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Trigonometric Formof Complex Numbers

Section 6.2

Complex Numbers, Polar Coordinates, and Parametric Equations

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6.2

Figure 6.2

The Complex Plane

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6.2

The absolute value or modulus of the complex number a + bi is defined by

.|| 22 babia

Definition: Absolute Valueor Modulus of a + bi

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6.2

Figure 6.4

Trigonometric Formof a Complex Number

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6.2

)sin(cos irz

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 .

22 bar

Definition: Trigonometric Formof a Complex Number

Copyright © 2011 Pearson Education, Inc. Slide 6-7

6.2

If z1 = r1(cos 1 + i sin 1) and z2 = r2(cos 2 + i sin 2), then

z1z2 = r1r2 [cos (1 + 2) + i sin (1 + 2)]

and)].sin()[cos( 2121

2

1

2

1 irr

zz

Theorem: The Product andQuotient of Complex Numbers

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6.2

The conjugate of the complex number r (cos + i sin ) is

)).sin()(cos( ir

Theorem: Complex Conjugates