Stelle die folgenden komplexen Zahlen in der arithmetischen Form ´a + bi´ (mit ´a, b in RR´) und in der trigonometrischen Form ´r * e^(i alpha)´ (´r in R^+´, Winkel im Bogenmaß, ´0 <= alpha < 2pi´) dar.

  1. ´(−12 + 18 bb i)/(1 + 5 bb i)´
  2. ´2e^(bb i (7pi)/6) * (sqrt(3) + bb i)´
  3. ´(4 + sqrt(48) bb i)^8´

(Zweierpotenzen müssen nicht ausmultipliziert werden.)

Approach

´r * e^(i alpha) = r(cos(alpha) + i sin(alpha)) = r * cos(alpha) + r * sin(alpha)´

´a_1 + a_2 i = sqrt(a_1^2 + a_2^2) * (a_1/(sqrt(a_1^2 + a_2^2)) + a_2/(sqrt(a_1^2 + a_2^2)) i)´ mit ´0 <= alpha <= 2pi´

Teil 1

´z = (-12 + 18i)/(1 + 5i) = ((-12 + 18i)(1-5i))/((1 + 5i)(1-5i))´ ´= (-12 + 60i + 18i + 90i)/(1 + 25) = (78 + 78i)/26 = 3 + 3i´

´|z| = sqrt(3^2 + 3^2) = sqrt(18)´

´z = sqrt(18) (3/sqrt(18) + 3/sqrt(18) i)´ ´= sqrt(18) (3/(3 sqrt(2)) + 3/(3 sqrt(2)) i)´ ´= sqrt(18) (1/sqrt(2) + 1/sqrt(2) i)´ ´= sqrt(18) (sqrt(2)/2 + sqrt(2)/2 i)´ ´= sqrt(18) (cos(pi/4) + sin(pi/4) i)´ ´= 3 sqrt(2) * e^(i pi/4)´

Teil 2

´z_0 = sqrt(3) + i´ ´|z_0| = sqrt(3 + 1) = 2´

´z_0 = 2 (sqrt(3)/2 + 1/2 i)´ ´= 2 (cos(pi/6) + sin(pi/6) i)´ ´= 2e^(i pi/6)´

´2e^(i (7pi)/6) * 2e^(i pi/6)´ ´= 4e^(i((7pi)/6 + pi/6))´ ´= 4e^(i (8pi)/6)´ ´= 4e^(i (4pi)/3)´

´= 4 (cos((4pi)/3) + sin((4pi)/3) i)´ ´= 4 (- 1/2 - 1/2 i)´ ´= -2 - 2 i´

<mark>Solution seems to be wrong!</mark>

Teil 3

´(4 + sqrt(48)i)^8 = sum_(n=0)^8 ((8),(k)) 4^k (sqrt(48)i)^(8-k)´

Zu aufwendig!

´4 + sqrt(48)i = 8(1/2 + (4 sqrt(3))/8 i)´ ´= 8(cos(pi/3) + sin(pi/3) i)´ ´= 8e^(i pi/3)´

´=> 4 + sqrt(48)i = 8e^(i pi/3)^8 = 8^8 e^(i (8pi)/3)´ ´= 8^8 (cos(8/3 pi) + sin(8/3 pi) i)´ ´= 8^8 (-1/2 + sqrt(3)/2 i)´ ´= 2^24 (-1/2 + sqrt(3)/2 i)´ ´= -2^23 + 2^23 sqrt(3) i)´


Solution
      • ´3 + 3i´
      • ´3 sqrt(2) * e^(i pi/4)´
      • ´2e^(i pi/6)´
      • ´-2 - 2 i´ <mark>Solution seems to be wrong!</mark>
      • ´8e^(i pi/3)´
      • ´-2^23 + 2^23 sqrt(3) i)´
  • URL:
  • Language:
  • Subjects: math
  • Type: Calculate
  • Duration: 35min
  • Credits: 3
  • Difficulty: 0.8
  • Tags: hpi complex number
  • Note:
    HPI, 2014-07-07, Mathe 2, Aufgabe 54
  • Created By: adius
  • Created At:
    2014-07-26 21:32:36 UTC
  • Last Modified:
    2014-07-27 23:20:32 UTC