Section 6.3

Polar Coordinates

Another Way of Graphing

• The rectangular coordinate system we regularly use is not the only means we have of graphing points.

• Start with a horizontal ray that extends to the right.

• This ray is called the polar axis and the endpoint is called the pole.

Polar Coordinates

• A point P in the polar coordinate system is in the form

• Where r is the distance from the pole to the point P, and

• is the angle between the polar axis and the line segment that contains P.

,r

Graphing polar coordinates

• Start by graphing a ray at an angle of “theta” with the polar axis (counterclockwise if “theta” if positive, clockwise if “theta” if negative).

• If r is positive, point P is on the ray. If r is negative, point P is on the line segment opposite the ray.

Examples

• Graph each of the following:

120,1

3

5,2

2,3

45,4

One point, at least four different names

• Because both r and “theta” can be either positive or negative, it is possible to name a point P in at least four different ways.

• Consider the point

4

3,5

P

Converting from polar to rectangular

• Start by superimposing the coordinate grid so that the pole of the polar system lies on top of the origin of the rectangular system.

• Angle “theta” will be in standard position in the rectangular system because the polar axis will lie along the positive x-axis.

Important formulas

x

y

ryx

ry

rx

tan

sin

cos

222

From Polar to Rectangular

2

3,3

4

7,4

180,2

From Rectangular to Polar

0,9

5,35

6,6

Polar and Rectangular Equations

• The same formulas we use to convert polar coordinates to rectangular coordinates (and vice versa) can also be used to convert polar equations (they have r’s and θ’s) to rectangular equations (they have x’s and y’s).

• Don’t be afraid to be a little “creative” in your thinking.

Convert Each of the Following

sin3cos6

csc9

64

10

22

r

r

yx

y