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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.

37. r = 2 cos 4θ

To determine

To sketch: The curves for the polar equation r=2cos4θ and its Cartesian coordinates..

Explanation

Given:

The polar equation is as below.

r=2cos4θ .

Calculation:

The polar equation is as below.

r=2cos4θ .

Calculation:

Substitute (0) for θ in equation r=2cos4θ .

r=2cos4θ=2cos4(0)=2

For the polar equation r=2cos4θ values for the Cartesian coordinates curve is tabulated below.

t r
0 2
0.174611111 1.5083962
0.261916667 0.9651105
0.349222222 0.3251756
0.436527778 -0.316124

Graph:

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

From the figure 1, when θ increases from 0 to π8 ., the polar curve r decreases  from 2 to 0 in polar graph.

When θ increases from π8 to π4 . The polar curve r decreases from 0 to 2 .

When θ increases from π4 to 3π8 . The polar curve r increases from 2 to 0

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