Answer:
The complex numbers computed are:
A) [tex]q=0.8752+j0.4838=1e^{-j0.5049}[/tex]
B) [tex]r=-8-j8\sqrt{3} =16e^{j\pi \frac{4}{3}}[/tex]
The sketches are attached to this answer
Step-by-step explanation:
To compute these complex numbers you have to remember these rules:
[tex]Z=a+jb=(a^2+b^2)^{\frac{1}{2}}e^{jtan^{-1}(b/a)}[/tex] (a)
[tex]Z=|z|e^{j\alpha}=|z|cos(\alpha)+j|z|sin(\alpha)[/tex] (b)
Also for multiplication, division, and powers, if W and U are complex numbers and k is a real number:
[tex]{W}\cdot{U}={|W|e^{j\alpha}}{|U|e^{j\beta}}={|W|}{|U|}e^{j(\alpha+\beta)}[/tex] Â (1)
[tex]\frac{W}{U}=\frac{|W|e^{j\alpha}}{|U|e^{j\beta}}=\frac{|W|}{|U|}e^{j(\alpha-\beta)}[/tex] Â Â Â Â Â Â Â Â Â Â Â Â (2)
[tex]W^{k}=|w|^{k}e^{j(\alpha\cdot k)}[/tex] Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â (3)
With these rules we will do the followings steps:
for A:
1) We solve first the divition, writing the 2 complex numbers exponential form (equation (a)).
2) With the rule (2) we solve the division.
3) with rule (3) we solve the power.
For B:
1)We write the numbers a, b, c, d, and f in exponential form (equation (a)).
2) We use the rule (1) for the product.
