Copyright © 2011 Pearson Education, Inc. Trigonometric Form of Complex Numbers Section 6.2 Complex Numbers, Polar Coordinates, and Parametric Equations

6.2 Copyright © 2011 Pearson Education, Inc. Slide 6-3 Figure 6.2 The Complex Plane

6.2 Copyright © 2011 Pearson Education, Inc. Slide 6-4 The absolute value or modulus of the complex number a + bi is defined by Definition: Absolute Value or Modulus of a + bi

6.2 Copyright © 2011 Pearson Education, Inc. Slide 6-5 Figure 6.4 Trigonometric Form of a Complex Number

6.2 Copyright © 2011 Pearson Education, Inc. Slide 6-6 If z = a + bi is a complex number, then the trigonometric form of z is where and  is an angle in standard position whose terminal side contains the point (a, b). An abbreviation for r(cos  + i sin  ) is r cis . Definition: Trigonometric Form of a Complex Number

6.2 Copyright © 2011 Pearson Education, Inc. Slide 6-7 If z 1 = r 1 (cos  1 + i sin  1 ) and z 2 = r 2 (cos  2 + i sin  2 ), then z 1 z 2 = r 1 r 2 [cos (  1 +  2 ) + i sin (  1 +  2 )] and Theorem: The Product and Quotient of Complex Numbers

6.2 Copyright © 2011 Pearson Education, Inc. Slide 6-8 The conjugate of the complex number r (cos  + i sin  ) is Theorem: Complex Conjugates