Problem 28 - Precalculus Mathematics for Calculus

Question

Sketch a graph of the polar equation. $$r=\cos \theta-1$$

Step-by-Step Solution

Answer

The graph is a circle centered at (-1, 0) with radius 1, tangent to the origin.

Step 1: Understand Polar Coordinates

In polar coordinates, a point on the plane is defined by the distance from the origin \( r \) and the angle \( \theta \) from the positive x-axis. The equation \( r = \cos \theta - 1 \) is in polar form.

Step 2: Analyze the Equation

The given polar equation is \( r = \cos \theta - 1 \). Notice that it is similar to a basic circle equation in polar form, but shifted. For \( \theta = 0 \), we have \( r = \cos(0) - 1 = 0 \), meaning the point is at the origin. For \( \theta = \pi \), \( r = \cos(\pi) - 1 = -2 \), which means the point is along the negative x-axis but 2 units from the origin.

Step 3: Determine Key Points

Calculate a few key values for \( \theta \):- For \( \theta = 0 \), \( r = 0 \).- For \( \theta = \frac{\pi}{2} \), \( r = \cos\left(\frac{\pi}{2}\right) - 1 = -1 \).- For \( \theta = \pi \), \( r = -2 \).- For \( \theta = \frac{3\pi}{2} \), \( r = \cos\left(\frac{3\pi}{2}\right) - 1 = -1 \).These points suggest symmetry and the graph involves negative values of \( r \), causing points to be plotted as positive \( 180^\circ \) from the angle.

Step 4: Plot the Graph

Using the values from Step 3, plot the points on polar axes:- For \( \theta = 0 \), the point is \( (0,0) \).- For \( \theta = \frac{\pi}{2} \), the point is plotted at \( (1, 270^\circ) \).- For \( \theta = \pi \), it is \( (2, 180^\circ) \).- For \( \theta = \frac{3\pi}{2} \), the point is \( (1, 90^\circ) \).Join these points in a smooth, symmetric curve that resembles a circle.

Step 5: Describe the Graph

The graph of the equation \( r = \cos \theta - 1 \) forms a circle. This circle is centered at \((-1, 0)\) in rectangular coordinates with a radius of 1. The circle is tangent to the origin and lies entirely in the third quadrant of the Cartesian plane.

Key Concepts

Polar Equations

In mathematics, polar equations are a delightful way to represent curves using polar coordinates. Instead of describing a location with an x and y coordinate, polar equations involve two components: the radius, denoted as \( r \), which tells you how far away a point is from the origin, and the angle, \( \theta \), which determines the direction from the positive x-axis.
This contrasts with the Cartesian system, where we use (x, y) to mark a point on a plane.

• In polar systems, \( \theta \) is usually measured in radians but can also be in degrees.
• The polar equation given, \( r = \cos \theta - 1 \), uses \( \cos \theta \) to define the radius's length, shifting it based on the angle.

Recognizing how the radius changes with angle in polar equations helps in visualizing unique curves that Cartesian coordinates may not easily show.
In the case of our exercise, understanding this polar form helps us envision the transformation of points around the origin.

Graphing in Polar Coordinates

Graphing in polar coordinates requires some adjustment if you're more familiar with Cartesian plots. Instead of drawing points using \( x \) and \( y \), you plot points based on their distance from the origin (\( r \)) at a specific angle (\( \theta \)).
Here's how to graph the given polar equation effectively:

• Start by creating a polar grid with concentric circles representing different radius values.
• Mark angles starting from the positive x-axis, forming a radial pattern.
• Convert key values from the polar equation: For \( \theta = 0 \), \( \theta = \frac{\pi}{2} \), \( \theta = \pi \), and \( \theta = \frac{3\pi}{2} \).
• For \( r \lt 0 \), plot the corresponding point by moving in the opposite direction (adding 180 degrees).

This specific graph for \( r = \cos \theta - 1 \) will give you a circular shape not centered at the origin. Ensuring symmetry and awareness of angle direction is crucial for accuracy.
You'll need to follow these principles closely to achieve an accurate and visually coherent graph of a polar equation.

Circle in Polar Coordinates

Circles have a special and straightforward representation in polar coordinates, but sometimes equations might seem a bit shifted or transformed like in our given exercise. While a simple circle might follow the equation \( r = a \), the equation \( r = \cos \theta - 1 \) introduces a transformation that shifts the circle based on its center.
Let's break it down:

• The standard form \( r = a \cos \theta \) or \( r = a \sin \theta \) represents circles centered at the origin, with a radius \( a \).
• In our equation, \( r = \cos \theta - 1 \), we notice a horizontal shift leftward by 1 unit in the graph.
• This results in a circle with a center at \((-1, 0)\) and radius 1.

Visualizing circles using polar coordinates allows us to appreciate subtle shifts and transformations just by modifying our equations slightly.
This comprehension transports easily into sketching precise circles on polar plots, revealing intricate relationships between equations and graph shapes.