Question

Complex number Help!
Find the real and imaginary parts of the following numbers and plot them in the complex
plane.
z=e^(2-pi*i/4)

Answer 1

\[z=e^{2-\frac{\pi}{4} i} =e^2 e^{\frac{-\pi}{4}i} \text{ by law of exponents } \]

Answer 2

I had that so far, but I don't know where to go from there.

Answer 3

\[e^{ \theta i }=\cos(\theta)+i \sin(\theta) \\ \text{ so } a \cdot e^{i \theta} =a \cos(\theta)+ i a \sin(\theta)\]

Answer 4

real part is a cos(theta)
imaginary part is a sin(theta)

Answer 5

cool or not cool?

Answer 6

So would the real part be e^(2)cos(theta)?

Answer 7

and the imaginary part be -pi/4sin(theta)?

Answer 8

I think you are working too fast

Answer 9

your theta here is definitely -pi/4
and your a is what was in front of the e^(i theta) thing

Answer 10

I don't really understand then.

Answer 11

e^2 then?

Answer 12

I mean e^2 is a.

Answer 13

e^2 is a
-pi/4 is theta

Answer 14

so the real part is e^2*cos(-pi/4)?

Answer 15

yep

Answer 16

and you can evaluate the cos(-pi/4) part

Answer 17

\[e^2 \frac{\sqrt{2}}{2} \text{ is real part }\]

Answer 18

oh okay. so the imaginary would be e^2*sin(-pi/4)?

Answer 19

and you would evaulate that too.

Answer 20

so in general
assume x and y are real
\[e^{x+yi}=e^{x}e^{yi}=e^x (\cos(y)+i \sin(y))= e^x \cos(y)+i e^x \sin(y) \\ \text{ where the real part is } e^x \cos(y) \\ \text{ and the imgainry part is } e^x \sin(y)\]

Answer 21

and yes to your question
you can also evaluate that

Answer 22

Okay. That makes a lot more sense! Thank you!

Answer 23

Can I ask you about the next one I have if you're still there?

Answer 24

yes i was gone but not i'm here

Answer 25

now*

Answer 26

The next one is \[z=e^{-1-i}\]
I set it up like the last one where \[z=\frac{ e^{-1} }{ e^{i}}\]

Answer 27

hey are we doing the same thing?

Answer 28

yeah, sorry I didn't say that.

Answer 29

The thing that confuses me is there is no theta

Answer 30

Unless 1 is theta

Answer 31

you can do the exact same I did above

Answer 32

\[e^{-1-i}=e^{-1}e^{-i}\]

Answer 33

your theta is -1 in this case

Answer 34

\[e^{-1}e^{-1 \cdot i}\]

Answer 35

Oh okay. I can do it from there. thanks!