Problem 63 - Essential Calculus

Question

Sketch a graph of the polar equation and find the tangents at the pole. $$ r=2 \cos 3 \theta $$

Step-by-Step Solution

Answer

The graph of the polar equation \( r = 2 \cos 3 \theta \) resembles a three-leaved flower. The tangents at the pole occur whenever the curve passes through the pole, that is, for the cases when \( \theta = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6} \). Radiating lines from the origin to points where \( \theta \) has these values act as potential tangents.

Step 1: Graphing the Polar Equation

Plot different points by taking various values for \( \theta \) between 0 and \( \pi \) and then find the corresponding \( r \) using the given equation. Start with \( \theta = 0 \) and continue with small increments. Remember that in polar coordinates a point is expressed as (r, \( \theta \)), where \( r \) is the distance from the origin and \( \theta \) the angle from the positive x-axis. This should provide a clear graph of the given polar equation.

Step 2: Identifying Pole Locations

The pole of a graph in polar coordinates is the origin of the coordinate system, where the value of \( r \) is zero. In this case, evaluate the equation \( r = 2 \cos 3 \theta \) to find for which values of \( \theta \), if any, \( r \) becomes zero.

Step 3: Find the Tangents at the Pole

For those \( \theta \) values where \( r \) is zero, the curve passes through the pole. For such points, multiple tangents can be formed; with the pole being the origin, radiating lines (or rays) from the origin to points where \( \theta = \text{(constant)} \) function as potential tangents. These lines represent the different directions in which the curve leaves the pole.