11-1: Polar Coordinates and Complex Numbers

Important things to remember!!
 Converting from polar to rectangular:
1) x = r cos 0
2) y = r sin 0
Converting from rectangular to polar:
 3) r2 = x2 + y2
4) tan 0 = (y/x)
Click for polar graphing calculator! (Manipula Math)
Example problems
1) Change (3, 4) to polar coordinates.
Solution: Using property 3 from above, find r.   32 + 42 = 25 and take the square root.
                Therefore, r = 5
Using property 4 from above, tan 0 = (4/3). Use your calculator set to degree mode, the answer is: 53.1 degrees. (rounded to nearest tenth.
                Therefore, the point is ( 5, 53.1o)
2) Change ( 4, 150o) to rectangular coordinates
Solution: Using property 1 from above x = 4 cos 150. Using your calculator you get x = -3.46 rounded to hundredths.
Using property 2 from above y = 4 sin 150. Using your calculator you get y = 2
             Therefore, the point is ( -3.46, 2)
 3) Change (-3, -7) to polar coordinates
Solution: Using property 3 from above find r. Square -3 and add to the square of -7 you get 58 and taking the square root on your calculator means r = 7.6   Piece of cake!
To find theta we use property 4, tan 0 = 7/3. Notice I used positive values! I did this to find the reference angle in quad I. The reference angle is equal to 66.8 degrees. Since we are in quadrant III, (look at the signs of the original problem) we should add 180. Why? ( we are in quad III and we memorized we add 180 in that quad, right?) So the angle is 246.8o.
            Therefore, the point is (7.6, 246.8o) ( all to simple!!!)
You can use the special functions on the TI-82 to find all the above answers.  Make sure your calculator is set in degree mode(if you want the answer in degrees) and press 2nd function angle.  This takes you to a menu that has choice 5 and 6 to change to polar form.  Choice 5 gives you the radius and choice 6 gives you the theta value.  Example (1,3) Get to choice 5 on your calculator: You should see this R arrow Pr ( you type 1, 3) and the calculator gives you 3.16227766. This is r.  Round it and your all set.  Then go back to the angle menu and choose 6.  You should see R arrow P0 type 1,3) again and get 71.56505118 round this and the point would be ( 3.2, 71.6o)
   (Yikes stripes this easy!!)
Do basically the same thing to find x and y by using choices 7 and 8. If the problem is in degrees set calc to degrees and if in rads make sure calc is in rads.
Sketching graphs in polar form using the TI-82
Graphs are sketched at the bottom of the page!
To use your graphing calculator is extremely easy to graph in polar form.  First, get to the mode screen.  Make sure you highlight the radian button and drop down one line and highlight the pol button.  Now hit the y= key. Instead of y= it's now r= !!  Amazing what modern technology will do!!  When you hit the x, t, theta button, it now shows a theta!!  Double wowie!
Try a few problems
1) r = 3 sin 0 Go ahead, type it in and hit graph.  Did you get something that sort of looked like a circle?  Hit zoom then choice 5.  Look better?  It should be a circle tangent at the origin and translated up.  The top point is (0, 3)
2) r = 4 cos 0 Try this one.  Hit y= and type it in.  Hit graph.  This time, you should get a circle tangent again to the origin but moved to the right.  The point farthest right is (2, 0).  Wonder if this is a pattern?  Sine up or down and cosine right or left?
3) r = - 5 sin 0    Circle moved down
4) r = -3 cos 0    Circle moved left.
5) r = 3 sin 4 Interesting graph.   Looks like 8 leaves on a rose.  Hit zoom and take choice 1. Move the cursor to a corner of the picture and hit enter.  Hit the down arrow and make the line bigger than the picture.  Hit enter when it is.  Now stretch it across until it makes a box around the picture.  Hit enter  Now you see a better picture of it.  The number of leaves depend on the number multiplying 0.  If it is odd that's how many leaves you get.  If it is even like this one, you get double the number of leaves.  Let's see, the number for this problem was 4, I double it and I should have 8 leaves.   Count them on the screen.   How about that.  To be a picture like this, the number multiplying 0 must be bigger than 1 and a whole number.
 6) r = 4 cos 30    Same as above with 3 leaves.
 7) r = 3 cos 20   Another leaved rose.  This time 4 leaves.  Remember after each problem to reset the calculator to standard zoom by hitting zoom then 6
 8) r = 2 + 2 cos 0.  A cardiod.  (heart- shaped)
 9) r = 1 + 2 sin 0    A limicon
 10) r = sec 0      What is this?   A vertical line if you remember sec is 1/cos

Let's head on toward the next section.  Geometric representation of complex numbers!!
Sounds like fun!