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MAC 1114

MAC 1114. Module 10 Polar Form of Complex Numbers. Rev.S08. Learning Objectives. Upon completing this module, you should be able to: Identify and simplify imaginary and complex numbers. Add and subtract complex numbers. Simplify powers of i . Multiply complex numbers.

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MAC 1114

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  1. MAC 1114 • Module 10 • Polar Form of Complex Numbers Rev.S08

  2. Learning Objectives • Upon completing this module, you should be able to: • Identify and simplify imaginary and complex numbers. • Add and subtract complex numbers. • Simplify powers of i. • Multiply complex numbers. • Use property of complex conjugates. • Divide complex numbers. • Solve quadratic equations for complex solutions. • Convert between rectangular form and trigonometric (polar) form. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  3. Polar Form of Complex Numbers There are two major topics in this module: - Complex Numbers - Trigonometric (Polar) Form of Complex Numbers http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  4. Quick Review on Complex Numbers • and • iis the imaginary unit • Numbers in the forma + biare called complex numbers • a is the real part • b is the imaginary part http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  5. Examples • a) b) • c) • d) e) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  6. Example of Solving Quadratic Equations • Solve x = −25 • Take the square root on both sides. • The solution set is {±5i}. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  7. Another Example • Solve: x2 + 54 = 0 • The solution set is http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  8. Multiply and Divide • Divide: • Multiply: http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  9. (4 − 6i) + (−3 + 7i) = [4 + (−3)] + [−6 + 7]i = 1 + i (10 − 4i) − (5 − 2i) = (10 − 5) + [−4 − (−2)]i = 5 − 2i Addition and Subtraction of Complex Numbers • For complex numbersa + bi and c + di, • Examples http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  10. Multiplication of Complex Numbers • For complex numbersa + 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  11. Let’s Practice Some Multiplication of Complex Numbers • (7 + 3i)2 • (2 − 4i)(3 + 5i) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  12. Powers of i • i1 = ii5 = i i9 = i • i2 = −1 i6 = −1 i10 = −1 • i3 = −i i7 = −i i11 = −i • i4 = 1 i8 = 1 i12 = 1 • and so on. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  13. Simplifying Examples • i−4 • i17 • Since i4 = 1, • i17 = (i4)4 • i • = 1 • i • = i http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  14. Properties of Complex Conjugates • Example • 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  15. Complex Plane • 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). http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  16. Relationships Among z, y, r, and θ http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  17. 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 θ + i sin θis sometimes abbreviated cis θ. • Using this notation http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  18. 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. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  19. How to Convert from Rectangular Form to Polar 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 θ by using the equation • choosing the quadrant indicated in Step 1. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  20. Example • Example: Find trigonometric notation for −1 −i. • First, find r. • Thus, http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  21. What have we learned? • We have learned to: • Identify and simplify imaginary and complex numbers. • Add and subtract complex numbers. • Simplify powers of i. • Multiply complex numbers. • Use property of complex conjugates. • Divide complex numbers. • Solve quadratic equations for complex solutions. • Convert between rectangular form and trigonometric (polar) form. http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

  22. Credit • Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: • Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th Edition http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Rev.S08

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