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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.

36. r = −sin 5θ

To determine

To sketch: The curves for the polar equation r=sin5θ and its Cartesian coordinates.

Explanation

Given:

The polar equation is as below.

r=sin5θ .

Calculation:

Substitute (0) for θ in equation r=sin5θ .

r=sin5θ=sin5(0)=0

For the polar equation r=sin5θ values for the Cartesian coordinates curve is tabulated below.

t r
0 0
0.087305556 –0.421185
0.174611111 –0.754644
0.261916667 –0.93313
0.349222222 –0.925236

Graph:

The curve for the Cartesian coordinates is shown below in figure1.

From the figure 1, when θ increases from 0 to π10 , the polar curve r decreases from 0 to 1 in polar graph.

When θ increases from π10 to π5 , the polar curve r increases from 1 to 0 .

When θ increases from π5 to 3π10 , the polar curve r increases from 0 to 1 .

When θ increases from 3π10 to 2π5 , the polar curve r decreases from 1 to 0

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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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