Problem 62 - Essential Calculus

Question

Sketch a graph of the polar equation and find the tangents at the pole. $$ r=3(1-\cos \theta) $$

Step-by-Step Solution

Answer

The tangents to the graph of \(r = 3(1 - \cos \theta)\) at the pole are the positive and negative x-axes, i.e., the lines at \(\theta = 0\) and \(\theta = \pi\).

Step 1: Sketch the graph

Start by plotting values for \(\theta\) from 0 to \(2 \pi\) in increments, say every \( \pi /4\), and calculating the corresponding values of r using the given equation \(r = 3(1 - \cos \theta)\). A collection of points is obtained, which can be plotted and then connected smoothly to reveal the graph. Notice that the graph is discontinuous at \(\theta = \pi\).

Step 2: Find the pole

Look for the point on the graph where r=0. This is the pole. According to the equation, \(r = 3(1 - \cos \theta)\), the pole is located at \(\theta = 0\).

Step 3: Determine the tangents at the pole

The tangent lines at the pole are lines that barely touch the graph at that point, without crossing it. Usually, this involves determining the slope of the graph at that point, however in the case of a pole the graph is discontinuous at the pole. Because the tangent cannot cross the graph, the tangents to the graph at \(\theta = 0\) only occur along the positive and negative x-axes, i.e., the lines \(\theta = 0\) and \(\theta = \pi\).