1. Polar Form and Complex Numbers

2. In a rectangular coordinate system, There is an x and a y-axis. In polar coordinates, there is one axis, called the polar axis, and its vertex is called the pole.

4. While Cartesian Coordinates depend on x and y values, Polar Coordinates depend on r and

5. For any point plotted, it can be represented by 4 different polar coordinates. For example: let’s plot

7. Now, plot the ordered pair (3, -210 degrees) Next, write three other ordered pairs that represent the same point

8. Distance Formula in the Polar Plane How can we find the DISTANCE between two points Defined in the Polar Plane? Well, we can use the Law of Cosines…

9. Distance Formula in the Polar Plane

10. The following relationships exist between Polar Coordinates (r,  ) and Rectangular Coordinates (x, y): Polar vs. Rectangular Forms

11. Rewrite the following polar coordinates in rectangular form: Polar vs. Rectangular Forms

12. Now, rewrite the following rectangular coordinates in polar form: (5, 5). Polar vs. Rectangular Forms

13. An equation whose variables are polar coordinates is called a polar equation. The graph of a polar equation consists of all points whose polar coordinates satisfy the equation.

14. Identify and graph the equation: r = 2 Circle with center at the pole and radius 2.

20. is a horizontal line a units above the pole if a > 0 and units below the pole if a < 0.

21. is a vertical line a units to the right of the pole if a > 0 and units to the left of the pole if a < 0.

25. Circle: radius a; center at (a, 0) in rectangular coordinates. Circle: radius a; center at (-a, 0) in rectangular coordinates.

26. Circle: radius a; center at (0, a) in rectangular coordinates. Circle: radius a; center at (0, -a) in rectangular coordinates.

29. In order to use your graphing calculator to graph Polar Equations, change your MODE to POLAR (instead of Function). Also, change your viewing window as follows… For DEGREES:  min = 0  max = 360  step = 10 Xmin = -8 Xmax = 8 Xscl = 1 Ymin = -8 Ymax = 8 Yscl = 1 For RADIANS:  min = 0  max = 2   step =  /18 Xmin = -8 Xmax = 8 Xscl = 1 Ymin = -8 Ymax = 8 Yscl = 1

30. Now that you have your graphing calculator set up to graph Polar Equations, graph the following equations and see if you can identify the shape and how the numbers affect the graph itself… r = 2 + 2sin  r = 2 + 2cos  r = 1 + sin  r = -2 + -2cos  r = 3 + 3sin  r = 3 + 3cos 

31. Is the graph of a CARDIOID (heart) shape, symmetric to either the x axis (for cosine) or y axis (for sine) or

32. Now graph the following equations and see if you can identify the shape and how the numbers affect the graph itself… r = 2 + 3sin  r = 1 + 2cos  r = 1 + 4sin  r = 3 + 2cos  r = 2 + sin  r = 4 + 2cos 

33. Is the graph of a Limacon (pronounced “lee- ma-sahn”) shape, symmetric to either the x axis (for cosine) or y axis (for sine) or

34. Notice how the graph of a limacon changes depending on whether a > b or a < b a < b a > b

35. Now graph the following equations and see if you can identify the shape and how the numbers affect the graph itself… r = 3sin2  r = 2cos4  r = 4sin3  r = 5cos2  r = 3sin  r = -3cos3 

36. Is the graph of a ROSE shape, symmetric to either the x axis (for cosine) or y axis (for sine) or

37. Notice how the ‘b’ value affects the graph: if b is even, then there are ‘2*b’ number of rose petals (loops); if ‘b’ is odd, there are ‘b’ number of petals and Below are the graphs of the roses for

38. The next type of graph we are going to look at involves the following formats for the equation: and However, with the graphing calculator, we cannot Type the equations in this fashion. Instead, we take the square root of both sides of the Equation and type that equation into the calculator. For example: is typed in as

39. Now graph the following equations and see if you can identify the shape and how the numbers affect the graph itself…

40. Is the graph of a lemniscate (pronounced “lem-nah-scut”) shape, symmetric to either the x axis (for cosine) or the line y = x (for sine) or

41. The next type of graph we are going to look at involves the following format for the equation: However, with the graphing calculator, we will not be able to see much of the graph if we work with degrees, because r keeps increasing as the angle measure does. So switch to RADIAN MODE and be sure to modify the X and Y values in WINDOW to accommodate each graph.

42. Now graph the following equations and see if you can identify the shape and how the numbers affect the graph itself…

43. Is the graph of a Spiral of Archimedes (pronounced “Ar-cah-mee-dees”) shape.