(r, )

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.

(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

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.

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

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:

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

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:

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

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°)

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.

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.

When trying to figure out the graphs of polar equations we can convert them to rectangular equations particularly if we recognize the graph in rectangular coordinates. We could square both sides Now use our conversion: We recognize this as a circle with center at (0, 0) and a radius of 7. On polar graph paper it will centered at the origin and out 7

Let's try another: Take the tangent of both sides To graph on a polar plot we'd go to where and make a line. Now use our conversion: Multiply both sides by x We recognize this as a line with slope square root of 3.

Let's try another: Now use our conversion: We recognize this as a horizontal line 5 units below the origin (or on a polar plot below the pole)

I still don't know what the graph looks like! Sometimes converting to rectangular equations doesn't help us figure out what the graph would look like or it is not necessary. The only way we know how to convert is if there is an r in front of the sin  term so we'll multiply both sides by r. Now use our conversions: I still don't know what the graph looks like! UGH! In these cases we'll plot points, choosing a  from the polar form and finding a corresponding r value.

This type of graph is called a cardioid. Let's let each unit be 1/4. Let's plot the symmetric points

r = a(1 + cos) r = a(1 + sin) r = a(1 - cos) r = a(1 - sin) Equations of cardioids would look like one of the following: r = a(1 + cos) r = a(1 + sin) r = a(1 - cos) r = a(1 - sin) where a > 0 All graphs of cardioids pass through the pole.

This type of graph is called a limacon without an inner loop. Let's let each unit be 1. Let's plot the symmetric points

r = a +b cos r = a +b sin r = a - b cos r = a - b sin Equations of limacons without inner loops would look like one of the following: r = a +b cos r = a +b sin r = a - b cos r = a - b sin where a > 0, b > 0, and a > b These graphs DO NOT pass through the pole.

This type of graph is called a limacon with an inner loop. Let's let each unit be 1/2. Let's plot the symmetric points

r = a +b cos r = a +b sin r = a - b cos r = a - b sin Equations of limacons with inner loops would look like one of the following: r = a +b cos r = a +b sin r = a - b cos r = a - b sin where a > 0, b > 0, and a < b These graphs will pass through the pole twice.

This type of graph is called a rose with 4 petals. Let's let each unit be 1/2. Let's plot the symmetric points

Where n even has 2n petals and n odd has n petals (n  0 or  1) Equations of rose curves would look like one of the following: r = a cos(n) r = a sin(n) Where n even has 2n petals and n odd has n petals (n  0 or  1)

This type of graph is called a lemniscate Let's let each unit be 1/4.

Equations of lemniscates would look like one of the following: r2 = a2 cos(2) r2 = a2 sin(2) These graphs will pass through the pole and are propeller shaped.

Function Gallery in your book on page 352 summarizes all of the polar graphs. You can graph these on your calculator. You'll need to change to polar mode and also you must be in radians. If you are in polar function mode when you hit your button to enter a graph you should see r1 instead of y1. Your variable button should now put in  on TI-83's and it should be a menu choice in 85's & 86's.

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