Answer:
The solution is [tex]z = -2.853 - i 0.927[/tex].
Step-by-step explanation:
Complex power is determined by means of the De Moivre's Theorem, whose expression is:
[tex]z^{n} = r^{n} \cdot (\cos n\theta + i \sin n\theta)[/tex]
Where [tex]r[/tex] is the norm of the complex number. In this case, expression can be written as:
[tex]-i 243 = 3^{5} \cdot (\cos 5\theta + i \sin 5\theta)[/tex]
The real component must be equal to zero and complex component must be equal to -1. That is to say:
[tex]\cos 5\theta = 0[/tex]
[tex]\sin 5\theta = -1[/tex]
Possible solutions for each component are, respectively:
Real component
[tex]5\theta = \cos^{-1}0[/tex]
[tex]5\theta = \frac{\pi}{2} \pm \pi\cdot j[/tex], [tex]\forall j \in \mathbb{N}_{O}[/tex]
[tex]\theta = \frac{\pi}{10} \pm \frac{\pi}{5} \cdot j[/tex], [tex]\forall j \in \mathbb{N}_{O}[/tex]
Possible solutions: [tex]\frac{11\pi}{10}[/tex], [tex]\frac{13\pi}{10}[/tex], [tex]\frac{3\pi}{2}[/tex]
Complex component
[tex]5\theta = \sin^{-1}(-1)[/tex]
[tex]5\theta = \frac{3\pi}{2} \pm 2\pi \cdot j[/tex], [tex]\forall j \in \mathbb{N}_{O}[/tex]
[tex]\theta = \frac{3\pi}{10} \pm \frac{2\pi}{5} \cdot j[/tex], [tex]\forall j \in \mathbb{N}_{O}[/tex]
Possible solutions: [tex]\frac{11\pi}{10}[/tex], [tex]\frac{3\pi}{2}[/tex]
There is one solution whose argument is strictly between 180 degrees ([tex]\pi[/tex]) and 270 degrees ([tex]1.5\pi[/tex]).
[tex]z = 3 \cdot \left( \cos \frac{11\pi}{10} + i \sin \frac{11\pi}{10} \right)[/tex]
[tex]z = -2.853 - i 0.927[/tex]
