Problem 63 - Calculus Early Transcendentals

Question

Graph the following equations. Then use arrows and labeled points to indicate how the curve is generated as \(\theta\) increases from 0 to \(2 \pi\). $$r=\frac{3}{1-\cos \theta}$$

Step-by-Step Solution

Answer

Answer: The general approach to graph this polar equation is to: 1. Convert the polar equation to Cartesian coordinates for better understanding of the curve. 2. Graph the polar equation by plotting points for various values of \(\theta\) from 0 to \(2\pi\). 3. Label the points and add arrows to indicate the curve's direction as \(\theta\) increases from 0 to \(2\pi\).

Step 1: Convert Polar Equation to Cartesian Coordinates

To gain a better understanding of the curve, let's first convert the polar equation to Cartesian coordinates. Using the following conversion formulas: $$x = r\cos(\theta)$$ $$y = r\sin(\theta)$$ First, find the value of r: $$r = \frac{3}{1 - \cos \theta}$$ Now, multiply the equation by \(\cos(\theta)\): $$x = \frac{3\cos(\theta)}{1 - \cos \theta}$$ Similarly, multiply the equation by \(\sin(\theta)\): $$y = \frac{3\sin(\theta)}{1 - \cos \theta}$$ Finally, divide both equations by y: $$\frac{x}{y} = \frac{\cos(\theta)}{\sin(\theta)}$$ Now, we can clearly see that this curve represents a conic section.

Step 2: Graph the Polar Equation

Now that we have a clearer understanding of the polar equation, we can graph it. To graph the polar equation, plot the points corresponding to various values of \(\theta\) from 0 to \(2\pi\). Note that when \(\theta=0\) or \(\theta=2\pi\), we have a singularity in our fraction (denominator becomes zero). Consequently, the graph will have an asymptote. Here is a list of some points to plot on the graph: - For \(\theta=\frac{\pi}{4}\), \(r = \frac{3}{1-\cos(\frac{\pi}{4})}\) - For \(\theta=\frac{\pi}{2}\), \(r = \frac{3}{1-\cos(\frac{\pi}{2})}\) - For \(\theta=\frac{3\pi}{4}\), \(r = \frac{3}{1-\cos(\frac{3\pi}{4})}\) - For \(\theta=\pi\), \(r = \frac{3}{1-\cos(\pi)}\) Plot these points and connect them to create the curve.

Step 3: Label the Points and Add Arrows

Lastly, use arrows and labels to indicate how the curve is generated as \(\theta\) increases from 0 to \(2\pi\). Place an arrow on the graph following the curve's direction and label each point with its corresponding value of \(\theta\) to complete the polar graph. For example, label the point at \(\frac{\pi}{4}\), then draw an arrow in the direction of the point at \(\frac{\pi}{2}\), and so on. Continue this process for all the points until \(\theta=2\pi\). The final graph will have the curve with labels and arrows indicating how it is generated as \(\theta\) increases from 0 to \(2\pi\).