Polar Coordinates and Graphs of Polar Equations. Copyright by Houghton Mifflin Company, Inc. All...

Polar Coordinates and Graphs of Polar

Equations

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

The polar coordinate system is formed by fixing a point, O, which is the pole (or origin).

= directed angle Polar axis

r = directed distance

OPole (Origin)

The polar axis is the ray constructed from O.

Each point P in the plane can be assigned polar coordinates (r, ).

P = (r, )

r is the directed distance from O to P. is the directed angle (counterclockwise) from the polar axis to OP.


The point lies two units from the pole on the

terminal side of the angle

1 2 3 0

3 units from the pole

Plotting Points

The point lies three

units from the pole on the terminal

side of the angle


There are many ways to represent the point

1 2 3 0

2, 3

additional ways

to represent the

point

Find the other representations for the point


)4

3,3(

1 2 3 0

)4

3,3(

)4

5,3(

)4

7,3(

)4

,3(

Stop

• Warm Up. Graph and find the other 3 representations.


)3

2,2(

1 2 3 0


(r, )(x, y)

Pole x

y

(Origin)

yr

x

The relationship between rectangular and polar coordinates is as follows.

The point (x, y) lies on a circle of radius r, therefore,

r2 = x2 + y2.

Definitions of trigonometric functions


Coordinate Conversion

(Pythagorean Identity)

Example:Convert the point into rectangular coordinates (x, y).

1cos co 3 24 s 4 2x r

3sin sin 4 23 24 3y r


Example:Convert the point (1,1) into polar coordinates.

, 1,1x y

1tan 11yx

4

2 2 2 21 1 2r x y

set of polar coordinates is ( , ) 2, .4One r

5Another set is ( , ) 2, .4r Stop

Warm Up

•Convert the following point from polar to rectangular

•Convert the following point from rectangular to polar: (-4, 1)


)3

2,1(

Convert rectangular to polar equations and polar to rectangular equations.


Graph polar equations


Example:Convert the polar equation into a rectangular equation.

4sinr

4sinr 2 4 sinr r Multiply each side by r.2 2 4x y y Substitute rectangular

coordinates.

22 2 4x y Equation of a circle with center (0, 2) and radius of 2

Polar form

2 2 4 0x y y


Example:Convert the polar equation into a rectangular equation. 3

5

yx

tan

33

5tan

xy

3 yx 3


Example:Convert the rectangular equation x2 + y2 – 6x = 0 into a polar equation.


Example:Graph the polar equation r = 2cos .

1 2 3 0

2

0

–2

–101

20r

6

3

2

23

56

76

32

116

2

3

3

3

3 The graph is a circle of radius 1 whose center is at

point (x, y) = (1, 0).


If substitution leads to equivalent equations, the graph of a polar equation is symmetric with respect to one of the following.

1. The line 2

2. The polar axis 3. The pole

Replace (r, ) by (r, – ) or (–r, – ).

Replace (r, ) by (r, – ) or (–r, – ).

Replace (r, ) by (r, + ) or (–r, ).

Example:In the graph r = 2cos , replace (r, ) by (r, – ).

r = 2cos(–) = 2cos

The graph is symmetric with respect to the polar axis. cos(–) = cos


Example:Find the zeros and the maximum value of r for the graph of r = 2cos .

1 2 3 0

The maximum value of r is 2.

It occurs when = 0 and 2. 0 when 3 and .2 2

r

These are the zeros of r.


Each polar graph below is called a Limaçon.

1 2cosr 1 2sinr

–3

–5 5

3

–5 5

3

–3Note the symmetry of each graph.

What does the symmetry have in common with the trig function?


Each polar graph below is called a Rose curve.

The graph will have n petals if n is odd, and 2n petals if n is even. And, again, note the symmetry.

–5 5

3

–3

–5 5

3

–3

a

a

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