De MoivreDe Moivre’’s s TheoremTheoremPowers of Complex NumbersPowers of Complex Numbers

De MoivreDe Moivre’’s Theorems Theorem

• We use this theorem to:

• I. Simplify complex numbers raised to a power.

• II. Solve certain types of equations, or find the nth roots of a complex number.

• I. POWERS

De MoivreDe Moivre’’s Theorem:s Theorem:

You need to know:You need to know:A. Convert rectangular form into trigonometric formA. Convert rectangular form into trigonometric formB. Simplify fractionsB. Simplify fractionsC. Simplify radicalsC. Simplify radicals

Problem 1Problem 1

Then, find the angle:Then, find the angle:

Your turn…Your turn…

Practice: Practice:

Use De MoivreUse De Moivre’’s theorem to find (-1 + i√3 )s theorem to find (-1 + i√3 )1212

a. Convert the complex number to trig form:a. Convert the complex number to trig form:

b. Then use De Moivreb. Then use De Moivre’’s Theoem to find the values Theoem to find the value

Answer: 4096Answer: 4096

De MoivreDe Moivre’’s s TheoremTheoremPowers of Complex NumbersPowers of Complex Numbers

Grab your book and calculatorGrab your book and calculator

Who was De MoivreWho was De MoivreA brilliant French mathematician who was persecuted in France because of his religious beliefs. De Moivre moved to England where he tutored mathematics privately and became friends with Sir Isaac Newton.

De Moivre made a breakthrough in the fields of probability (writing the Doctrine of Chance), but more importantly for IB HL students he moved trigonometry into the field of analysis through complex numbers with De Moivre’s theorem.

Warm - upWarm - up

1. (3 - 2i)1. (3 - 2i)55

2. (√5 - 2. (√5 - 4i)4i)33

= - 597- 122i = - 597- 122i

= - 43√5 + 4i = - 43√5 + 4i