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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.

38. r = 2 sin 6θ

To determine

To sketch: The curves for the polar equation r=2sin6θ and its Cartesian coordinates.

Explanation

Given:

The polar equation is as below.

r=2sin6θ .

Calculation:

Substitute (0) for θ in equation r=2sin6θ .

r=2sin6θ=2sin6(0)=0

For the polar equation r=2sin6θ values for the Cartesian coordinates curve is tabulated below.

t r
0 0
0.087305556 0.996596
0.174611111 1.7061753
0.261916667 1.9317905
0.349222222 1.6266204

Graph:

The curve for the Cartesian coordinates is shown below in figure 1.

From the figure 1 , when θ increases from 0 to π12 , the polar curve r increases from 0 to 2 in polar graph.

When θ increases from π12 to π6 , the polar curve r decreases from 2 to 0 .

When θ increases from π6 to π4 , the polar curve r decreases from 0 to 2

When θ increases from π4 to π3 , the polar curve r increases from 2 to 0 .

When θ increases from π3 to 5π12 , the polar curve r increases from 0 to 2 .

When θ increases from 5π12 to π2 , the polar curve r decreases from 2 to 0

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