1. (r, )

2. We are going to look at a new coordinate system called the polar coordinate system.
You are familiar with plotting with a rectangular coordinate system.

3. (r, ) 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
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. Polar coordinates can also be given with the angle in degrees.
(8, 210°)
330
315
300
270
240
225
210
180
150
135
120 0
90
60
30
45
(6, -120°)
(-5, 300°)
(-3, 540°)

11. Here each r unit is 1/2 and we went out 3 and did all angles.
Convert the rectangular coordinate system equation to a polar coordinate system equation. 
Here each r unit is 1/2 and we went out 3 and did all angles.
r must be  3 but there is no restriction on  so consider all values.
Before we do the conversion let's look at the graph.

12. substitute in for x and y
Convert the rectangular coordinate system equation to a polar coordinate system equation.
What are the polar conversions we found for x and y?
substitute in for x and y
We wouldn't recognize what this equation looked like in polar coordinates but looking at the rectangular equation we'd know it was a parabola.

13. Graphing Polar Equations on the TI-84
Hit the MODE key.
Arrow down to where it says Func (short for "function" which is a bit misleading since they are all functions).
Now, use the right arrow to choose Pol.
Hit ENTER. (*It's easy to forget this step, but it's crucial: until you hit ENTER you have not actually selected Pol, even though it looks like you have!)

14. Polar Graphs
You will notice that polar equations have graphs like the following:

15. Graphing Polar Equations on the TI-84
The calculator is now in polar coordinates mode. To see what that means, try this.
Hit the Y= key. Note that, instead of Y1=, Y2=, and so on, you now have r1= and so on.
In the r1= slot, type 5-5sin(θ)
Now hit the familiar X,T,θ,n key, and you get an unfamiliar result. In polar coordinates mode, this key gives you a θ instead of an X.
Finally, close off the parentheses and hit GRAPH.

16. Graphing Polar Equations on the TI-84
If you did everything right, you just asked the calculator to graph the polar equation r=5-5sin(θ). The result looks a bit like a valentine.

17. Graphing Polar Equations on the TI-84
The WINDOW options are a little different in this mode too. You can still specify X and Y ranges, which define the viewing screen. But you can also specify the θ values that the calculator begins and ends with.

18. Graphing Polar Equations on the TI-84
For instance, you may limit the graph to 0<θ<π/2. This would not change the viewing window, but it would only draw part of the graph.

19. Graphing Polar Equations on the TI-84
Graph r = 3 sin 2θ
Enter the following window values:
Θmin = 0 Xmin = -6 Ymin = -4
θmax = 2π Xmax = 6 Ymax = 4
Θstep = π/24 Xscl = 1 Yscl = 1

20. Rose with 7 petals made with graphing program on computer
Limacon With Inner Loop made with TI Calculator
Have fun plotting pretty pictures!

21. Acknowledgement
I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.
Shawna has kindly given permission for this resource to be downloaded from and for it to be modified to suit the Western Australian Mathematics Curriculum.
Stephen Corcoran
Head of Mathematics
St Stephen’s School – Carramar