Section 5.2 – Polar Equations and Graphs

An equation whose variables are polar coordinates is called a polar equation. The graph of a polar equation consists of all points whose polar coordinates satisfy the equation.

Identify and graph the equation: r = 2

r 2

r2 4

x y2 2 4

Circle with center at the pole and radius 2.

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Identify and graph the equation: =3

tan tan

3

31

yx

31

y x 3

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3

Identify and graph the equation: r sin 2

sin sin yr

y r

y 2

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Theorem

Let a be a nonzero real number, the graph of the equation

r asin

is a horizontal line a units above the pole if a > 0 and units below the pole if a < 0.

a

Theorem

Let a be a nonzero real number, the graph of the equation

r acos

is a vertical line a units to the right of the pole if a > 0 and units to the left of the pole if a < 0.

a

Identify and graph the equation: r 4cos

r r2 4 cos

x y x2 2 4

x x y2 24 0

x x y2 24 4 4

x y 2 42 2

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Theorem

Let a be a positive real number. Then,

r a2 sin Circle: radius a; center at (0, a) in rectangular coordinates.

r a 2 sin Circle: radius a; center at (0, -a) in rectangular coordinates.

Theorem

Let a be a positive real number. Then,

r a2 cos Circle: radius a; center at (a, 0) in rectangular coordinates.

r a 2 cos Circle: radius a; center at (-a, 0) in rectangular coordinates.

Theorem Tests for Symmetry

Symmetry with Respect to the Polar Axis (x-axis):

In a polar equation, replace by If

an equivalent equation results, the graph

is symmetric with respect to the polar

axis.

.

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r,

r,

Theorem Tests for Symmetry

Symmetry with Respect to the Line (y-axis):

In a polar equation, replace by

If an equivalent equation results, the

graph is symmetric with respect to the

line = 2

.

.

2

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r, r,

Theorem Tests for Symmetry

Symmetry with Respect to the Pole (Origin):

In a polar equation, replace by If

an equivalent equation results, the graph

is symmetric with respect to the pole.

r r .

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Other Graphs

You will run into some other graphs

• Cardioids • Limacons – without inner loop• Limacons – with inner loop• Lemniscate• Rose

See table 7 on pages 341 & 342