Problem 33 - Precalculus

Question

Sketch the graph of the polar equation using symmetry, zeros, maximum \(r\) -values, and any other additional points. $$r=4+3 \cos \theta$$

Step-by-Step Solution

Answer

The curve of \(r=4+3 \cos \theta\) is symmetric around the x-axis. It has no zeros, reaches a maximum r-value of 7 at \(\theta = 0\) or \(\theta = 2\pi\), and a minimum r-value of 4 at \(\theta = \pi/2\). The graph of the equation will look like a circle with its center moved to the right of the origin in the polar plane.

Step 1: Identify Symmetry

We identify symmetry first. The given equation is symmetric about the x-axis since, if we replace \(\theta\) with \(-\theta\), we have \(r=4+3 \cos(-\theta)=4+3 \cos \theta\). Thus, if \((r, \theta)\) is a point on the graph, so is \((r, -\theta)\).

Step 2: Find Zeros

Let's find out when \(r=0\). Solving for \(\theta\), we get \(4+3 \cos \theta = 0\) which results in \(\cos \theta = -4/3\). But \(-1 \leq \cos \theta \leq 1\), thus there are no zeros.

Step 3: Find Maximum r-value

Maximum \(r\) value occurs at \(\cos(\theta)=1\). For this case, \(\theta = 0\) or \(\theta = 2\pi\), which gives the maximum value of \(r = 4 + 3(1) = 7\).

Step 4: Plot the Graph

The graph starts at \((r, \theta) = (7, 0)\). Using symmetry, we know the graph will reflect across the x-axis. The graph goes to the point at angle \(\pi/2\), and the value of \(r\) at \(\theta = \pi/2\) is \(4+3 \cos(\pi/2)=4+3(0)=4\). This procedure continues for all values of \(\theta\) from 0 to \(2\pi\) to get a complete plot of the polar equation.

Key Concepts

Polar Coordinates

Polar coordinates offer an alternative way to locate points on a plane, using angles and distances relative to a fixed center point, usually referred to as the pole or origin. In this system, a point in the plane is defined by an ordered pair \( (r, \theta) \), where \( r \) is the radial distance from the origin to the point, and \( \theta \) is the counterclockwise angle from the positive x-axis to the line segment that joins the origin with the point.

In contrast to Cartesian coordinates which use a grid of vertical and horizontal lines, the beauty of polar coordinates lies in their simplicity when dealing with problems involving angles and curves that are concentric about a point. As such, they are ideally suited for graphing equations like \( r=4+3 \cos \theta \) where the radius depends on the angle.

By understanding how polar coordinates work, students can more easily approach the graphing of polar equations, considering not just the points but also the paths between them as the angle \( \theta \) varies.

Cosine Function

The cosine function is a fundamental trigonometric function, oscillating between -1 and 1. It relates to the x-coordinate of a point on the unit circle for a given angle. In the polar equation \( r=4+3 \cos \theta \), the cosine term dictates how the radial distance \( r \) changes as the angle \( \theta \) varies.

When \( \theta = 0 \), the cosine function has a maximum value of 1, making the radius maximum as well, as pointed out in the textbook solution. As \( \theta \) increases, \( \cos \theta \) decreases, thereby decreasing \( r \) until \( \theta = \pi \), where the cosine function is at its minimum, and hence, \( r \) is minimized.

Students must also appreciate that the cosine function is even, meaning that \( \cos(-\theta) = \cos(\theta) \). This attribute of cosine is fundamental in identifying symmetry in polar graphs and explains why there are no zeros in the given equation—cosine never reaches -4/3.

Symmetry in Polar Graphs

Symmetry is a key concept in graphing polar equations because it allows for easier plotting by reducing the workload. Recognizing symmetry can simplify the graphing process, as it did in the given exercise, where identification of x-axis symmetry prevented redundant calculations and plotting.

There are several types of symmetry to consider in polar coordinates: symmetry with respect to the polar axis (x-axis in Cartesian coordinates), symmetry with respect to the line \( \theta = \frac{\pi}{2} \) (y-axis in Cartesian), and symmetry with respect to the pole (origin). For the given exercise, the equation \( r=4+3 \cos \theta \) possesses x-axis symmetry. This occurs because the cosine function, as part of the polar equation, is even.

When students plot polar graphs with recognized symmetry, they can focus on calculating points for just half of the plane and then reflect those points across the line or point of symmetry. This strategy is incredibly efficient and results in an accurate plot with less effort, thereby enhancing students’ understanding and proficiency with polar equations.