Answer:
A
Step-by-step explanation:
Begin with Euler's formula. I am assuming youre aware of this if youre taking complex algebra? You can prove Euler's formula by doing a Maclaurin expansion of cosine, sine, and e^x. Euler's formula states that:
[tex]e^{ix}=cos(x)+isin(x)[/tex]
We can put the first complex number in exponetial form by noticing the input is 2pi/3. The second complex number has an input of pi/3. Therefore:
[tex]z_1=8e^{i(2\pi /3)}[/tex]
[tex]z_2=0.5e^{i(\pi /3)}[/tex]
Then:
[tex]\frac{z_1}{z_2} =\frac{8e^{i(2\pi /3)}}{0.5e^{i(\pi /3)}}[/tex]
Simplify the coefficients to get:
[tex]\frac{z_1}{z_2} =\frac{16e^{i(2\pi /3)}}{e^{i(\pi /3)}}[/tex]
When you divide exponentials, you subtract the exponents. Therefore:
[tex]\frac{z_1}{z_2} =16e^{i(2\pi /3)-i(\pi /3)}=16e^{i(\pi /3)[/tex]
Put it back into trigonemtric form using Euler's formula:
[tex]16e^{i(\pi /3)}=16cos(\pi /3)+i16sin(\pi /3)[/tex]
Cosine of pi/3 is 0.5, and sine of pi/3 is square root of 3 over 2. We have:
[tex]16e^{i(\pi /3)}=16cos(\pi /3)+i16sin(\pi /3)=16*0.5+16*\frac{\sqrt{3} }{2} *i=8+8\sqrt{3} i[/tex]
