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Chapter 8: Problem 71

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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 \(\theta=n\pi\).
03

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.

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Most popular questions from this chapter

In Exercises \(17-20,\) use a graphing utility to graph the polar equation. Identify the graph. \(r=\frac{2}{2+3 \sin \theta}\)

Consider the circle \(r=3 \sin \theta\) (a) Find the area of the circle. (b) Complete the table giving the areas \(A\) of the sectors of the circle between \(\theta=0\) and the values of \(\theta\) in the table. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline \boldsymbol{\theta} & 0.2 & 0.4 & 0.6 & 0.8 & 1.0 & 1.2 & 1.4 \\ \hline \boldsymbol{A} & & & & & & & \\ \hline \end{array} $$ (c) Use the table in part (b) to approximate the values of \(\theta\) for which the sector of the circle composes \(\frac{1}{8}, \frac{1}{4},\) and \(\frac{1}{2}\) of the total area of the circle. (d) Use a graphing utility to approximate, to two decimal places, the angles \(\theta\) for which the sector of the circle composes \(\frac{1}{8}, \frac{1}{4},\) and \(\frac{1}{2}\) of the total area of the circle.

Use a graphing utility to graph the curve represented by the parametric equations. Indicate the direction of the curve. Identify any points at which the curve is not smooth. $$ \text { Witch of Agnesi: } x=2 \cot \theta, \quad y=2 \sin ^{2} \theta $$

Conjecture (a) Use a graphing utility to graph the curves represented by the two sets of parametric equations. \(x=4 \cos t \quad x=4 \cos (-t)\) \(y=3 \sin t \quad y=3 \sin (-t)\) (b) Describe the change in the graph when the sign of the parameter is changed. (c) Make a conjecture about the change in the graph of parametric equations when the sign of the parameter is changed. (d) Test your conjecture with another set of parametric equations.

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=2 t, \quad y=|t-2| $$
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