Pre-Calc Exam Notes 144

Pre-Calc Exam Notes 144 - 144 Chapter 6 • Additional...

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Unformatted text preview: 144 Chapter 6 • Additional Topics §6.3 Example 6.12 Find (1 + i ) 10 . Solution: Since 1 + i = radicallow 2 (cos 45 ◦ + i sin 45 ◦ ) (why?), by De Moivre’s Theorem we have (1 + i ) 10 = ( radicallow 2) 10 (cos 450 ◦ + i sin 450 ◦ ) = 2 10/2 (0 + i (1)) = 2 5 · i = 32 i . We can use De Moivre’s Theorem to find the n th roots of a complex number. That is, given any complex number z and positive integer n , find all complex numbers w such that w n = z . Let z = r (cos θ + i sin θ ). Since the cosine and sine functions repeat every 360 ◦ , we know that z = r (cos ( θ + 360 ◦ k ) + i sin ( θ + 360 ◦ k )) for k = 0, ± 1, ± 2, .... Now let w = r (cos θ + i sin θ ) be an n th root of z . Then w n = z ⇒ [ r (cos θ + i sin θ )] n = r (cos ( θ + 360 ◦ k ) + i sin ( θ + 360 ◦ k )) ⇒ r n (cos n θ + i sin n θ ) = r (cos ( θ + 360 ◦ k ) + i sin ( θ + 360 ◦ k )) ⇒ r n = r and n θ = θ + 360 ◦ k ⇒ r = r 1/ n and θ = θ + 360 ◦ k n ....
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This note was uploaded on 01/21/2012 for the course MAC 1130 taught by Professor Dr.cheun during the Fall '11 term at FSU.

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