This document provides instruction on graphing polar equations. It begins by reviewing polar coordinate planes and then demonstrates how to graph the equation r = 3sinθ by hand. Values for θ and the corresponding r are calculated and plotted. The graph is then drawn on polar graph paper. The document then shows how to graph the same equation using a calculator in both radian and degree mode. Several other example polar equations are presented with the appropriate window settings. Tomorrow's lesson will cover specific families of polar equations.

1. 8.2 Graphs of Polar
Equations
2 Timothy 3:16 "All Scripture is breathed out by God
and profitable for teaching, for reproof, for correction,
and for training in righteousness."

2. We graph on the r-θ plane.
(Review the design of the polar graph paper)

3. We graph on the r-θ plane.
(Review the design of the polar graph paper)
Just as we did when learning how to graph using
a Cartesian Coordinate plane in Algebra, we will
start off by graphing a polar equation “by hand”.

4. We graph on the r-θ plane.
(Review the design of the polar graph paper)
Just as we did when learning how to graph using
a Cartesian Coordinate plane in Algebra, we will
start off by graphing a polar equation “by hand”.
We will set up a table of values for this equation:
r = 3sin θ
“pick a θ , find r “

5. r = 3sin θ
θ r

6. r = 3sin θ
θ r
0°

7. r = 3sin θ
θ r
0° 0

8. r = 3sin θ
θ r
0° 0
30°

9. r = 3sin θ
θ r
0° 0
30° 1.5

10. r = 3sin θ
θ r
0° 0
30° 1.5
45°

11. r = 3sin θ
θ r
0° 0
30° 1.5
45° 2.12

12. r = 3sin θ
θ r
0° 0
30° 1.5
45° 2.12
60°

13. r = 3sin θ
θ r
0° 0
30° 1.5
45° 2.12
60° 2.6

14. r = 3sin θ
θ r
0° 0
30° 1.5
45° 2.12
60° 2.6
90°

15. r = 3sin θ
θ r
0° 0
30° 1.5
45° 2.12
60° 2.6
90° 3

16. r = 3sin θ
θ r θ r
0° 0
30° 1.5
45° 2.12
60° 2.6
90° 3

17. r = 3sin θ
θ r θ r
0° 0 120°
30° 1.5
45° 2.12
60° 2.6
90° 3

18. r = 3sin θ
θ r θ r
0° 0 120° 2.6
30° 1.5
45° 2.12
60° 2.6
90° 3

19. r = 3sin θ
θ r θ r
0° 0 120° 2.6
30° 1.5 135°
45° 2.12
60° 2.6
90° 3

20. r = 3sin θ
θ r θ r
0° 0 120° 2.6
30° 1.5 135° 2.12
45° 2.12
60° 2.6
90° 3

21. r = 3sin θ
θ r θ r
0° 0 120° 2.6
30° 1.5 135° 2.12
45° 2.12 150°
60° 2.6
90° 3

22. r = 3sin θ
θ r θ r
0° 0 120° 2.6
30° 1.5 135° 2.12
45° 2.12 150° 1.5
60° 2.6
90° 3

23. r = 3sin θ
θ r θ r
0° 0 120° 2.6
30° 1.5 135° 2.12
45° 2.12 150° 1.5
60° 2.6 180°
90° 3

24. r = 3sin θ
θ r θ r
0° 0 120° 2.6
30° 1.5 135° 2.12
45° 2.12 150° 1.5
60° 2.6 180° 0
90° 3

25. r = 3sin θ
θ r θ r
0° 0 120° 2.6
30° 1.5 135° 2.12
45° 2.12 150° 1.5
60° 2.6 180° 0
90° 3 210°

26. r = 3sin θ
θ r θ r
0° 0 120° 2.6
30° 1.5 135° 2.12
45° 2.12 150° 1.5
60° 2.6 180° 0
90° 3 210° −1.5

27. r = 3sin θ
θ r θ r
0° 0 120° 2.6
30° 1.5 135° 2.12
45° 2.12 150° 1.5
60° 2.6 180° 0
90° 3 210° −1.5
225°

28. r = 3sin θ
θ r θ r
0° 0 120° 2.6
30° 1.5 135° 2.12
45° 2.12 150° 1.5
60° 2.6 180° 0
90° 3 210° −1.5
225° −2.12

29. Now, sketch these points on your graph paper.
When done, compare your graph with the person
next to you.

30. r = 3sin θ
θ r θ r
0° 0 120° 2.6
30° 1.5 135° 2.12
45° 2.12 150° 1.5
60° 2.6 180° 0
90° 3 210° −1.5
225° −2.12

31. Let’s graph it on your calculator ...
mode: Polar
format: Polar GC
Graph in Radian mode ... and trace
then
Graph in Degree mode ... and trace

32. Let’s graph it on your calculator ...
mode: Polar
format: Polar GC
Graph in Radian mode ... and trace
then
Graph in Degree mode ... and trace
You need to be able to graph polar equations using
your calculator and the trace feature.
Let’s do some examples ...

33. 1. r = 5 zoom standard
zoom square

34. 1. r = 5 zoom standard
zoom square
cos ( 3θ )
2. r = [ −1, 1], [ −1, 1] graph
2

35. 1. r = 5 zoom standard
zoom square
cos ( 3θ )
2. r = [ −1, 1], [ −1, 1] graph
2
3. r = 8 cos ( 2θ ) [ −8, 8 ], [ −8, 8 ]
so zoom standard
zoom square
works ...

36. 4. pg. 592 figure 12
3 ⎛ 5θ ⎞
θ max : 720°
r = sin θ + sin ⎜ ⎟
⎝ 2 ⎠ [ −2, 2 ], [ −2, 2 ]

37. 4. pg. 592 figure 12
3 ⎛ 5θ ⎞
θ max : 720°
r = sin θ + sin ⎜ ⎟
⎝ 2 ⎠ [ −2, 2 ], [ −2, 2 ]
5. pg. 593 figure 13
⎛ 2θ ⎞ θ max : 1080°
r = cos ⎜ ⎟
⎝ 3 ⎠ [ −1, 1], [ −1, 1]
graph
zoom square

38. Tomorrow we will look at some specific “families”
of polar equations.
Do you know how to get your calculator back to
“normal”?
No HW!!!
The price of greatness is responsibility.
Winston Churchill