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**Trigonometric Form of a Complex Number**

Digital Lesson Trigonometric Form of a Complex Number

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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Trigonometric Form of Complex Numbers 6.6a The first stuff in our last section of the chapter!

Trigonometric Form of Complex Numbers 6.6a The first stuff in our last section of the chapter!

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