Problem 28 - Precalculus

Question

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=4 \cos \theta$$

Step-by-Step Solution

Answer

The polar equation \(r = 4 \cos \theta\) has symmetry about the polar axis and has zeros at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\). The maximum \(r\)-values are \(r=4\) at \(\theta = 0\) and \(r=-4\) at \(\theta = \pi\). Plotting these points along with additional ones gives a graph that resembles a circle with radius 2 units, centered at (2,0) on the polar plane.

Step 1: Identify Symmetry

Check if the polar equation \(r=4 \cos \theta\) is symmetric. If replacing \(\theta\) with \(-\theta\), yields the same equation, it means the graph is symmetric about the polar axis. The equation \(r=4 \cos (-\theta) = 4 \cos \theta\) shows that it is symmetric about the polar axis.

Step 2: Identify Zeros

The zeros of this polar equation are the values of \(r\) when \(\theta\) makes \(r = 0\). These occur when \(\cos \theta = 0\), or \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\).

Step 3: Identify Maximum \(r\)-values

The maximum values of \(r\) occur where \(\cos \theta = 1\) or \(\cos \theta = -1\). These occur at \(\theta = 0\) and \(\theta = \pi\). Hence, the maximum \(r\)-values are \(r=4\) and \(r=-4\) respectively.

Step 4: Plot Additional Points

Other points can be plotted by calculating the \(r\)-values for additional \(\theta\)-values. For example, at \(\theta = \frac{\pi}{4}\), \(r=4 \cos \frac{\pi}{4} = 2\sqrt{2}\), and at \(\theta = \frac{3\pi}{4}\), \(r=4 \cos \frac{3\pi}{4} = -2\sqrt{2}\). These additional points help to provide more accuracy to the final graph.

Step 5: Draw the Graph

Use the identified points of symmetry, zeros, maximum \(r\)-values, and additional points to draw the sketch. This results in a complete graph of \(r = 4 \cos \theta\) which looks like a circle with radius 2 units, centered at (2,0) on the polar plane.

Key Concepts

Polar Coordinate System

Imagine a plane where every point is defined not by the familiar x and y coordinates, but by how far away and at what angle it is from a central point, called the pole (similar to the origin in the Cartesian system). This is the essence of the polar coordinate system, where each point is described by a pair \( (r, \theta) \), with \( r \) being the radial distance from the pole, and \( \theta \) the angle made with the positive x-axis – referred to as the polar axis in this context.

Unlike the linear movement from the origin in Cartesian coordinates, \( r \) can be any non-negative number, and \( \theta \) can range from 0 to \( 2\pi \) radians (or 0 to 360 degrees), making a full circle around the pole. This unique framework is especially useful for dealing with situations that naturally involve circular motion or patterns, such as the orbits of planets or the ripples from a pebble thrown into a pond.

Symmetry in Polar Graphs

Symmetry in polar graphs simplifies the graphing process and provides visual harmony. A polar equation can depict three types of symmetry: symmetry about the polar axis (the positive x-axis in Cartesian coordinates), symmetry about the line \( θ = \frac{\pi}{2} \) (the positive y-axis), or symmetry about the pole.

For instance, the symmetrical nature of the polar equation \( r = 4 \cos \theta \) about the polar axis means that for every point \( (r, \theta) \) on the graph, there is a corresponding point \( (r, -\theta) \). This reflective property allows us to calculate only half of the points and still correctly visualize the graph's entirety. Understanding symmetry will enable students to efficiently construct polar graphs with confidence and precision.

Finding Zeros in Polar Equations

Finding zeros in polar equations involves determining the angles \( \theta \) at which the radial distance \( r \) is zero. Equivalently, these are the points at which the graph intersects the pole. For the equation \( r = 4 \cos \theta \) provided in our example, you would set the equation to zero and solve for \( \theta \: \)

• \( 0 = 4 \cos \theta \)
• \( \cos \theta = 0 \)
• \( \theta = \frac{\pi}{2}, \frac{3\pi}{2} \)

At these angles, the graph will pass through the pole. Identifying zeros is a crucial step in sketching an accurate representation of the polar graph, as it highlights the points where the curve either begins, pivots, or ends.

Determining Maximum r-values

The concept of maximum \( r \-values \) in a polar equation relates to the farthest distance from the pole reached by the curve, within the range of \( \theta \) considered. For our equation \( r = 4 \cos \theta \), we seek the angles where \( \cos \theta \) is at its maximum and minimum, which occur at \(\text{{0 and }} \pi \) respectively, manifesting as \( r \) values of 4 and -4. In polar notation, negative \( r \) values mean that the point is on the opposite side of the pole relative to where it would be for the positive \( r \) value. Therefore, maximum \( r \-values \) are essential for capturing the extent of the curve's reach in all directions, a factor that informs the amplitude and therefore the overall size and orientation of the graph on the polar plane.