1. DeMoivre’s Theorem Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 To write a complex number in trigonometric form, let  be the angle from the real axis 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) 

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 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

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 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 150.26 °

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 To raise a complex number to a power, the trigonometric form of a complex number is used. Powers of Complex Numbers

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 DeMoivre’s Theorem Definition: DeMoivre’s Theorem If z = r(cos  + i sin  ) is a complex number and n is a positive integer, then z n = [r(cos  + i sin  )] n = r n (cos n  + i sin n  )]. Example:

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: DeMoivre’s Theorem Example: Use DeMoivre’s Theorem to write (1 + i) 5 in standard form a + bi. Convert the complex number into trigonometric form. Example continues.

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Example Continued Example continued:

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Example: DeMoivre’s Theorem Example: Write (3 + 4i) 3 in standard form a + bi.