Question

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

Short answer

Answer: As θ increases from 0 to \(2\pi\), the curve exhibits two different behaviors. For \(0 < \theta \leq \pi\), the value of r is positive, whereas for \(\pi \leq \theta < 2\pi\), the value of r is negative. At θ = π/2, r = 3, and at θ = π and θ = 3π/2, r values are 1 and -3, respectively. The graph of the equation forms a smooth curve with properly labeled points and arrows showing the direction of the curve as θ increases.

Step-by-step solution

Analyze the equation

The given equation is in polar coordinates, with \(r\) being the radial distance from the origin and \(\theta\) being the angle. The equation is given by: $$r=\frac{3}{1-\cos \theta}$$

Determine the behavior of the curve

When \(\theta = 0\), \(\cos \theta = 1\) and the denominator becomes zero, which makes \(r\) undefined. Therefore, as \(\theta\) increases from 0 to \(2\pi\), we can study the behavior of the curve for the equation.

Identify critical points or regions

Notice that the value of \(r\) will be positive when the denominator (1 - cosθ) is negative and vice versa. The denominator equals zero at both θ = 0 and θ = 2π. So, we have 2 cases to consider: 1. When \(0 < \theta \leq \pi\), the value of \(r\) will be positive. 2. When \(\pi \leq \theta < 2\pi\), the value of \(r\) will be negative. Let's evaluate the function at some critical points: - θ = π/2: \(r = \dfrac{3}{1 - \cos\frac{\pi}{2}} = 3\) - θ = π: \(r = \dfrac{3}{1 - \cos\pi} = 1\) - θ = 3π/2: \(r = \dfrac{3}{1 - \cos\frac{3\pi}{2}} = - 3\)

Graph the equation

Plot the curve for the given equation. Start with the points we found above, such as \((3, \frac{\pi}{2})\), \((1, \pi)\) and \((-3, \frac{3\pi}{2})\). Once you have a few points plotted, connect the points with a smooth curve to create the graph.

Label points and use arrows to indicate how the curve is generated

Now that the curve is sketched, you should indicate how the curve is generated as \(\theta\) increases from 0 to \(2\pi\). Label the points found earlier, such as \((3, \frac{\pi}{2})\), \((1, \pi)\), and \((-3, \frac{3\pi}{2})\). Add arrows to show the direction of the curve, indicating how it is generated as \(\theta\) increases. With properly labeled points and arrows, the graph is now complete.