Answer:
a. [tex]z_{1} = \frac{3\sqrt{2}}{2}+j\cdot 3\frac{\sqrt{2}}{2}[/tex], b. [tex]z_{3} = -j\cdot 2[/tex], c. [tex]z_{4} = -j[/tex], d. [tex]z_{5} = 1[/tex]
Step-by-step explanation:
All given complex numbers are in polar form, which are characterized by:
[tex]z = r\cdot e^{j\cdot \theta}[/tex]
Where:
[tex]r[/tex] - Magnitude, dimensionless.
[tex]\theta[/tex] - Direction, measured in radians.
The rectangular form of complex numbers is represented by:
[tex]z = r\cdot (\cos \theta + j\cdot \sin \theta)[/tex]
Now, each complex number is evaluated:
a. [tex]z_{1} = 3\cdot e^{j\cdot \frac{ \pi}{4} }[/tex]
[tex]r = 3[/tex] and [tex]\theta = \frac{\pi}{4}[/tex]
[tex]z_{1} = 3\cdot \left(\cos \frac{\pi}{4}+j\cdot \sin \frac{\pi}{4} \right)[/tex]
[tex]z_{1} = \frac{3\sqrt{2}}{2}+j\cdot 3\frac{\sqrt{2}}{2}[/tex]
b. [tex]z_{3} = 2\cdot e^{-j\cdot \frac{\pi}{2} }[/tex]
[tex]r = 2[/tex] and [tex]\theta = -\frac{\pi}{2}[/tex]
[tex]z_{3} = 2\cdot \left[\cos \left(-\frac{\pi}{2}\right) + j\cdot \sin \left(-\frac{\pi}{2} \right) \right][/tex]
[tex]z_{3} = -j\cdot 2[/tex]
c. [tex]z_{4} = j^{3}[/tex]
By definition of complex number, [tex]j = \sqrt{-1}[/tex], which means that [tex]j^{2} = -1[/tex]. Hence:
[tex]z_{4} = j^{2+1}[/tex]
[tex]z_{4} = j^{2}\cdot j[/tex]
[tex]z_{4} = j\cdot j^{2}[/tex]
[tex]z_{4} = -j[/tex]
d. [tex]z_{5} = j^{-4}[/tex]
By definition of complex number, [tex]j = \sqrt{-1}[/tex], which means that [tex]j^{2} = -1[/tex].
Hence:
[tex]z_{5} = \frac{1}{j^{4}}[/tex]
[tex]z_{5} = \frac{1}{j^{2+2}}[/tex]
[tex]z_{5} = \frac{1}{j^{2}\cdot j^{2}}[/tex]
[tex]z_{5} = \frac{1}{(-1)\cdot (-1)}[/tex]
[tex]z_{5} = 1[/tex]
