2. Copyright © 2009 Pearson Education, Inc. CHAPTER 8: Applications of Trigonometry 8.1The Law of Sines 8.2The Law of Cosines 8.3Complex Numbers: Trigonometric Form 8.4Polar Coordinates and Graphs 8.5Vectors and Applications 8.6Vector Operations

3. Copyright © 2009 Pearson Education, Inc. 8.3 Complex Numbers: Trigonometric Form  Graph complex numbers.  Given a complex number in standard form, find trigonometric, or polar, notation; and given a complex number in trigonometric form, find standard form.  Use trigonometric notation to multiply and divide complex numbers.  Use DeMoivre’s theorem to raise a complex number to powers.  Find the nth roots of a complex number.

4. Slide 8.3 - 4 Copyright © 2009 Pearson Education, Inc. Graphical Representation Just as real numbers can be graphed on a line, complex numbers can be graphed on a plane. We graph a complex number a + bi in the same way that we graph an ordered pair of real numbers (a, b). However, in place of an x-axis,we have a real axis, and in place of a y-axis, we have an imaginary axis. Horizontal distances correspond to the real part of a number. Vertical distances correspond to the imaginary part.

5. Slide 8.3 - 5 Copyright © 2009 Pearson Education, Inc. Example Graph each of the following complex numbers. a) 3 + 2ib) –4 – 5ic) –3id) –1 + 3ie) 2 Solution:

6. Slide 8.3 - 6 Copyright © 2009 Pearson Education, Inc. Absolute Value The absolute value of a complex number a + bi is

7. Slide 8.3 - 7 Copyright © 2009 Pearson Education, Inc. Example Find the absolute value of each of the following. Solution:

8. Slide 8.3 - 8 Copyright © 2009 Pearson Education, Inc. Trigonometric Notation If we let  be an angle in standard position whose terminal side passes through the point (a, b), then

9. Slide 8.3 - 9 Copyright © 2009 Pearson Education, Inc. Trigonometric Notation Trigonometric Notation for Complex Numbers To find trigonometric notation for a complex number given in standard notation a + bi, we must find r and determine the angle  for which sin  = b/r and cos  = a/r. r is called the absolute value of a + bi.  is called the argument of a + bi. This notation is also called polar notation.

10. Slide 8.3 - 10 Copyright © 2009 Pearson Education, Inc. Example Find trigonometric notation for each of the following complex numbers. Solution: a) Note that a = 1 and b = 1. Then  in Q I,  = π/4 or 45º

11. Slide 8.3 - 11 Copyright © 2009 Pearson Education, Inc. Example Solution continued b) a = and b = –1. Then  in Q IV,  = 11π/6 or 330º

12. Slide 8.3 - 12 Copyright © 2009 Pearson Education, Inc. Example Find standard notation, a + bi, for each of the following complex numbers. Solution:

13. Slide 8.3 - 13 Copyright © 2009 Pearson Education, Inc. Example Solution continued

14. Slide 8.3 - 14 Copyright © 2009 Pearson Education, Inc. Complex Numbers: Multiplication For any complex numbers r 1 (cos  1 + i sin  1 ) and, r 1 (cos  1 + i sin  1 ),

15. Slide 8.3 - 15 Copyright © 2009 Pearson Education, Inc. Example Convert to trigonometric notation and multiply. Solution: First find trigonometric notation, then multiply.

16. Slide 8.3 - 16 Copyright © 2009 Pearson Education, Inc. Complex Numbers: Division For any complex numbers r 1 (cos  1 + i sin  1 ) and, r 1 (cos  1 + i sin  1 ), r 2 ≠ 0,

17. Slide 8.3 - 17 Copyright © 2009 Pearson Education, Inc. Example Divide Solution: and express the answer in standard notation.

18. Slide 8.3 - 18 Copyright © 2009 Pearson Education, Inc. Powers of Complex Numbers DeMoivre’s Theorem For any complex number r(cos  + i sin  ) and any natural number n,

19. Slide 8.3 - 19 Copyright © 2009 Pearson Education, Inc. Example Find each of the following. Solution: First find trigonometric notation, then raise to the power.

20. Slide 8.3 - 20 Copyright © 2009 Pearson Education, Inc. Example Solution continued First find trigonometric notation, then raise to the power.

21. Slide 8.3 - 21 Copyright © 2009 Pearson Education, Inc. Roots of Complex Numbers The nth roots of a complex number r(cos  + i sin  ) r ≠ 0, are given by where k = 0, 1, 2, …, n –1.

22. Slide 8.3 - 22 Copyright © 2009 Pearson Education, Inc. Example Find the cube roots of 1. Then locate them on a graph. Solution: First find trigonometric notation, then raise to the power. Then n = 3, 1/n = 1/3, and k = 0, 1, 2; so,

23. Slide 8.3 - 23 Copyright © 2009 Pearson Education, Inc. Example Solution continued The roots are The graphs of the cube roots lie equally spaced about a circle of radius 1. The roots are 360º/3, or 120º apart, as shown on the next slide.

24. Slide 8.3 - 24 Copyright © 2009 Pearson Education, Inc. Example Solution continued