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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

Sketch the curve with the given polar equation by first sketching the graph of r as a function of θ in Cartesian coordinates.

43. r = 2 + sin 3θ

To determine

To sketch: The curves for the given Cartesian and polar coordinates r=2+sin3θ .

Explanation

Given:

The polar equation is as below.

r=2+sin3θ .

Calculation:

Calculation:

Substitute (0) for θ in equation r=2+sin3θ .

r=2+sin3θ=2+sin3(0)=2

For the polar equation r=2+sin3θ values for the Cartesian coordinates curve is tabulated below.

θ

(degrees)

θ

(rad)

r
0 0 2
10 0.174611111 2.500203119
20 0.349222222 2.866259866
30 0.523833333 2.999999752
40 0.698444444 2.865555907

The curve for the Cartesian coordinates is shown below in figure1

From the figure 1, when θ increases from 0 to π6 , polar curve r increases from 2 to 3 in polar graph.

When θ increases from π6 to π2 , polar curve r decreases from 3 to 1

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P-52RECh-10 P-53RECh-10 P-54RECh-10 P-55RECh-10 P-56RECh-10 P-57RECh-10 P-58RECh-10 P-1PCh-10 P-2PCh-10 P-3PCh-10 P-4PCh-10 P-5PCh-10 P-6P

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