 4: Write as a Sum or Difference

The expression \( \sin 4x \sin 5x \) is now written as a difference of two cosine functions using the identity: \[ \sin 4x \sin 5x = \frac{1}{2} \cos(x) - \frac{1}{2} \cos(9x) \] Thus, it is expressed as a sum/difference of trigonometric functions.

Key Concepts

Product-to-Sum Formulas

In trigonometry, product-to-sum formulas are incredibly handy. They help simplify complex expressions involving the products of sine and cosine functions. When you have two sine functions, such as \( \sin A \sin B \), the product-to-sum formula lets us rewrite this as a sum of cosine functions:\[\sin A \sin B = \frac{1}{2} [\cos(A-B) - \cos(A+B)]\]Using these formulas:

• Makes integrals and limits easier to solve.
• Simplifies expressions in solving equations.
• Turns products into sums or differences, making calculations more straightforward.

In our exercise, we used this formula to transform \( \sin 4x \sin 5x \) into a sum of cosine terms. By doing so, we can easily manage and solve the expression.

Trigonometric Functions

Trigonometric functions are essential in math due to their ability to represent periodic phenomena. The main trigonometric functions—sine, cosine, and tangent—are based on right-angled triangles. Here's a brief overview:

• Sine (\( \sin \)): The length of the opposite side divided by the hypotenuse.
• Cosine (\( \cos \)): The length of the adjacent side divided by the hypotenuse.
• Tangent (\( \tan \)): The length of the opposite side divided by the adjacent side.

These functions help us understand angles and distances in circles. For instance, the cosine function is even, meaning that \( \cos(-x) = \cos(x) \), a property we used in the exercise to simplify \( \cos(-x) \) to \( \cos(x) \). Being familiar with these functions' properties is crucial when applying identities like the product-to-sum formulas.

Sum or Difference of Functions

Transforming a product of trigonometric functions into a sum or difference is a common task in math. This approach:

• The simplification makes expressions easier to integrate or differentiate.
• Helps in finding the exact values of trigonometric expression products.
• Makes angle measurements clearer when analyzing wave patterns or signals.

In our example, we used the formula: \[\sin 4x \sin 5x = \frac{1}{2} \cos(x) - \frac{1}{2} \cos(9x)\]Here, the difference of cosine functions replaces the product, simplifying the expression. This conversion helps in various mathematical problems, including physics and engineering, where wave interference patterns need analysis. Understanding how to switch between product and sum forms enhances problem-solving skills across disciplines.