Problem 71 - Essential Calculus

Question

Sketch a graph of the polar equation. $$ r=3 \csc \theta $$

Step-by-Step Solution

Answer

The graph of \(r=3 \csc (\theta)\) results in a pattern that repeats every \(\pi\) units with arms approaching the vertical lines \(\theta=n\pi\), for integer values of \(n\), as asymptotes.

Step 1: Understanding the function and domain

The csc function is the reciprocal of the sine function. Given the polar equation \(r=3 \csc \theta\), it implies that \(r\) approaches infinity as sine of \(\theta\) approaches zero. That is, the domain will be \(\theta=n\pi\) for \(n\neq0\), where \(n\) is an integer.

Step 2: Graphing the function

Plot the graph of \(r=3 \csc (\theta)\) in the polar plane. This can be constructed as a series of points equally distanced from the pole at angles that are multiples of \(\pi\), and copying the pattern for all \(n\) integer values. Because \(\csc(\theta)\) is undefined at multiples of \(\pi\), and distance \(r\) cannot be negative, this means that there is an asymptote at \(\theta=n\pi\).

Step 3: Understanding the graph

The graph consists of a curve with arms which approach the line \(\theta=n\pi\), with \(n\) being an integer, as asymptotes from both sides as \(r\) goes to infinity. The curve appears to give a multiple leaf pattern which essentially repeats every \(\pi\) units around the circle and forms a series of vertical lines. The distance between the asymptotes and the arms decreases as the angle \(\theta\) moves away from the asymptotes.