Section 9.1 Polar Coordinates

1. Section 9.1Polar Coordinates

2. Polar axis x Origin Pole

3. r Polar axis O Pole

4. The Polar Plane Coordinates (r, θ)

5. 4 Polar axis O Pole

7. 4 O 7

8. 8

9. Section 9.2Polar Equations and Graphs 9

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

13. Let a be a nonzero real number, the graph of the equation is a horizontal line y = a

14. Let a be a nonzero real number, the graph of the equation is a vertical line x = a

15. 15

16. 16

17. Let a be a positive or negative real number. Then, Circle: radius a ; center at (0, a) Circle: radius a ; center at (a, 0). 17

18. 18 Symmetry with Respect to the Polar Axis (x-axis):

19. Symmetry with Respect to the Line (y-axis)

20. Symmetry with Respect to the Pole (Origin):

21. Tests for Symmetry Symmetry with Respect to the Polar Axis (x-axis): Replace θ by - θ Symmetry with Respect to the Line (y-axis): Replace θ by Π - θ Symmetry with Respect to the Pole (Origin): Replace r by -r If an equivalent equation results then the graph is symmetric with respect to the given pole or line. 21

22. Specific Types of Polar Graphs Cardioids (heart shaped) where a > 0. The graph passes through the pole. Limacons without an inner loop (French word for snail) where a > 0, b > 0, and a > b. The graph does not pass through the pole. 22

23. Limacons with an inner loop (French word for snail) where a > 0, b > 0, and a < b. The graph passes through the pole twice. Rose Curves If n is even and not zero, the graph has 2n petals. If n is odd and not one or negative one, the graph has n petals.

24. Lemniscates (Greek word for propeller) where a is non-zero. The graph will be propeller shaped. 24

25. Section 9.3The Complex Plane 25

26. The Complex Plane Imaginary Axis Real Axis O

27. Imaginary Axis y Real Axis O x z is the magnitude of z = x + yi 27

28. Cartesian Form Polar Form AND 28

29. Imaginary Axis 4 Real Axis -3 29

32. DeMoivre’s Theorem 32

35. 35

36. Section 9.4Vectors 36

37. A vector is a quantity that has both magnitude and direction. Vectors in the plane can be represented by arrows. The length of the arrow represents the magnitude of the vector. The arrowhead indicates the direction of the vector.

38. Q Terminal Point Initial Point P

39. if they have the same magnitude and direction. The vector v whose magnitude is 0 is called the zero vector, 0. Two vectors v and w are equal, written

40. Vector Addition Terminal point of w Initial point of v

41. Vector addition is commutative. Vector addition is associative.

42. Properties of Scalar Products 42

43. Use the vectors illustrated below to graph each expression. 43

46. 46

47. 47

48. Let i be a unit vector along the pos. x-axis; Let j be a unit vector along the pos. y-axis. If v has initial point at the origin O and terminal point at P = (a, b), then 48

49. P = (a, b) b v = ai + bj a The scalars a and b are called components of the vector v = ai + bj.

50. Position Vector The position vector re-positions the vector so that the initial point is the origin. 50