9.2 Polar Equations and Graphs

Steps for Converting Equations from Rectangular to Polar form and vice versa

Four critical equivalents to keep in mind are:

ytan

x

Convert the equation: r = 2 to rectangular form

Since we know that , square both sides of the equation.

We still need r2, but is there a better choice than squaring both sides?

Convert the following equation from rectangular to polar form.

2 2x y x

and

Since

x r cos2r r cosr cos

Convert the following equation from rectangular to polar form.

2 22x 2y 3 2 22(x y ) 3

2 2 3x y

2

3r

2

2 3r

2

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

Circle with center at the pole and radius 2.

The graph is a straight line at extending through the pole.

3

The graph is a horizontal line at y = -2

Theorem

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

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

Theorem

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

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

Graph: r cos 3

r 4cos

Theorem

Let a be a positive real number. Then,

Circle: radius ; center at ( , 0) in rectangular coordinates.

Circle: radius ; center at (- , 0) in rectangular coordinates.

r a cos

r a cos

a

2a

2

a

2a

2

r 6sin

Theorem

Let a be a positive real number. Then,

Circle: radius ; center at (0, ) in rectangular coordinates.

Circle: radius ; center at (0, ) in rectangular coordinates.

r a sin

r a sin

a

2a

2

a

2a

2

r 4 4cos r 4 4cos

Cardioids (heart-shaped curves) where a > 0 and passes through the origin

r a a cos r a a cos

a a sin a a sin

4 4sin 4 4sin

Limacons without the inner loop

are given by equations of the form

where a > 0, b > 0, and a > b. The graph of limacon without an inner loop does not pass through the pole.

r 5 4sin

r 5 4sin

r 5 3cos

r 5 4cos

Limacons with an inner loop

are given by equations of the form

where a > 0, b > 0, and a < b. The graph of limacon with an inner loop will pass through the pole twice.

r 2 5sin

r 4 5sin r 3 5cos

r 4 5cos

Rose curvesare given by equations of the form

and have graphs that are rose shaped. If n is even and not equal to zero, the rose has 2n petals; if n is odd not equal to +1, the rose has n petals. a represents the length of the petals.

r 4sin 3

r 4sin 4

r 4cos3

r 4cos 4

Lemniscates

are given by equations of the form

and have graphs that are propeller shaped.

4

2

2

4

5 5

2r 9sin 2

4

2

2

4

5 5

2r 9cos 2