   Chapter 10.4, Problem 42E

Chapter
Section
Textbook Problem

Find all points of intersection of the given curves.42. r2 = sin 2θ, r2 = cos 2θ

To determine

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

Explanation

Given:

The given polar equations are as below.

r2=sin2θ (1)

r2=cos2θ (2)

Calculation:

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

r2=sin2θ

Substitute 0 for θ in the equation (1).

r2=sin2(0)r=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 90 and 180 to 270 .

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

 θ r=sin2θ x=rcosθ y=rsinθ 0.00 0.00 0.00 0.00 10.00 0.58 0.58 0.10 20.00 0.80 0.75 0.27 30.00 0.93 0.81 0.47 40.00 0.99 0.76 0.64 50.00 0.99 0.64 0.76 60.00 0.93 0.47 0.81 70.00 0.80 0.27 0.75 80.00 0.58 0.10 0.58 90.00 0.00 0.00 0.00 180.00 0.01 -0.01 0.00 190.00 0.58 -0.58 -0.10 200.00 0.80 -0.75 -0.27 210.00 0.93 -0.81 -0.47 220.00 0.99 -0.76 -0.64 230.00 0.99 -0.64 -0.76 240.00 0.93 -0.47 -0.81 250.00 0.80 -0.27 -0.75 260.00 0.58 -0.10 -0.58 270.00 0.00 0.00 0.00

Calculate the value of r using the equation (2).

r2=cos2θ

Substitute 0 for θ in the equation (2).

r2=cos2(0)r=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 45 to 45 and 225 to 225 .

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

 θ r=cos2θ x=rcosθ y=rsinθ -45.00 0.00 0.00 0.00 -44.00 0.19 0.13 -0.13 -43.00 0.26 0.19 -0.18 -42.00 0.32 0.24 -0.22 -41.00 0.37 0.28 -0.24 -40.00 0.42 0.32 -0.27 -35.00 0.58 0.48 -0.34 -30.00 0.71 0.61 -0.35 -25.00 0.80 0.73 -0.34 -20.00 0.88 0.82 -0.30 -15.00 0.93 0.90 -0.24 -10.00 0.97 0.95 -0.17 -5

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