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De Moivre’s Theorem Powers of Complex Numbers De Moivre’s 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 Moivre’s Theorem: You need to know: A. Convert rectangular form into trigonometric form B. Simplify fractions C. Simplify radicals Problem 1 Then, find the angle: Your turn… Practice: Use De Moivre’s theorem to find (-1 + i√3 )12 a. Convert the complex number to trig form: b. Then use De Moivre’s Theoem to find the value Answer: 4096 Grab your book and calculator De Moivre’s Theorem Powers of Complex Numbers Who was De Moivre A 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 - up 1. (3 - 5 2i) 2. (√5 - 3 4i) = - 597- 122i = - 43√5 + 4i