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Trigonometric Form of a Complex Number Digital Lesson

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

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

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 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). Trigonometric Form of a Complex Number a = r cos b = r sin Imaginary axis Real axis b r a (a, b)

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 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

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Example: Write the complex number z = –7 + 4i in trigonometric form. Example: Trigonometric Form of a Complex Number Imaginary axis Real axis z = –7 + 4i °

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: Standard Form of a Complex Number in Radians Standard form Write the complex number in standard form a + bi. Example:

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example: Standard Form of a Complex Number in Degrees Write the complex number in standard form a + bi. Example: Standard form

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Graphing Utility: Standard Form of a Complex Number Graphing Utility: Write the complex number in standard form a + bi. [2nd] [decimal point]

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