Trigonometric Form of a Complex Number Digital Lesson Trigonometric Form of a Complex Number

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

Definition: Absolute Value 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 + 6i and find its absolute value. Imaginary axis Real axis 4 – 2 – 4 2 6 8 z = 3 + 6i Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Absolute Value

Trigonometric Form of a Complex Number 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 Real axis b r a (a, b)  a = r cos  b = r sin  Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Trigonometric Form of a Complex Number

Definition: Trigonometric Form of a Complex Number 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 argument Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Definition: Trigonometric Form of a Complex Number

Example: Trigonometric Form of a Complex Number Write the complex number z = –7 + 4i in trigonometric form. Imaginary axis Real axis z = –7 + 4i 150.26° Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Trigonometric Form of a Complex Number

Example: Standard Form of a Complex Number in Radians Write the complex number in standard form a + bi. Example: Standard form Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Standard Form of a Complex Number in Radians

Example: Standard Form of a Complex Number in Degrees Write the complex number in standard form a + bi. Example: Standard form Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Standard Form of a Complex Number in Degrees

Graphing Utility: Standard Form of a Complex Number Write the complex number in standard form a + bi. [2nd] [decimal point] Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Graphing Utility: Standard Form of a Complex Number