10.5 Powers of Complex Numbers. Exploration! Let z = r(cos θ + isin θ) z 2 = z z = [r(cos θ + isin θ)] [r(cos θ + isin θ)] from yesterday… = r 2 (cos.

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Presentation transcript:

10.5 Powers of Complex Numbers

Exploration! Let z = r(cos θ + isin θ) z 2 = z z = [r(cos θ + isin θ)] [r(cos θ + isin θ)] from yesterday… = r 2 (cos 2θ + isin 2θ) how about z 3 = z z 2 = [r(cos θ + isin θ)] [r 2 (cos 2θ + isin 2θ)] = r 3 (cos 3θ + isin 3θ) this pattern keeps happening… let’s generalize! DeMoivre’s Theorem If z = r(cos θ + isin θ) is a complex number in polar form, then for any integer n, z n = r n (cos nθ + isin nθ)

Ex 1) Evaluate each power and express answer in rectangular form. A) B) try on your own 10

Ex 2) Evaluate z 7 for. Express answer in rectangular form. *Hint: Convert to polar  do DeMoivre’s  convert back

A nautilus shell has been traced on the complex plane. z  (1, 1)  1 + i rect complex Ex 3) Calculate z 2, z 3, and z 4 and see if they lie on the curve of the shell wall. (1, 1)  polar (divide class into 3 groups, just do one) Yes, they lie on the sketch of the shell wall!

Ex 4) Evaluate z 4 for z = 2(cos 40° – isin 40°) Do DeMoivre’s … WAIT is something wrong? it has to be in complex polar  can’t subtract! change to z = 2(cos (–40°) + isin (–40°)) z 4 = (2) 4 (cos (–160°) + isin (–160°)) z 4 ≈ – – i (calculator! )

*Remember: z 0 = 1 and *DeMoivre’s Theorem doesn’t just have to be positive powers. Ex 5) Use DeMoivre’s Thm to evaluate and express in rectangular form. A)B) try on your own

Homework #1006 Pg 520 #1, 3, 5, 9, 11, 13, 16, 20, 21, 23, 29, 31, 32