   Chapter 10, Problem 44RE

Chapter
Section
Textbook Problem

A family of curves has polar equations ra = |sin 2θ| where a is a positive number. Investigate how the curves change as a changes.

To determine

To find: The variation of curve by changing the value of a from the polar equation.

Explanation

Given:

The polar equation is ra=|sin2θ| .

Calculation:

Calculate the value of r .

ra=|sin2θ|r=(sin2θ)1a

Take the value of a=0.01 .

Calculate the value of r .

r=(sin2θ)1a

Substitute 0° for θ and 0.01 for a in the above equation.

r=(sin2θ)1a=[sin(2×0×π180)]10.01=0

Calculate the value of x.

x=rcosθ

Substitute the value of 0 for r and 0° for θ in the above equation.

x=rcosθ=0(cos0×π180)=0

Calculate the value of y.

y=rsinθ

Substitute the value of 0 for r and 0° for θ in the above equation.

y=rsinθ=0(sin0×π180)=0

Tabulate the calculated values as shown in below table (1).

 θ r=(sin2θ)10.01 x=rcosθ y=rsinθ 0 0.00 0.00 0.00 10 0.00 0.00 0.00 20 0.00 0.00 0.00 30 0.00 0.00 0.00 40 0.22 0.17 0.14 50 0.22 0.14 0.17 60 0.00 0.00 0.00 70 0.00 0.00 0.00 80 0.00 0.00 0.00 90 0.00 0.00 0.00 100 0.00 0.00 0.00 110 0.00 0.00 0.00 120 0.00 0.00 0.00 130 0.22 -0.14 0.17 140 0.22 -0.17 0.14 150 0.00 0.00 0.00 160 0.00 0.00 0.00 170 0.00 0.00 0.00 180 0.00 0.00 0.00 190 0.00 0.00 0.00 200 0.00 0.00 0.00 210 0.00 0.00 0.00 220 0.22 -0.17 -0.14 230 0.22 -0.14 -0.17 240 0.00 0.00 0.00 250 0.00 0.00 0.00 260 0.00 0.00 0.00 270 0.00 0.00 0.00 280 0.00 0.00 0.00 290 0.00 0.00 0.00 300 0.00 0.00 0.00 310 0.22 0.14 -0.17 320 0.22 0.17 -0.14 330 0.00 0.00 0.00 340 0.00 0.00 0.00 350 0.00 0.00 0.00 360 0.00 0.00 0.00

Plot the graph for the above values as shown in figure (1).

Take the value of a=1 .

Calculate the value of r .

r=(sin2θ)1a

Substitute 0° for θ and 1 for a in the above equation.

r=(sin2θ)1a=[sin(2×0×π180)]11=0

Calculate the value of x.

x=rcosθ

Substitute the values of 0 for r and 0° for θ in the above equation.

x=rcosθ=0(cos0×π180)=0

Calculate the value of y.

y=rsinθ

Substitute the values of 0 for r and 0° for θ in the above equation.

y=rsinθ=0(sin0×π180)=0

Tabulate the calculated values as shown in below table (2).

 θ r=(sin2θ)11 x=rcosθ y=rsinθ 0 0.00 0.00 0.00 10 0.34 0.34 0.06 20 0.64 0.60 0.22 30 0.87 0.75 0.43 40 0.98 0.75 0.63 50 0.98 0.63 0.75 60 0.87 0.43 0.75 70 0.64 0.22 0.60 80 0.34 0.06 0.34 90 0.00 0.00 0.00 100 -0.34 0.06 -0.34 110 -0.64 0.22 -0.60 120 -0.87 0.43 -0.75 130 -0.98 0.63 -0.75 140 -0.98 0.75 -0.63 150 -0.87 0.75 -0.43 160 -0.64 0.60 -0.22 170 -0.34 0.34 -0.06 180 0.00 0.00 0.00 190 0.34 -0.34 -0.06 200 0.64 -0.60 -0.22 210 0.87 -0.75 -0.43 220 0.98 -0.75 -0.63 230 0.98 -0.63 -0.75 240 0.87 -0.43 -0.75 250 0.64 -0.22 -0.60 260 0.34 -0.06 -0.34 270 0.00 0.00 0.00 280 -0.34 -0.06 0.34 290 -0.64 -0.22 0.60 300 -0.87 -0.43 0.75 310 -0.98 -0.63 0.75 320 -0.98 -0.75 0.63 330 -0.87 -0.75 0.43 340 -0

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