Problem 80 - Precalculus

Question

For the following exercises, use a graph to determine whether are the same or different. If they are the same, show why. If they are different, replace the second function with one that is identical to the first. (Hint: think \(2 x=x+x . )\) $$ f(\theta)=\cos (2 \theta), g(\theta)=\cos ^{2} \theta-\sin ^{2} \theta $$

Step-by-Step Solution

Answer

The functions are the same; they both simplify to \( \cos(2\theta) \).

Step 1: Understand the given functions

We are given two functions: \( f(\theta) = \cos(2\theta) \) and \( g(\theta) = \cos^2 \theta - \sin^2 \theta \). Our task is to determine if these two functions are identical by simplifying and comparing them.

Step 2: Recall Trigonometric Identities

Recall the double-angle formula for cosine: \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \). This identity directly reveals that the expression for \( g(\theta) \) is a form of \( \cos(2\theta) \).

Step 3: Compare the Functions

Compare the expression for \( g(\theta) = \cos^2 \theta - \sin^2 \theta \) with the known identity \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \). Since they are identical, \( g(\theta) \) is indeed the same as \( f(\theta) \).

Step 4: Conclude the Solution

Since \( f(\theta) = \cos(2\theta) \) and \( g(\theta) = \cos^2 \theta - \sin^2 \theta \) both simplify to the same expression using the double-angle identity, they are the same. Thus, no replacement of \( g(\theta) \) is necessary.

Key Concepts

Double-Angle Formulas

Double-angle formulas in trigonometry are particularly handy for solving problems involving angles that are multiples of a simpler angle. The double-angle formula for cosine, for instance, is used to express \( \cos(2\theta) \) in terms of trigonometric functions of \( \theta \). This specific identity is given by:

• \( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) \)

This formula allows us to transform a complex angle into more manageable parts using squares of sine and cosine. In the problem given, this formula is key. By recognizing that both functions, \( f(\theta) \) and \( g(\theta) \), simplify to the same double-angle expression, one can conclude their equivalence. This approach simplifies the problem and shows the utility of such trigonometric identities.

Function Comparison

When comparing functions, the goal is often to determine if they are equivalent over a given domain. This can involve simplifying expressions using known identities, as seen in trigonometric functions. For instance, in the given task, both functions \( f(\theta) = \cos(2\theta) \) and \( g(\theta) = \cos^2(\theta) - \sin^2(\theta) \) simplify to the same expression using the double-angle formula for cosine.

• This outcome indicates that these functions are indeed the same, demonstrating the importance of recognizing and applying mathematical identities in function comparison.
• When two functions simplify to the same expression, they are said to be equivalent, meaning they produce the same output for any input value within their domains.

Function comparison plays a crucial role in confirming mathematical relationships and ensuring the accuracy of calculations or transformations.

Graphical Analysis

Graphical analysis is a visual method to analyze or validate mathematical functions and equations. It involves plotting functions on a graph to see their behavior and verify any equivalencies. For trigonometric functions such as \( f(\theta) = \cos(2\theta) \) and \( g(\theta) = \cos^2(\theta) - \sin^2(\theta) \), graphical analysis helps illustrate that these are indeed the same function since their graphs will completely overlap.

• This visual overlap confirms that at every point \( \theta \), the values produced by both functions are identical, thereby offering a concrete verification beyond symbolic manipulation.
• By plotting both functions on a graph, students can see the immediate equivalence, even without solving algebraic equations, which reinforces their understanding of trigonometric identities and functions.

Graphical analysis not only aids in understanding mathematical relationships but also provides an intuitive way to grasp concepts that are mathematically proven. It's a wonderful way to appreciate the beauty and consistency of mathematics.