Problem 65 - Precalculus

Question

Determine whether the statement is true or false. Justify your answer. For values of \(e>1\) and \(0 \leq \theta \leq 2 \pi,\) the graphs of the following equations are the same. $$r=\frac{e x}{1-e \cos \theta} \quad \text { and } \quad r=\frac{e(-x)}{1+e \cos \theta}$$

Step-by-Step Solution

Answer

The statement is true. For values of \( e > 1 \) and for any \( \theta \) between 0 and \( 2\pi \), the graphs of the two equations are indeed the same.

Step 1: Understand the Variables and Equations

Firstly, make sure to understand the problem. The variable \( \theta \) ranges from 0 to \( 2\pi \) and \( e \) is greater than 1. Two equations are provided and the task is to analyze whether these equations give out the same results for these variable ranges.

Step 2: Test the Equations with Sample Values

The most straightforward way to check the claim is to substitute some values into each equation and see whether they give the same results. For instance, using \( \theta = \pi \) and \( e = 2 \), for the first equation we get: \( r = \frac{2x}{1 - 2 \cos \pi} = \frac{-x}{1}\). For the second equation we get: \( r = \frac{2(-x)}{1 + 2 \cos \pi} = \frac{-x}{1}\). As we can see, both equations gives the same output.

Step 3: Evaluate the General Case

Even though the equations gave the same result for a specific set of values, we can't say that it's true for all values. Thus, we need to apply a general approach. We know that \( \cos(-\theta) = \cos \theta \) and \( -\cos \theta = \cos(-\theta) \). So making \( \theta \) to \( -\theta \) in the second equation, we get \( r = \frac{e(-x)}{1 + e \cos(-\theta)} = \frac{ex}{1 - e \cos \theta} \) which is identical to the first equation.

Key Concepts

Polar Equations

Polar equations are mathematical expressions used to describe curves in polar coordinates. In polar coordinates, each point on a plane is determined by a distance from a reference point and an angle from a reference direction. Instead of the usual Cartesian coordinates \((x, y)\), polar coordinates use \((r, \theta)\), where:

• \(r\) is the radius or distance from the origin
• \(\theta\) is the angle measured from the positive x-axis

These equations can represent many beautiful curves, like spirals and circles.
In this exercise, two polar equations involve a parameter \(e\), which helps shape the graph. The task is to see if both equations describe the same graph under certain conditions.

Parameter Range

The parameter range defines the allowable values for parameters within an equation. In this context, \(\theta\) varies between \(0\) and \(2\pi\), encompassing a complete circle around the origin. This allows us to examine the behavior of the equations over a full rotation.
The parameter \(e\) must be greater than 1. This ensures that we are dealing with a specific type of conic section, often a hyperbola. Analyzing how the equation behaves under these constraints helps determine if two polar equations yield the same graph.

Trigonometric Identities

Trigonometric identities are useful mathematical tools that help simplify complex expressions involving trigonometric functions. In the analysis of the given polar equations, understanding these identities plays a crucial role. For example:

• The identity \(\cos(-\theta) = \cos \theta\) shows that cosine is an even function.
• Using \(-\cos \theta = \cos(-\theta)\) reveals symmetry properties that help in manipulating and comparing equations.

By substituting \(-\theta\) for \(\theta\) and using these identities, we transformed the second equation into a form matching the first equation.
This shows how exploring the relationships between trigonometric terms can unveil equivalences between seemingly different equations.