Question

Z^5=-243i, find the solution to the equation whose argument is strictly between 180 degrees and 270 degrees. Round your answer to the nearest 10th

Answer 1

Answer:

The solution is $z = -2.853 - i 0.927$.

Step-by-step explanation:

Complex power is determined by means of the De Moivre's Theorem, whose expression is:

$z^{n} = r^{n} \cdot (\cos n\theta + i \sin n\theta)$

Where $r$ is the norm of the complex number. In this case, expression can be written as:

$-i 243 = 3^{5} \cdot (\cos 5\theta + i \sin 5\theta)$

The real component must be equal to zero and complex component must be equal to -1. That is to say:

$\cos 5\theta = 0$

$\sin 5\theta = -1$

Possible solutions for each component are, respectively:

Real component

$5\theta = \cos^{-1}0$

$5\theta = \frac{\pi}{2} \pm \pi\cdot j$, $\forall j \in \mathbb{N}_{O}$

$\theta = \frac{\pi}{10} \pm \frac{\pi}{5} \cdot j$, $\forall j \in \mathbb{N}_{O}$

Possible solutions: $\frac{11\pi}{10}$, $\frac{13\pi}{10}$, $\frac{3\pi}{2}$

Complex component

$5\theta = \sin^{-1}(-1)$

$5\theta = \frac{3\pi}{2} \pm 2\pi \cdot j$, $\forall j \in \mathbb{N}_{O}$

$\theta = \frac{3\pi}{10} \pm \frac{2\pi}{5} \cdot j$, $\forall j \in \mathbb{N}_{O}$

Possible solutions: $\frac{11\pi}{10}$, $\frac{3\pi}{2}$

There is one solution whose argument is strictly between 180 degrees ($\pi$) and 270 degrees ($1.5\pi$).

$z = 3 \cdot \left( \cos \frac{11\pi}{10} + i \sin \frac{11\pi}{10} \right)$

$z = -2.853 - i 0.927$