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

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

C.) If we let and then, where

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

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

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

B.) Ex. 2 – Given. find

III. Powers of Complex Numbers A.) DeMoivre’s (di-’mo ̇ i-vərz) Theorem – If z = r(cosθ + i sinθ) and n is a positive integer, then, Why??? – Let’s look at z 2 -

B.) Ex. 3 – Find by “Foiling”

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

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

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

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

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

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

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

Equidistant from the origin and equally spaced about the origin.

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.