Problem 38 - Calculus Early Transcendentals

Question

Tabulate and plot enough points to sketch a graph of the following equations. \(r=4+4 \cos \theta\)

Step-by-Step Solution

Answer

Based on the given polar equation \(r=4+4\cos\theta\), we calculated the corresponding \(r\) and \(\theta\) values, converted them to Cartesian coordinates, and plotted the points on a graph. The points to be plotted were \((8,0)\), \((2\sqrt{2},6)\), \((0,4)\), \((-2\sqrt{2},6)\), \((0,0)\), \((-2\sqrt{2},-6)\), \((0,-4)\), and \((2\sqrt{2},-6)\). The graph of the polar equation is a smooth curve that passes through these points.

Step 1: Understanding the polar equation

We are given the polar equation \(r=4+4\cos\theta\). This equation describes how the radius \(r\) is related to the angle \(\theta\). Our task is to find the coordinates \((r,\theta)\) and then convert them into Cartesian coordinates \((x,y)\) to plot the graph.

Step 2: Choose values for \(\theta\)

We will choose different values for \(\theta\) and calculate the corresponding \(r\) values using the polar equation \(r=4+4\cos\theta\). To ensure we cover all parts of the graph, we choose \(\theta\) values from \(0\) to \(2\pi\) radians (or \(0\) to \(360^\circ\)), because the graph will repeat after \(2\pi\) or \(360^\circ\). Let's choose the following values for \(\theta\): \(0^\circ\), \(45^\circ\), \(90^\circ\), \(135^\circ\), \(180^\circ\), \(225^\circ\), \(270^\circ\), and \(315^\circ\). You could choose other values or more values, but these should be enough to give a good understanding of how the graph will look like.

Step 3: Calculate corresponding \(r\) values

Now we will calculate the corresponding \(r\) values using the polar equation \(r=4+4\cos\theta\) for each chosen angle. Be careful, to convert the degree values to radians: \(\text{radian}=\text{degree}*\frac{\pi}{180}\). \(\theta=0^\circ\): \(r=4+4\cos(0)=4+4(1)=8\) \(\theta=45^\circ\): \(r=4+4\cos\left(\frac{\pi}{4}\right)=4+4\left(\frac{\sqrt{2}}{2}\right)=4+2\sqrt{2}\) \(\theta=90^\circ\): \(r=4+4\cos\left(\frac{\pi}{2}\right)=4+4(0)=4\) \(\theta=135^\circ\): \(r=4+4\cos\left(\frac{3\pi}{4}\right)=4+4\left(-\frac{\sqrt{2}}{2}\right)=4-2\sqrt{2}\) \(\theta=180^\circ\): \(r=4+4\cos(\pi)=4+4(-1)=0\) \(\theta=225^\circ\): \(r=4+4\cos\left(\frac{5\pi}{4}\right)=4+4\left(-\frac{\sqrt{2}}{2}\right)=4-2\sqrt{2}\) \(\theta=270^\circ\): \(r=4+4\cos\left(\frac{3\pi}{2}\right)=4+4(0)=4\) \(\theta=315^\circ\): \(r=4+4\cos\left(\frac{7\pi}{4}\right)=4+4\left(\frac{\sqrt{2}}{2}\right)=4+2\sqrt{2}\) Now we have the polar coordinates \((r,\theta)\) for each chosen angle.

