chapter  24
De Moivre’s theorem
Pages 6

Problem 2. Determine the value of (−7 + j5)4, expressing the result in polar and rectangular forms.

(−7 + j5) = √

[(−7)2 + 52]∠ tan−1 5−7 = √74∠144.46◦

(Note, by considering the Argand diagram, −7 + j5 must represent an angle in the second quadrant and not in the fourth quadrant.) Applying De Moivre’s theorem:

(−7 + j5)4 = [√74∠144.46◦]4

= √

744∠4 × 144.46◦ = 5476∠577.84◦ = 5476∠217.84◦or

5476∠217◦15′ in polar form Since r∠θ = r cos θ + jr sin θ,

5476∠217.84◦ = 5476 cos 217.84◦ + j5476 sin 217.84◦

= −4325 − j3359 i.e. (−7 + j5)4 = −4325 − j3359

in rectangular form

Now try the following exercise.