Question

Evaluate each of the following complex numbers and express the result in rectangular form:

Answer 1

Answer:

a. $z_{1} = \frac{3\sqrt{2}}{2}+j\cdot 3\frac{\sqrt{2}}{2}$, b. $z_{3} = -j\cdot 2$, c. $z_{4} = -j$, d. $z_{5} = 1$

Step-by-step explanation:

All given complex numbers are in polar form, which are characterized by:

$z = r\cdot e^{j\cdot \theta}$

Where:

$r$ - Magnitude, dimensionless.

$\theta$ - Direction, measured in radians.

The rectangular form of complex numbers is represented by:

$z = r\cdot (\cos \theta + j\cdot \sin \theta)$

Now, each complex number is evaluated:

a. $z_{1} = 3\cdot e^{j\cdot \frac{ \pi}{4} }$

$r = 3$ and $\theta = \frac{\pi}{4}$

$z_{1} = 3\cdot \left(\cos \frac{\pi}{4}+j\cdot \sin \frac{\pi}{4} \right)$

$z_{1} = \frac{3\sqrt{2}}{2}+j\cdot 3\frac{\sqrt{2}}{2}$

b. $z_{3} = 2\cdot e^{-j\cdot \frac{\pi}{2} }$

$r = 2$ and $\theta = -\frac{\pi}{2}$

$z_{3} = 2\cdot \left[\cos \left(-\frac{\pi}{2}\right) + j\cdot \sin \left(-\frac{\pi}{2} \right) \right]$

$z_{3} = -j\cdot 2$

c. $z_{4} = j^{3}$

By definition of complex number, $j = \sqrt{-1}$, which means that $j^{2} = -1$. Hence:

$z_{4} = j^{2+1}$

$z_{4} = j^{2}\cdot j$

$z_{4} = j\cdot j^{2}$

$z_{4} = -j$

d. $z_{5} = j^{-4}$

By definition of complex number, $j = \sqrt{-1}$, which means that $j^{2} = -1$.

Hence:

$z_{5} = \frac{1}{j^{4}}$

$z_{5} = \frac{1}{j^{2+2}}$

$z_{5} = \frac{1}{j^{2}\cdot j^{2}}$

$z_{5} = \frac{1}{(-1)\cdot (-1)}$

$z_{5} = 1$