   Chapter 10.4, Problem 40E

Chapter
Section
Textbook Problem

# Find all points of intersection of the given curves.40. r = cos 3θ, r = sin 3θ

To determine

To Find: All points of intersection of the given curve.

Explanation

Given:

The given polar equations are as below.

r=cos3θ (1)

r=sin3θ (2)

Calculation:

Calculate the value of r using the equation (1).

r=cos3θ

Substitute 0 for θ in the equation (1).

r=cos3(0)=1

Calculate the value of x.

x=rcosθ

Substitute 1 for r and 0 for θ .

x=1×cos(0×π180)=1

Calculate the value of y.

y=rsinθ

Substitute 1 for r and 0 for θ .

y=1×sin(0×π180)=0

Similarly, calculate the values of x and y using the value of θ from 0 to 180 .

Tabulate the values of x and y in table (1).

 θ r=cos3θ x=rcosθ y=rsinθ 0.00 1.00 1.00 0.00 5.00 0.97 0.96 0.08 10.00 0.87 0.85 0.15 15.00 0.71 0.68 0.18 20.00 0.50 0.47 0.17 25.00 0.26 0.23 0.11 30.00 0.00 0.00 0.00 35.00 -0.26 -0.21 -0.15 40.00 -0.50 -0.38 -0.32 45.00 -0.71 -0.50 -0.50 50.00 -0.87 -0.56 -0.66 55.00 -0.97 -0.55 -0.79 60.00 -1.00 -0.50 -0.87 65.00 -0.97 -0.41 -0.88 70.00 -0.87 -0.30 -0.81 75.00 -0.71 -0.18 -0.68 80.00 -0.50 -0.09 -0.49 85.00 -0.26 -0.02 -0.26 90.00 0.00 0.00 0.00 95.00 0.26 -0.02 0.26 100.00 0.50 -0.09 0.49 105.00 0.71 -0.18 0.68 110.00 0.87 -0.30 0.81 115.00 0.97 -0.41 0.88 120.00 1.00 -0.50 0.87 125.00 0.97 -0.55 0.79 130.00 0.87 -0.56 0.66 135.00 0.71 -0.50 0.50 140.00 0.50 -0.38 0.32 145.00 0.26 -0.21 0.15 150.00 0.00 0.00 0.00 155.00 -0.26 0.23 -0.11 160.00 -0.50 0.47 -0.17 165.00 -0.71 0.68 -0.18 170.00 -0.87 0.85 -0.15 175.00 -0.97 0.96 -0.08 180.00 -1.00 1.00 0.00

Calculate the value of r using the equation (1).

r=sin3θ

Substitute 0 for θ in the equation (1).

r=sin3(0)=0

Calculate the value of x.

x=rcosθ

Substitute 0 for r and 0 for θ .

x=0×cos(0×π180)=0

Calculate the value of y.

y=rsinθ

Substitute 0 for r and 0 for θ .

y=0×sin(0×π180)=0

Similarly, calculate the values of x and y using the value of θ from 0 to 180 .

Tabulate the values of x and y in table (2).

 θ r=sin3θ x=rcosθ y=rsinθ 0

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