   Chapter 10, Problem 3P

Chapter
Section
Textbook Problem

What is the smallest viewing rectangle that contains every member of the family of polar curves r = 1 + c sin θ, where 0 ≤ c ≤ 1 ? Illustrate your answer by graphing several members of the family in this viewing rectangle.

To determine

To Find: The different rectangle curves from the given polar equation and smallest viewing rectangle that contains every member of the family of polar curves.

Explanation

Given:

The given polar curve is r=1+csinθ (1)

Calculation:

Calculate the value of x.

x=rcosθ

Substitute 1+csinθ for r in the above equation.

x=rcosθx=(1+csinθ)cosθ=cosθ+csinθcosθ

Substitute 12sin2θ for sinθcosθ in the above equation.

x=cosθ+csinθcosθx=cosθ+c(12sin2θ)

Calculate the value of y.

y=rsinθ

Substitute 1+csinθ for r in the above equation.

y=rsinθy=(1+csinθ)sinθy=sinθ+csin2θ

Provide different values for c and plot the graph.

Take the value of c=1

 θ x=cosθ+1(12sin2θ) y=sinθ+(1)sin2θ 0 1.00 0.00 10 1.16 0.20 20 1.26 0.46 30 1.30 0.75 40 1.26 1.06 50 1.14 1.35 60 0.93 1.62 70 0.66 1.82 80 0.34 1.95 90 0.00 2.00 100 -0.34 1.95 110 -0.66 1.82 120 -0.93 1.62 130 -1.14 1.35 140 -1.26 1.06 150 -1.30 0.75 160 -1.26 0.46 170 -1.16 0.20 180 -1.00 0.00 190 -0.81 -0.14 200 -0.62 -0.23 210 -0.43 -0.25 220 -0.27 -0.23 230 -0.15 -0.18 240 -0.07 -0.12 250 -0.02 -0.06 260 0.00 -0.01 270 0.00 0.00 280 0.00 -0.01 290 0.02 -0.06 300 0.07 -0.12 310 0.15 -0.18 320 0.27 -0.23 330 0.43 -0.25 340 0.62 -0.23 350 0.81 -0.14 360 1.00 0.00

Take the value of c=23

 θ x=cosθ+(23)(12sin2θ) y=sinθ+(23)sin2θ 0 1.00 0.00 10 1.10 0.19 20 1.15 0.42 30 1.15 0.67 40 1.09 0.92 50 0.97 1.16 60 0.79 1.37 70 0.56 1.53 80 0.29 1.63 90 0.00 1.67 100 -0.29 1.63 110 -0.56 1.53 120 -0.79 1.37 130 -0.97 1.16 140 -1.09 0.92 150 -1.15 0.67 160 -1.15 0.42 170 -1.10 0.19 180 -1.00 0.00 190 -0.87 -0.15 200 -0.73 -0.26 210 -0.58 -0.33 220 -0.44 -0.37 230 -0.31 -0.37 240 -0.21 -0.37 250 -0.13 -0.35 260 -0.06 -0.34 270 0.00 -0.33 280 0.06 -0.34 290 0.13 -0.35 300 0.21 -0.37 310 0.31 -0.37 320 0.44 -0.37 330 0.58 -0.33 340 0.73 -0.26 350 0.87 -0.15 360 1.00 0.00

Take the value of c=13

 θ x=cosθ+(13)(12sin2θ) y=sinθ+(13)sin2θ 0 1.00 0.00 10 1.04 0.18 20 1.05 0.38 30 1.01 0.58 40 0.93 0.78 50 0.81 0.96 60 0.64 1.12 70 0.45 1.23 80 0.23 1.31 90 0.00 1.33 100 -0.23 1.31 110 -0.45 1.23 120 -0.64 1.12 130 -0.81 0.96 140 -0

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