Step 4: Convert polar coordinates to Cartesian coordinates

We will use the following formulas to convert polar coordinates \((r,\theta)\) to Cartesian coordinates \((x,y)\): \(x=r\cos\theta\) \(y=r\sin\theta\) Now, calculate the Cartesian coordinates for each \((r,\theta)\) pair: \((8,0^\circ)\): \((x,y)=(8\cos(0),8\sin(0))=(8,0)\) \((4+2\sqrt{2},45^\circ)\): \((x,y)=\left((4+2\sqrt{2})\cos\left(\frac{\pi}{4}\right),(4+2\sqrt{2})\sin\left(\frac{\pi}{4}\right)\right)=(2\sqrt{2},6)\) \((4,90^\circ)\): \((x,y)=(4\cos\left(\frac{\pi}{2}\right),4\sin\left(\frac{\pi}{2}\right))=(0,4)\) \((4-2\sqrt{2},135^\circ)\): \((x,y)=\left((4-2\sqrt{2})\cos\left(\frac{3\pi}{4}\right),(4-2\sqrt{2})\sin\left(\frac{3\pi}{4}\right)\right)=(-2\sqrt{2},6)\) \((0,180^\circ)\): \((x,y)=(0\cos(\pi),0\sin(\pi))=(0,0)\) \((4-2\sqrt{2},225^\circ)\): \((x,y)=\left((4-2\sqrt{2})\cos\left(\frac{5\pi}{4}\right),(4-2\sqrt{2})\sin\left(\frac{5\pi}{4}\right)\right)=(-2\sqrt{2},-6)\) \((4,270^\circ)\): \((x,y)=(4\cos\left(\frac{3\pi}{2}\right),4\sin\left(\frac{3\pi}{2}\right))=(0,-4)\) \((4+2\sqrt{2},315^\circ)\): \((x,y)=\left((4+2\sqrt{2})\cos\left(\frac{7\pi}{4}\right),(4+2\sqrt{2})\sin\left(\frac{7\pi}{4}\right)\right)=(2\sqrt{2},-6)\)

Step 5: Plot the points on the graph

Now we have enough points in Cartesian coordinates to sketch the graph. Plot the points: \((8,0)\), \((2\sqrt{2},6)\), \((0,4)\), \((-2\sqrt{2},6)\), \((0,0)\), \((-2\sqrt{2},-6)\), \((0,-4)\), and \((2\sqrt{2},-6)\). Connect these points to form a smooth curve, and you'll have a graph of the polar equation \(r=4+4\cos\theta\).

Key Concepts

Cartesian Coordinates

In mathematics, Cartesian Coordinates provide a system for uniquely specifying the location of a point in a plane. This system involves two perpendicular lines, called axes, which intersect at a point called the origin. The horizontal axis is usually referred to as the x-axis, and the vertical axis is the y-axis. Each point in the plane is given in terms of its x-coordinate (how far along it is on the x-axis) and y-coordinate (how far it is on the y-axis).

To convert from Polar to Cartesian coordinates involves using the following equations:

• The x-coordinate: calculated with the formula: \(x = r\cos\theta\), where \(r\) is the radius and \(\theta\) is the angle in radians.
• The y-coordinate: found using: \(y = r\sin\theta\).

By transforming polar coordinates \((r, \theta)\) into Cartesian coordinates, we gain a better understanding of their placement within the graphing space as traditional coordinates.

Graphing

Graphing involves plotting points in a plane to visually represent the relationships between variables. When working with polar equations, like the one given in the problem—\(r = 4 + 4 \cos \theta\)—the task is to convert polar coordinates into Cartesian coordinates for graphing.

Begin by selecting values for \(\theta\) from 0 to \(2\pi\) radians, ensuring you capture the full cycle of the graph. For each \(\theta\) value, calculate the corresponding \(r\) value and then convert these polar coordinates into Cartesian coordinates using the formulas mentioned earlier:

• Plot these \((x, y)\) points on a standard rectangular coordinate grid.
• Connect the points to reveal the shape of the graph.
• Look for symmetry and curves to understand the full pattern.

This technique helps in translating abstract mathematical expressions into visual forms that are easier to understand and analyze.

Cosine Function

The Cosine Function is a fundamental trigonometric function that appears in many mathematical contexts, including the given polar equation \(r = 4 + 4\cos\theta\). This equation involves adding a constant to a cosine term. To understand the effect of the cosine function, observe how it affects the radius \(r\) as the angle \(\theta\) changes.

The cosine function oscillates between -1 and 1. Therefore, as \(\theta\) varies, the term \(4\cos\theta\) causes fluctuations in the value of \(r\). This results in:

• At \(\theta = 0\), \(\cos 0 = 1\), giving maximum radius; \(r = 8\).
• As \(\theta = \frac{\pi}{2}\), \(\cos \frac{\pi}{2} = 0\), causing \(r = 4\).
• At \(\theta = \pi\), \(\cos \pi = -1\), leading to \(r = 0\), a minimum point.

The periodic behavior of the cosine function is fundamental in predicting the shape and periodicity of the resulting graph. Understanding this concept is critical in working with polar graphs and interpreting their forms accurately.