Digital Lesson Trigonometric Form of a Complex Number

In the complex plane, every complex number corresponds to a point. Example: Plot the points 3 + 4 i and – 2 i in the complex plane. Imaginary axis 4 (3, 4) or 3 + 4 i 2 – 2 (– 2, – 2) or – 2 i Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Real axis – 2 2

The absolute value of the complex number z = a + bi is the distance between the origin (0, 0) and the point (a, b). Example: Plot z = 3 + 6 i and find its absolute value. Imaginary axis 8 6 z = 3 + 6 i 4 2 – 4 – 2 4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Real axis 3

To write a complex number a + bi in trigonometric form, let be the angle from the positive real axis (measured counter clockwise) to the line segment connecting the origin to the point (a, b). Imaginary axis a = r cos (a, b) b = r sin r b a Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Real axis 4

The trigonometric form of a complex number z = a + bi is given by z = r(cos + i sin ) where a = r cos , b = r sin , The number r is the modulus of z, and is the argument of z. Example: modulus Copyright © by Houghton Mifflin Company, Inc. All rights reserved. argument 5