/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 69 Sketch a graph of the polar equa... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 69

Sketch a graph of the polar equation. $$ r=3-2 \cos \theta $$

Short Answer

Expert verified
The graph of the polar equation \( r = 3 - 2\cos\theta \) is a lima莽on without a loop shifted 3 units from the origin.

Step by step solution

01

Identify the type of polar graph

This is a type of lima莽on function which is a polar graph equation, where \( \textit{r} \) depends on the trigonometric function of \( \textit{胃} \). Lima莽on functions can vary in shape, some having loops and some do not. To determine this, we will need to inspect the coefficients in the equation more.
02

Determine if the graph has a loop or not

Whether it has a loop depends on the ratio of coefficients of \( r \) and the trigonometric function. Compare the absolute values of the independent term 3 (which is responsible for shifting the graph from the origin) and the coefficient of \( \cos \theta \), 2. Since \( |3| > |2| \), there will be no loop in the graph.
03

Plot the graph using points

To plot the graph of this equation, select a series of values for \( \theta \) and compute the corresponding values of \( r \). You might start with \( \theta = 0 \), \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \). Plot these \( (r,\theta) \) points in the polar plane and then smoothly connect the dots. Remember the point (r, 胃) is plotted a distance |r| from the origin and an angle 胃 from the polar axis (positive x-axis).
04

Complete the graph

After plotting some key points and understanding the symmetry of the function, complete the graph by sketching it smoothly. Make sure your graph represents the correct type of lima莽on function without a loop.

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Most popular questions from this chapter

Graphical Reasoning In Exercises 1-4, use a graphing utility to graph the polar equation when (a) \(e=1,\) (b) \(e=0.5\) and \((\mathrm{c}) e=1.5 .\) Identify the conic. \(r=\frac{2 e}{1+e \sin \theta}\)

Find two different sets of parametric equations for the rectangular equation. $$ y=3 x-2 $$

Use a graphing utility to graph the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation. $$ x=t^{3}, \quad y=3 \ln t $$

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \text { Circle: } x=h+r \cos \theta, \quad y=k+r \sin \theta $$

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. \(r=\frac{3}{2+6 \sin \theta}\)
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