1. Copyright © 2007 Pearson Education, Inc. Slide 10-1 7.5Trigonometric (Polar) Form of Complex Numbers The Complex Plane and Vector Representations Call the horizontal axis the real axis and the vertical axis the imaginary axis. Now complex numbers can be graphed in this complex plane. The sum of two complex numbers can be represented graphically by the vector that is the resultant of the sum of vectors corresponding to the two numbers.

2. Copyright © 2007 Pearson Education, Inc. Slide 10-2 7.5Expressing the Sum of Complex Numbers Graphically ExampleFind the sum of 6 – 2i and –4 – 3i. Graph both complex numbers and their resultant. Solution (6 – 2i) + (–4 – 3i) = 2 – 5i

3. Copyright © 2007 Pearson Education, Inc. Slide 10-3 7.5Trigonometric (Polar) Form Relationship Among x, y, r, and  The graph shows the complex number x + yi that corresponds to the vector OP.

4. Copyright © 2007 Pearson Education, Inc. Slide 10-4 7.5 Trigonometric (Polar) Form Substituting x = r cos  and y = r sin  into x + yi gives Trigonometric or Polar Form of a Complex Number The expression r(cos  + i sin  ) is called the trigonometric form or polar form of the complex number x + yi.

5. Copyright © 2007 Pearson Education, Inc. Slide 10-5 7.5 Trigonometric (Polar) Form Notation: cos  + i sin  is sometimes written cis . Using this notation, r(cos  + i sin  ) is written r cis . The number r is called the modulus or absolute value of the complex number x + yi. Angle  is called the argument of the complex number x + yi.

6. Copyright © 2007 Pearson Education, Inc. Slide 10-6 7.5Converting from Trigonometric Form to Rectangular Form ExampleExpress 2(cos 300º + i sin 300º) in rectangular form. Analytic Solution Graphing Calculator Solution

7. Copyright © 2007 Pearson Education, Inc. Slide 10-7 7.5Converting from Rectangular to Trigonometric Form Converting from Rectangular to Trigonometric Form 1.Sketch a graph of the number in the complex plane. 2.Find r by using the equation 3.Find  by using the equation tan  = y/x, x  0, choosing the quadrant indicated in Step 1.

8. Copyright © 2007 Pearson Education, Inc. Slide 10-8 7.5Converting from Rectangular to Trigonometric Form ExampleWrite each complex number in trigonometric form. Solution (a)Start by sketching the graph of in the complex plane. Then find r.

9. Copyright © 2007 Pearson Education, Inc. Slide 10-9 7.5Converting from Rectangular to Trigonometric Form Now find . Therefore, in polar form,  is in quadrant II and tan  = the reference angle in quadrant II is

10. Copyright © 2007 Pearson Education, Inc. Slide 10-10 7.5Converting from Rectangular to Trigonometric Form (b) From the graph,  = 270º. In trigonometric form, different way to determine .

11. Copyright © 2007 Pearson Education, Inc. Slide 10-11 7.5Deciding Whether a Number is in the Julia Set ExampleThe fractal called the Julia set is shown in the figure. To determine if a complex number z = a + bi is in this Julia set, perform the following sequence of calculations. Repeatedly compute the values of z 2 – 1, (z 2 – 1) 2 –1, [(z 2 – 1) 2 –1] 2 – 1,.... If the moduli of any of the resulting complex numbers exceeds 2, then z is not in the Julia set. Otherwise z is part of this set and the point (a, b) should be shaded in the graph.

12. Copyright © 2007 Pearson Education, Inc. Slide 10-12 7.5Deciding Whether a Number is in the Julia Set Determine if z = 0 + 0i belongs to the Julia set. Solution So, and so on. The calculations repeat as 0, –1, 0, –1, and so on. The moduli are either 0 or 1, therefore, 0 + 0i belongs to the Julia set.

13. Copyright © 2007 Pearson Education, Inc. Slide 10-13 7.5Products of Complex Numbers in Trigonometric Form Multiplying complex numbers in rectangular form. Multiplying complex numbers in trigonometric form.

14. Copyright © 2007 Pearson Education, Inc. Slide 10-14 7.5Products of Complex Numbers in Trigonometric Form Product Theorem If are any two complex numbers, then In compact form, this is written

15. Copyright © 2007 Pearson Education, Inc. Slide 10-15 7.5Using the Product Theorem ExampleFind the product of 3(cos 45º + i sin 45º) and 2(cos 135º + i sin 135º). Solution

16. Copyright © 2007 Pearson Education, Inc. Slide 10-16 7.5Quotients of Complex Numbers in Trigonometric Form The rectangular form of the quotient of two complex numbers. The polar form of the quotient of two complex numbers.

17. Copyright © 2007 Pearson Education, Inc. Slide 10-17 7.5Quotients of Complex Numbers in Trigonometric Form Quotient Theorem If r 1 (cos  1 + i sin  1 ) and r 2 (cos  2 + i sin  2 ) are complex numbers, where r 2 (cos  2 + i sin  2 )  0, then In compact form, this is written

18. Copyright © 2007 Pearson Education, Inc. Slide 10-18 7.5Using the Quotient Theorem ExampleFind the quotient Write the result in rectangular form. Solution