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COMPLEX NUMBERS AND POLAR COORDINATES 8.1 Complex Numbers 8.2 Trigonometric Form for Complex Numbers

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COMPLEX NUMBERS AND POLAR COORDINATES 8.1 Complex Numbers 8.2 Trigonometric Form for Complex Numbers

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    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

    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: x2 + 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

    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 i2 = ?1.

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

    10. Slide 8-10 Powers of i i1 = i i5 = i i9 = i i2 = ?1 i6 = ?1 i10 = ?1 i3 = ?i i7 = ?i i11 = ?i i4 = 1 i8 = 1 i12 = 1 and so on.

    11. Slide 8-11 Simplifying Examples i17 i4 = 1 i17 = (i4)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) = a2 + b2. 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).

    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 1 Sketch a graph of the number x + yi in the complex plane. Step 2 Find r by using the equation Step 3 Find ? 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

    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 ?

    39. Slide 8-39 Example Graph r = 2 cos 3?

    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

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