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Chapter 8: Problem 70

Sketch a graph of the polar equation. $$ r=5-4 \sin \theta $$

Short Answer

Expert verified
This lima莽on has an inner loop at \( \theta = \frac{\pi}{2} \) and an outer loop at \( \theta = \pi, 2\pi, \) and \( 0 \). It is symmetric with respect to the origin.

Step by step solution

01

Identify the circle or lima莽on

We start by analyzing the format of the equation to determine whether it's describing a circle or a lima莽on. In this particular case, \( r = 5 - 4 \sin \theta \), it does not represent a circle as the coefficient of the \( \sin \theta \) term does not equal to the constant. Hence, it represents a lima莽on.
02

Determine specific points

Evaluate the polar function for values of \( \theta \) that result in \( r = 0 \), as well as typical values such as \( \theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}, \) and \( 2\pi \). A few particular points could be: when \( \theta = 0, r = 5 \), when \( \theta = \frac{\pi}{2}, r = 1 \), when \( \theta = \pi, r = 5 \), when \( \theta = \frac{3\pi}{2}, r = 9 \), and when \( \theta = 2\pi, r = 5 \). These points help us define the outer and inner loop of the lima莽on.
03

Sketch the graph

Using the determined points, we can start sketching the graph. We see that the graph has an inner loop at \( \theta = \frac{\pi}{2} \) and comes out to the outer loop at \( \theta = \pi \), \( \theta = 2\pi \), and \( \theta = 0 \). This lima莽on is symmetric with respect to the origin.

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Most popular questions from this chapter

In Exercises 57 and \(58,\) let \(r_{0}\) represent the distance from the focus to the nearest vertex, and let \(r_{1}\) represent the distance from the focus to the farthest vertex. Show that the eccentricity of an ellipse can be written as \(e=\frac{r_{1}-r_{0}}{r_{1}+r_{0}} .\) Then show that \(\frac{r_{1}}{r_{0}}=\frac{1+e}{1-e}\).

Find two different sets of parametric equations for the rectangular equation. $$ y=x^{2} $$

True or False. Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The graph of the parametric equations \(x=t^{2}\) and \(y=t^{2}\) is the line \(y=x\).

In Exercises 43-46, find the area of the surface formed by revolving the curve about the given line. $$ \begin{array}{lll} \underline{\text { Polar Equation }} & \underline{\text { Interval }} & \underline{\text { Axis of Revolution }} \\ r=e^{a \theta} & 0 \leq \theta \leq \frac{\pi}{2} & \theta=\frac{\pi}{2} \end{array} $$

In Exercises \(7-16,\) find the eccentricity and the distance from the pole to the directrix of the conic. Then sketch and identify the graph. Use a graphing utility to confirm your results. \(r=\frac{5}{-1+2 \cos \theta}\)
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