Problem 18 - Algebra and Trigonometry

Question

Sketch a graph of the polar equation, and express the equation in rectangular coordinates. $$ r=-1 $$

Step-by-Step Solution

Answer

The graph is a circle of radius 1 centered at the origin. The rectangular equation is \( x^2 + y^2 = 1 \).

Step 1: Interpret the polar equation

The given polar equation is \( r = -1 \). This equation represents all the points in the polar coordinate plane that are exactly 1 unit away from the origin, but in the opposite direction of the angle \( \theta \).

Step 2: Understand the locus of points

In polar coordinates, \( r = -1 \) defines a circle centered at the origin with a radius of 1. However, since \( r \) is negative, the line sweeps one unit from the origin towards negative values of \( r \). This effectively reflects the point across the origin.

Step 3: Sketch the polar graph

The polar equation \( r = -1 \) describes a circle of radius 1 centered at the origin (the pole), but with each point reflected through the origin. The graph will be identical to a circle with a radius of 1 centered at the origin.

Step 4: Convert polar equation to rectangular form

In polar coordinates, \( r \) is related to the rectangular (Cartesian) coordinates \( x \) and \( y \) by the equation \( r^2 = x^2 + y^2 \). Given \( r = -1 \), we square both sides: \((-1)^2 = x^2 + y^2\), which simplifies to \(1 = x^2 + y^2\). This represents a circle centered at the origin with a radius of 1 in rectangular coordinates.

Key Concepts

Polar to Cartesian Conversion

When converting polar coordinates to Cartesian (rectangular) coordinates, we are essentially translating the way we look at points in a plane. Polar coordinates use a distance from the origin, \( r \), and an angle, \( \theta \), whereas Cartesian coordinates use the familiar \( (x, y) \) plane.
To perform this conversion, there are some fundamental relationships to understand:

• \( x = r \cos \theta \)
• \( y = r \sin \theta \)
• To find \( r \) from Cartesian, \( r = \sqrt{x^2 + y^2} \)
• To find \( \theta \), \( \theta = \tan^{-1} \left(\frac{y}{x}\right) \)

This conversion is particularly helpful when you need to express a polar equation in the Cartesian form, such as converting a polar circle equation enclosed by \( r = -1 \), which turns into \( 1 = x^2 + y^2 \) in rectangular coordinates. Recognizing and practicing with these conversions can significantly enhance your ability to navigate between different types of coordinate systems.

Rectangular Coordinates

Rectangular coordinates, often called Cartesian coordinates, depict points in a plane using two numbers: \( x \) and \( y \). These coordinates specify the horizontal and vertical displacements from the origin, respectively.
In every day geometry and algebra, we frequently use rectangular coordinates due to their straightforward nature.

• \( x \) represents distance along the horizontal axis.
• \( y \) represents distance along the vertical axis.
• Common rectangular equations represent shapes such as lines, ellipses, and in our specific case, circles.

For example, the rectangular equation \( x^2 + y^2 = 1 \) describes a circle centered at the origin with a radius of 1. Understanding the usefulness of rectangular coordinates helps when interpreting the more complex shapes that polar coordinates generate.

Graphing Polar Equations

Graphing in polar coordinates can seem challenging at first, but it provides a unique way to visualize data on a plane. Instead of focusing on horizontal and vertical axes as in rectangular coordinates, polar graphs focus on radial distance and angle.
To graph polar equations, you'll follow these steps:

• Identify the value of \( r \) for different angles \( \theta \).
• Plot each point at the corresponding angle, \( \theta \), and distance \( r \) from the origin (or pole).
• Connect the dots to observe the shape, keeping in mind that some shapes might loop, twist, or reflect through the origin.

In the example of \( r = -1 \), you'll find each point one unit opposite the direction of the angle \( \theta \). Therefore, the graph describes a circle of radius 1 centered at the origin but reflected through.
This unique functionality highlights the power of polar equations in representing symmetric and cyclic behaviors that aren't as easily managed with rectangular coordinates.