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Graph the cardioid r=2(1sinθ)For each pair of functions in Exercises 4045

Algebraically find all values of θwhere

f1(θ)=f2(θ)

Sketch the two curves in the same polar coordinate system.

Find all points of intersection between the two curves.

Short Answer

Expert verified

The graph is

Step by step solution

01

Given information

r=2(1sinθ)

02

Calculation

Consider the given cardioid, r=2(1-sinθ)

The goal is to create a graph of the equation.

Assume θ=0,π6,π2,π,3π2,2π

For different θvalues we find the values of the equation r=2(1-sinθ)

When θ=0

r=2(1-sin0)[sincer=2(1-sinθ)]r=2(1-0)[sincesin0=0]r=2

Then the coordinate (r,θ)=(2,0)

When θ=π6Open with

r=21-sinπ6[sincer=2(1-sinθ)]r=21-12sincesinπ6=12r=1

Then the coordinate (r,θ)=1,π6

When θ=π2

r=21-sinπ2[sincer=2(1-sinθ)]r=2(1-1)sincesinπ2=1r=0

Then the coordinate (r,θ)=0,π2

When θ=π

r=2(1-sinπ)[sincer=2(1-sinθ)]r=2(1-0)[sincesinπ=0]r=2

Then the coordinate (r,θ)=(2,π)

Whenθ=3π2

r=21-sin3π2[sincer=2(1-sinθ)]r=2(1-(-1))sincesin3π2=-1r=4

Then the coordinate (r,θ)=4,3π2

When θ=2π

r=2(1-sin2π)[since r=2(1-sinθ)]

r=2(1-0)[since sin2π=0]

r=2

Then the coordinate (r,θ)=(2,2π)

We have distinct coordinates for differentθ values.

To draw the graph, represent all the following points on the graph.

(2,0)1,π60,π2(2,π)4,3π2(2,2π)

03

Calculation

This is the required graph.

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