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6.6 DeMoivre’s Theorem

6.6 DeMoivre’s Theorem. I. Trigonometric Form of Complex Numbers. A.) The standard form of the complex number is very similar to the component form of a vector If we look at the trigonometric form of v , we can see.

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6.6 DeMoivre’s Theorem

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  1. 6.6 DeMoivre’s Theorem

  2. I. Trigonometric Form of Complex Numbers A.) The standard form of the complex number is very similar to the component form of a vector If we look at the trigonometric form of v, we can see

  3. B.) If we graph the complex z = a + bi on the complex plane, we can see the similarities with the polar plane. P (a, b) r b θ a z = a + bi

  4. C.) If we let and then, where

  5. D.) Def. – The trigonometric form of a complex number z is given by Where r is the MODULUS of z and θis the ARGUMENT of z.

  6. E.) Ex.1 - Find the trig form of the following:

  7. II. Products and Quotients A.) Let . Mult.- Div. - DERIVE THESE!!!!

  8. B.) Ex. 2 – Given . find

  9. III. Powers of Complex Numbers A.) DeMoivre’s (di-’mȯi-vərz) Theorem – If z = r(cosθ + i sinθ) and n is a positive integer, then, Why??? – Let’s look at z2-

  10. B.) Ex. 3 – Find by “Foiling”

  11. C.) Ex. 4– Now find using DeMoivre’s Theorem

  12. D.) Ex. 5 –Use DeMoivre’s Theorem to simplify

  13. IV. nth Roots of Complex Numbers A.) Roots of Complex Numbers – v= a + bi is an nth root of z iff vn = z . If z = 1, then v is an nth ROOT OF UNITY.

  14. B.) If , then the n distinct complex numbers Where k = 0, 1, 2, …, n-1 are the nth roots of the complex number z.

  15. C.) Ex. 6- Find the 4th roots of

  16. V. Finding Cube Roots A.) Ex. 7 - Find the cube roots of -1.

  17. Plot these points on the complex plane. What do you notice about them? Now....

  18. Equidistant from the origin and equally spaced about the origin.

  19. VI. Roots of Unity A.) Any complex root of the number 1 is also known as a ROOT OF UNITY. B.) Ex. 8 - Find the 6 roots of unity.

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