1. Polar Coordinates graphing on a circular coordinate system

2. This is a polar graph... The "pole" is the centre of this graph. A series of concentric circles use this pole as their common centre. The pole is sometimes referred to as "O" Extending to the right is the "polar axis" drawn from the pole, indicating the positive direction.

3. This is a polar graph... the polar axis indicates the radius of each concentric circle First circle: r = 1 Second circle: r = 2 Third circle: r = 3 etc., etc., etc...

4. Check out point "P" How do we express the coordinates of this point? Well, first off, let me ask you what is the radius of the circle that point P lies on? This is the first part of point P's coordinates (5, ?) P 5 The second part of the coordinate is the angular measure θ counter-clockwise between the radius OP and the polar axis θ = 130

5. So,point "P" is... ( 5, 130 ̊ ) Funny thing about polar graphs is that there is more than one way to express the same point (one being not in degrees). What's another way of expressing the coordinates of this point? ( 5, ) P Oh yeah......with RADIANS, baby!!! θ =130 ̊

6. This point...hmmm, wait. This point can be expressed by yet another set of coordinates. What if the θ was not positive but negative? Would the second coordinate still be 130 ̊ ? P θ = -230 NO, baby!!!

7. Point "P" Only a positive θ measures counter-clockwise from the polar axis. If you express the location of point "P" with a negative value as the second coordinate, you will always measures clockwise from the polar axis ( 5, -230 ̊ ) P Oh yeah... this can be expressed in pi radians as well θ =130 ̊ θ =-230 ̊

8. This point...it can be expressed even another way!!! Say, the θ returned to being positive again......but we changed the radius so that it was negative, not positive... (-5, ?) Would the second coordinate still be 130 ̊ ? P θ = 130 NO, baby!!!

9. This point (-5, ?) What do we do? See, a negative value of r will cause a reflection across the pole We start here If θ is positive, the angle is still measured counter-clockwise. P θ = 310 (-5, 310 ̊ )

10. Okay, so there are infinite ways to express that point actually... a circular graph allows you to express periodic behavior so, we can go on, and on, and on, if we like That same point can be expressed as... (5, 490 ̊ ) (5, 850 ̊ ) (5, 1210 ̊ ) etc., etc., etc...

11. (7, 850 ̊ )

12. Take out your polar coordinate paper Graph point Q(4, 235 ̊ ). Find other coordinates for the same point Q t hat satisfy each of the following conditions a) r is negative b) θ is negative c) θ > 360 degrees (-4, 415 ̊ ) (4, -125 ̊ ) (4, 595 ̊ )

13. What are the polar coordinates of the pole? Is there more than one possible set of coordinates? The most commonly used set of coordinates for the pole is (0, 0 ̊ ). However, since r = 0, the value of θ could be any angle. All values of θ will have a point at the pole if r = 0.