Problem 51 - Precalculus with Calculus Previews

Question

Find a rectangular equation that has the same graph as the given polar equation. $$ r=2 \sec \theta $$

Step-by-Step Solution

Answer

The rectangular equation is \( x = 2 \).

Step 1: Understand the Polar Equation

The given polar equation is \( r = 2 \sec \theta \). In this equation, \( r \) represents the radius, and \( \theta \) is the angle from the positive x-axis to the line connecting the origin to the point. The goal is to convert this into a rectangular (Cartesian) equation.

Step 2: Recall the Polar to Rectangular Conversion Formulas

We will use the conversion formulas between polar and rectangular coordinates: \( x = r \cos \theta \) and \( y = r \sin \theta \). Another useful identity is \( \sec \theta = \frac{1}{\cos \theta} \), and we know \( r = \frac{x}{\cos \theta} \).

Step 3: Substitute \( r = 2 \sec \theta \) into the Conversion Formulas

Since \( \sec \theta = \frac{1}{\cos \theta} \), we can rewrite the equation \( r = 2 \sec \theta \) as \( r = \frac{2}{\cos \theta} \). Therefore, \( r \cos \theta = 2 \).

Step 4: Substitute for \( r \cos \theta \) in Rectangular Equation

Since \( r \cos \theta = x \), substitute in the equation: \( x = 2 \).

Step 5: Simplify and Confirm the Rectangular Equation

The expression \( x = 2 \) is already in its simplest form and describes a vertical line in a Cartesian coordinate system where every point on the graph has an x-coordinate of 2.

Key Concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system, which is different from the familiar Cartesian (or rectangular) coordinates. Instead of defining a point by its horizontal and vertical distances from the origin, polar coordinates identify points based on:

• The distance from the origin (radius, denoted as \( r \)).
• The angle from the positive x-axis (denoted as \( \theta \)).

This system is particularly useful in scenarios involving circular or rotational symmetry. For example, points on a circle are easily described using the angle and radius rather than painstakingly calculating each x and y coordinate br>. Polar equations often look like \( r = f(\theta) \), where the function describes how the radius changes with different angles.

The polar equation from our exercise, \( r = 2 \sec \theta \), expresses how the distance of points from the origin varies depending on the angle br>. Understanding the concept is crucial when converting to rectangular coordinates.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, use an x-axis and a y-axis to define the position of a point in a plane. Every point in a Cartesian system is represented by two numbers:

• \( x \) (horizontal distance from the origin).
• \( y \) (vertical distance from the origin).

Together, these coordinates describe a precise location in the plane di>. The geometric interpretation of these points includes lines, circles, parabolas, etc., which is the basis of analytical geometry.

In the problem, we were tasked to convert the polar equation \( r = 2 \sec \theta \) into a rectangular one. Using the relationships:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)

we are able to find that \( r \cos \theta = x = 2 \). This rectangular equation tells us that for every angle \( \theta \), the x-coordinate remains constant br>. The result, therefore, is a vertical line at \( x=2 \). This illustrates how different coordinate systems can simplify different types of problems.

Secant Function

The secant function originates from trigonometry and is related to the cosine function. Defined as \( \sec \theta = \frac{1}{\cos \theta} \), it represents the reciprocal of the cosine angle di>. Recall that cosine is a key function in representing horizontal distances in a unit circle. Therefore, the secant function describes how waves or oscillations might behave when extended beyond a circle's bounds.

In our exercise, the polar equation \( r = 2 \sec \theta \) can be rewritten using the identity \( \sec \theta = \frac{1}{\cos \theta} \). This becomes \( r = \frac{2}{\cos \theta} \). When converted to rectangular coordinates, it simplifies to \( x = 2 \), reflecting that the secant function translates the dependency on an angle into a fixed horizontal position in Cartesian coordinates dd>. Understanding this relationship allows for a smoother transition between polar and rectangular equations, highlighting the versatility of trigonometric functions.