Polar Form and Complex Numbers

In a rectangular coordinate system,

There is an x and a y-axis.

In polar coordinates, there is one axis, called the polar axis, and its vertex is called the pole.

While Cartesian Coordinates depend

on x and y values, Polar Coordinates

depend on r and

For any point plotted, it can be

represented by 4 different polar

coordinates.

For example: let’s plot )60,4( 0

Now, plot the ordered pair (3, -210 degrees)

Next, write three other ordered pairs that represent the same point

Distance Formula in the Polar Plane

How can we find the DISTANCE between two pointsDefined in the Polar Plane? Well, we can use the Law ofCosines…

)cos(r2r-rr PP

then...plane,polar the

in points twoare ) ,P(r and ,rP If

12212

22

121

22111

Distance Formula in the Polar Plane

The following relationships exist between Polar Coordinates (r, ) and Rectangular Coordinates (x, y):

Polar vs. Rectangular Forms

x

ytan

222 ryx

cosrx

sinry

8

6

4

2

-2

-5 5 10x

yr

(x, y) ),( r

Rewrite the following polar coordinates in rectangular form:

Polar vs. Rectangular Forms

)120,4( o

Now, rewrite the following rectangular coordinates in polar form: (5, 5).

Polar vs. Rectangular Forms

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.

0

15

30

45

607590105

120

135

150

165

180

195

210

225

240255 270 285

300

315

330

345

43210

Identify and graph the equation: =3

tan tan

3

31

yx

31

y x 3

0

15

30

45

607590105

120

135

150

165

180

195

210

225

240255 270 285

300

315

330

345

43210

3

Identify and graph the equation: r sin 2

sin sin yr

y r

y 2

0

15

30

4560

7590105120

135

150

165

180

195

210

225240

255 270 285300

315

330

34543210

r asin

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

a

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

4

2

2

4

5 5

3cos r

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

0

15

30

4560

7590105120

135

150

165

180

195

210

225240

255 270 285300

315

330

34543210

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.

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.

4

2

2

5 5

sin4r

sin6r2

2

4

6

8

5 5

In order to use your graphing calculator to graph Polar Equations, change your MODE to POLAR (instead of Function). Also, change your viewing window as follows…

For DEGREES:

min = 0max = 360step = 10Xmin = -8Xmax = 8Xscl = 1Ymin = -8Ymax = 8Yscl = 1

For RADIANS:

min = 0max = 2 step = /18Xmin = -8Xmax = 8Xscl = 1Ymin = -8Ymax = 8Yscl = 1

Now that you have your graphing calculator set up to graph Polar Equations, graph the following equations and see if you can identify the shape and how the numbers affect the graph itself…

r = 2 + 2sin r = 2 + 2cosr = 1 + sinr = -2 + -2cosr = 3 + 3sinr = 3 + 3cos

cosaar

Is the graph of a CARDIOID (heart) shape, symmetric to either the x axis (for cosine) or y axis (for sine)

sinaar or

6

4

2

2

4

6

5 5 10

6

4

2

2

4

6

5 5 10

Now graph the following equations and see if you can identify the shape and how the numbers affect the graph itself…

r = 2 + 3sin r = 1 + 2cosr = 1 + 4sinr = 3 + 2cosr = 2 + sinr = 4 + 2cos

cosbar

Is the graph of a Limacon (pronounced “lee-ma-sahn”) shape, symmetric to either the x axis (for cosine) or y axis (for sine)

sinbar or

2

1

1

2

3

4 2 2 4

6

4

2

2

4

6

5 5

cos42 r

Notice how the graph of a limacon changes depending on whether a > b or a < b

sin23r

2

1

1

2

3

4 2 2 4

6

4

2

2

4

6

5 5

a < ba > b

Now graph the following equations and see if you can identify the shape and how the numbers affect the graph itself…

r = 3sin2 r = 2cos4r = 4sin3r = 5cos2r = 3sinr = -3cos3

bar cos

Is the graph of a ROSE shape, symmetric to either the x axis (for cosine) or y axis (for sine)

bar sinor

3

2

1

1

2

3

4 2 2 4

4

3

2

1

1

2

3

4

6 4 2 2 4 6

2cos3r

Notice how the ‘b’ value affects the graph: if b is even, then there are ‘2*b’ number of rose petals (loops); if ‘b’ is odd, there are ‘b’ number of petals

3sin4rand

3

2

1

1

2

3

4 2 2 4

4

3

2

1

1

2

3

4

6 4 2 2 4 6

Below are the graphs of the roses for

The next type of graph we are going to look at involves the following formats for the equation:

and

2cos2 ar 2sin2 ar

However, with the graphing calculator, we cannotType the equations in this fashion.Instead, we take the square root of both sides of the Equation and type that equation into the calculator.For example:

is typed in as2cos92 r 2cos9r

Now graph the following equations and see if you can identify the shape and how the numbers affect the graph itself…

2cos92 r

2sin42 r

2cos162 r

2sin52 r

Is the graph of a lemniscate (pronounced “lem-nah-scut”) shape, symmetric to either the x axis (for cosine) or the line y = x (for sine)

or2cos2 ar 2sin2 ar 4

3

2

1

1

2

3

4

6 4 2 2 4 6

4

3

2

1

1

2

3

4

6 4 2 2 4 6

The next type of graph we are going to look at involves the following format for the equation:

ar

However, with the graphing calculator, we will not beable to see much of the graph if we work with degrees,because r keeps increasing as the angle measure does.So switch to RADIAN MODE and be sure to modifythe X and Y values in WINDOW to accommodate each graph.

Now graph the following equations and see if you can identify the shape and how the numbers affect the graph itself…

r

3r

2r

Is the graph of a Spiral of Archimedes (pronounced “Ar-cah-mee-dees”) shape.

ar 12

10

8

6

4

2

2

4

6

8

10

12

15 10 5 5 10 15 20