Problem 49 - Algebra and Trigonometry

Question

47–52 Write the sum as a product. $$\cos 4 x-\cos 6 x$$

Step-by-Step Solution

Answer

The product form is \( 2 \sin(5x) \sin(x) \).

Step 1: Recall the Trigonometric Identity

To convert the given expression of the form \( \cos a - \cos b \) into a product, recall the trigonometric identity: \[\cos a - \cos b = -2 \sin \left( \frac{a+b}{2} \right) \sin \left( \frac{a-b}{2} \right)\]We will use this identity to express \( \cos 4x - \cos 6x \) as a product.

Step 2: Apply the Identity

Substitute \( a = 6x \) and \( b = 4x \) into the identity: \[\cos 4x - \cos 6x = -2 \sin \left( \frac{4x + 6x}{2} \right) \sin \left( \frac{4x - 6x}{2} \right)\]Simplify the terms inside the sine functions.

Step 3: Simplify the Expression

Calculate the expressions inside the sine function from Step 2:1. \( \frac{6x + 4x}{2} = \frac{10x}{2} = 5x \)2. \( \frac{4x - 6x}{2} = \frac{-2x}{2} = -x \)Substitute these back into the equation:\[\cos 4x - \cos 6x = -2 \sin(5x) \sin(-x)\]

Step 4: Use Properties of Sine

Use the property \( \sin(-x) = -\sin(x) \) to further simplify the expression:\[\cos 4x - \cos 6x = -2 \sin(5x) \cdot (-\sin(x))\]Simplify the negatives to obtain:\[\cos 4x - \cos 6x = 2 \sin(5x) \sin(x)\]

Key Concepts

Product-to-Sum Formulas

Understanding the concept of Product-to-Sum formulas is crucial when dealing with trigonometric identities like the one in the example. These formulas help to convert the product of sines and cosines into sums or differences. In particular, the situation with the expression \( \cos a - \cos b \) uses one of these formulas. The formula is:

• \( \cos a - \cos b = -2 \sin\left( \frac{a+b}{2} \right) \sin\left( \frac{a-b}{2} \right) \)

This conversion helps simplify complex trigonometric expressions, making them easier to work with, especially when dealing with integrals or solving equations.
In practical terms, the Product-to-Sum formulas are particularly useful in simplifying the expressions encountered in wave interference and signal processing, where understanding the detailed behavior of wave components is necessary. Remember that while these formulas appear complex, they are essentially breaking down changes in angles into symmetrical components involving sine functions.

Sine Properties

The sine function is fundamental in trigonometry, known for its wavelike properties and its pivotal role in oscillations. One key property is its odd symmetry:

• \( \sin(-x) = -\sin(x) \)
  This property simplifies expressions that include negative angles. In our solution, we used this to convert \( \sin(-x) \) to \(-\sin(x) \), streamlining the equation significantly.
  The sine function is periodic with a period of \( 2\pi \), meaning that its values repeat every \( 2\pi \) interval. This is useful for predicting behavior over cycles or waves.
• Sine is also an odd function, meaning it is symmetric about the origin in a graph: \( \sin(-x) \) reflects the wave about the y-axis to \( -\sin(x) \).

Understanding these properties is crucial for solving identities and equations that involve oscillating functions or circular motion.

Cosine Subtraction

The formula used in the exercise, where \( \cos 4x - \cos 6x \) is converted using the identity \( \cos a - \cos b \), reflects the concept of cosine subtraction.The identity is powerful because it transforms a simple difference of cosines into a multiplication of sine terms:

• \( \cos a - \cos b = -2 \sin\left( \frac{a+b}{2} \right) \sin\left( \frac{a-b}{2} \right) \)

This transformation allowed us to turn the expression into \( 2 \sin(5x) \sin(x) \), illustrating how the difference of angles becomes a product of sine terms.
Using cosine subtraction makes it possible to reveal underlying periodic behaviors in trigonometric expressions. It simplifies the finding of maxima, minima, and intersections, especially in graphs and modelling contexts. This is pivotal in fields like engineering and physics, where predicting oscillations or vibrations precisely is essential.