1. Slide 6-1 COMPLEX NUMBERS AND POLAR COORDINATES 8.1 Complex Numbers 8.2 Trigonometric Form for Complex Numbers Chapter 8

2. Slide 8-2 and i is the imaginary unit Numbers in the form a + bi are called complex numbers a is the real part b is the imaginary part 8.1 Complex Numbers

3. Slide 8-3 Examples a)b) c) d) e)

4. Slide 8-4 Example: Solving Quadratic Equations Solve x = 25 Take the square root on both sides. The solution set is { 5i}.

5. Slide 8-5 Another Example Solve: x 2 + 54 = 0 The solution set is

6. Slide 8-6 Example: Products and Quotients Multiply: Divide:

7. Slide 8-7 Addition and Subtraction of Complex Numbers For complex numbers a + bi and c + di, Examples (10 4i) (5 2i) = (10 5) + [ 4 ( 2)]i = 5 2i (4 6i) + ( 3 + 7i) = [4 + ( 3)] + [ 6 + 7]i = 1 + i

8. Slide 8-8 Multiplication of Complex Numbers For complex numbers a + bi and c + di, The product of two complex numbers is found by multiplying as if the numbers were binomials and using the fact that i 2 = 1.

9. Slide 8-9 Examples: Multiplying (2 4i)(3 + 5i) (7 + 3i) 2

10. Slide 8-10 Powers of i i 1 = ii 5 = ii 9 = i i 2 = 1i 6 = 1i 10 = 1 i 3 = ii 7 = i i 11 = i i 4 = 1 i 8 = 1 i 12 = 1 and so on.

11. Slide 8-11 Simplifying Examples i 17 i 4 = 1 i 17 = (i 4 ) 4 i = 1 i = i i 4

12. Slide 8-12 Property of Complex Conjugates For real numbers a and b, (a + bi)(a bi) = a 2 + b 2. The product of a complex number and its conjugate is always a real number. Example

13. Slide 8-13 We modify the familiar coordinate system by calling the horizontal axis the real axis and the vertical axis the imaginary axis. Each complex number a + bi determines a unique position vector with initial point (0, 0) and terminal point (a, b). 8.2 Trigonometric Form for Complex Numbers

14. Slide 8-14 Relationships Among x, y, r, and

15. Slide 8-15 Trigonometric (Polar) Form of a Complex Number The expression is called the trigonometric form or (polar form) of the complex number x + yi. The expression cos + i sin is sometimes abbreviated cis. Using this notation

16. Slide 8-16 Example Express 2(cos 120 + i sin 120 ) in rectangular form. Notice that the real part is negative and the imaginary part is positive, this is consistent with 120 degrees being a quadrant II angle.

17. Slide 8-17 Converting from Rectangular Form to Trigonometric Form Step 1Sketch a graph of the number x + yi in the complex plane. Step 2Find r by using the equation Step 3Find by using the equation choosing the quadrant indicated in Step 1.

18. Slide 8-18 Example Example: Find trigonometric notation for 1 i. First, find r. Thus,

19. Slide 8-19 Product Theorem If are any two complex numbers, then In compact form, this is written

20. Slide 8-20 Example: Product Find the product of

21. Slide 8-21 Quotient Theorem If are any two complex numbers, where then

22. Slide 8-22 Example: Quotient Find the quotient.

23. Slide 8-23 De Moivres Theorem If is a complex number, and if n is any real number, then In compact form, this is written

24. Slide 8-24 Example: Find ( 1 i) 5 and express the result in rectangular form. First, find trigonometric notation for 1 i Theorem

25. Slide 8-25 nth Roots For a positive integer n, the complex number a + bi is an nth root of the complex number x + yi if

26. Slide 8-26 nth Root Theorem If n is any positive integer, r is a positive real number, and is in degrees, then the nonzero complex number r(cos + i sin ) has exactly n distinct nth roots, given by where

27. Slide 8-27 Example: Square Roots Find the square roots of Trigonometric notation: For k = 0, root is For k = 1, root is

28. Slide 8-28 Example: Fourth Root Find all fourth roots of Write the roots in rectangular form. Write in trigonometric form. Here r = 16 and = 120. The fourth roots of this number have absolute value

29. Slide 8-29 Example: Fourth Root continued There are four fourth roots, let k = 0, 1, 2 and 3. Using these angles, the fourth roots are

30. Slide 8-30 Example: Fourth Root continued Written in rectangular form The graphs of the roots are all on a circle that has center at the origin and radius 2.

31. Slide 8-31 Polar Coordinate System The polar coordinate system is based on a point, called the pole, and a ray, called the polar axis.

32. Slide 8-32 Rectangular and Polar Coordinates If a point has rectangular coordinates (x, y) and polar coordinates (r, ), then these coordinates are related as follows.

33. Slide 8-33 Example Plot the point on a polar coordinate system. Then determine the rectangular coordinates of the point. P(2, 30 ) r = 2 and = 30, so point P is located 2 units from the origin in the positive direction making a 30 angle with the polar axis.

34. Slide 8-34 Example continued Using the conversion formulas: The rectangular coordinates are

35. Slide 8-35 Example Convert (4, 2) to polar coordinates. Thus (r, ) =

36. Slide 8-36 Rectangular and Polar Equations To convert a rectangular equation into a polar equation, use and and solve for r. you will get the polar equation For the linear equation

37. Slide 8-37 Example Convert x + 2y = 10 into a polar equation. x + 2y = 10

38. Slide 8-38 Example Graph r = 2 sin 1 1.414 2 0 - 1.414 r 330 315 270 180 150 135 - 1.732 120 -290 - 1.732 60 - 1.414 0 r 45 30 0

39. Slide 8-39 Example Graph r = 2 cos 3 0 1.41 2 0 2 r 9075604530150

40. Slide 8-40 Example Convert r = 3 cos sin into a rectangular equation.

41. Slide 8-41 Circles and Lemniscates

42. Slide 8-42 Limacons

43. Slide 8-43 Rose Curves