Problem 65 - Precalculus: Functions and Graphs

Question

Sketch the graph of the polar equation. $$r=3 \cos \theta$$

Step-by-Step Solution

Answer

The graph of \( r = 3\cos\theta \) is a circle centered at (1.5, 0) with radius 1.5.

Step 1: Understand the polar equation form

The given equation is in the form of \( r = a \cos \theta \), where \( a = 3 \). This is a specific type of polar equation known as a limaçon, and when \( a \) is positive, the graph is symmetric about the polar axis (the horizontal axis in a polar coordinate system).

Step 2: Identify key points

To sketch the graph, determine the values of \( r \) for standard angles such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \).- When \( \theta = 0 \), \( r = 3\cos(0) = 3 \).- When \( \theta = \frac{\pi}{2} \), \( r = 3\cos(\frac{\pi}{2}) = 0 \).- When \( \theta = \pi \), \( r = 3\cos(\pi) = -3 \). Since \( r \) is negative, this point is plotted at \( 3 \) units in the opposite direction.- When \( \theta = \frac{3\pi}{2} \), \( r = 3\cos(\frac{3\pi}{2}) = 0 \).- When \( \theta = 2\pi \), \( r = 3\cos(2\pi) = 3 \).

Step 3: Sketch the graph

Using the key points calculated, the graph can be plotted. Begin by drawing the entire pole (origin) and the radial line for \( \theta = 0 \). Plot the point at \( 3 \) along this line. Then plot the points at \( 0 \), \(-3\), and \( 0 \) at \( \theta = \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) respectively.The graph will be a circle with a radius of 1.5 centered at \( (1.5, 0) \) in the polar coordinate plane, corresponding to a circle with center \( (1.5, 0) \) and radius \( 1.5 \) in Cartesian coordinates.

Key Concepts

Limaçon

A limaçon is a unique type of curve that can be defined using polar coordinates. The name "limaçon" comes from the French word for "snail," aptly describing its snail-like shape.

• It is defined by the polar equation form: \( r = a + b \cos(\theta) \) or \( r = a + b \sin(\theta) \).
• The appearance of a limaçon can vary widely depending on the relation between the constants \(a\) and \(b\).
• When \( b = a \), the limaçon takes the form of a cardioid.

In the specific case of \( r = 3 \cos(\theta) \), the polar equation resembles a limaçon with only a cosine term. This indicates the symmetry about the polar axis, as well as how the curve will appear relative to the circle centered at the pole. Understanding how the parameters influence the shape is crucial as it determines whether the limaçon will have an inner loop, be dimpled, or appear convex.

Polar Equation Graphing

Polar equation graphing involves plotting points in a coordinate system that uses the radius and angle, rather than the traditional x and y Cartesian coordinates. Here's how to graph such equations effectively:

• Start with the given polar equation, such as \( r = 3 \cos(\theta) \).
• Choose standard angles—these are usually \( 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi \).
• Use the equation to calculate the radius \( r \) for each of these angles.

After these calculations, plot the points. For example, when \( \theta = 0 \) and \( r = 3 \cos(0) \), the point is plotted three units away from the origin along the positive x-axis. For \( \theta = \frac{\pi}{2} \), \( r \) equals 0, meaning the point remains at the origin. This approach helps connect the dots visually, creating curves like circles or limaçons easily represented in polar coordinates.

Cosine Function

The cosine function is a fundamental trigonometric function that was crucial in plotting the given polar equation, \( r = 3 \cos(\theta) \). This function oscillates between -1 and 1 as \( \theta \) varies from 0 to \( 2\pi \).

• It is important to understand how the cosine function modifies the radius \( r \) across different angles.
• The maximum value of \( \cos(\theta) \) is 1, occurring at \( \theta = 0, 2\pi \), contributing the positive maximum radius of 3 in this example.
• At \( \theta = \pi \),\( \cos(\theta) \) is -1, reflecting r=-3, but in polar terms, this represents a point at 3 units opposite to the angle's direction.

This oscillation forms the characteristic shape of the graph in polar coordinates. Understanding how cosine influences the radial distance from the pole to these points is essential to drawing precise curves and interpreting their behavior properly.