1. 11.1 Polar Coordinates and Graphs

2. Where is it?
Coordinate systems are used to locate the position of a point.
(3,1)
(4,/6)
In rectangular coordinates:
We break up the plane into a grid of horizontal and vertical line lines.
We locate a point by identifying it as the intersection of a vertical and a horizontal line.
In polar coordinates:
We break up the plane with circles centered at the origin and with rays emanating from the origin.
We locate a point as the intersection of a circle and a ray.

3. (r, ) Polar axis The center of the graph is called the pole.
Angles are measured from the positive x axis.
Points are represented by a radius and an angle
radius
angle
(r, )
To plot the point
Polar axis
First find the angle
Then move out along the terminal side 5

4. A negative angle would be measured clockwise like usual.
To plot a point with a negative radius, find the terminal side of the angle but then measure from the pole in the negative direction of the terminal side.

5. Let's plot the following points:
Notice unlike in the rectangular coordinate system, there are many ways to list the same point.

6. Let's take a point in the rectangular coordinate system and convert it to the polar coordinate system.
(3, 4)
Based on the trig you know can you see how to find r and ? r 4  3
r = 5
We'll find  in radians
(5, 0.93)
polar coordinates are:

7. Let's generalize this to find formulas for converting from rectangular to polar coordinates.
(x, y) r y  x

8. Now let's go the other way, from polar to rectangular coordinates.
Based on the trig you know can you see how to find x and y? 4 y x
rectangular coordinates are:

9. Let's generalize the conversion from polar to rectangular coordinates.y x

10. (8, 210°) (8, 210°) (6, -120°) (-5, 300°) (-5, 300°) (-5, 300°)
Polar coordinates can also be given with the angle in degrees.
(8, 210°)
(8, 210°)
(6, -120°)
(-5, 300°)
(-5, 300°)
(-5, 300°)
(-3, 540°)
(-3, 540°)
(-3, 540°)

11. Try plotting the following points in polar coordinates and find
three additional polar representations of the point:
Make sure to go over “sliding” through the pole to opposite ray when negative r is used.

12. Graphing a Polar Equation
circle
Note that some points are “retraced” as you mark them.

13. R = sin t

14. R = 2+3cos(ϴ)

15. R = 4 cos 2t

16. Hwk. Pg , 3-9, 11