Question

I need help to express these as a+bi...

1) (4(cos(7pi/9)+i sin(7pi/9)))^3
2) (1/2(cos(72degrees)+sin(72degrees)))^5
3) complex 5th roots of 5-5root(3)i
4) all seventh roots of unity

Answer 1

for the first one, take \(4^3\) and get \(64\) then multiply each angle by 3 and compute
\[64\left( \cos(\frac{7\pi}{3})+i\sin(\frac{7\pi}{3})\right)\]

Answer 2

Oh ok and I would do the same thing for #2?

Answer 3

yes

Answer 4

ok thanks so much! but I still don't understand the last two

Answer 5

complex 7th roots of unity
divide the circle up in to 7 equal parts, one of which is at 1 since 1 is a seventh root of 1

Answer 6

this is very easily expressed in trig form as
\[\cos(\frac{2\pi}{7})+i\sin(\frac{2\pi}{7})\] and so on

Answer 7

by "so on" i mean
\[\cos(\frac{4\pi}{7})+i\sin(\frac{4\pi}{7})\] etc etc

Answer 8

to write in the form of \(a+bi\) is i guess a calculator exercise, since i have no idea what \(\cos(\frac{2\pi}{7})\) is

Answer 9

I know you're supposed to multiply integers 0 - 6 by something... is that something 2pi?

Answer 10

as for #3 you need to write
\[5-5\sqrt3 i\] it polar form first

Answer 11

yeah i guess it is \(2\pi\) really what you are doing is dividing the unit circle (in the complex plane) in to seven equal parts

Answer 12

it is a little hard to draw 7 equal parts, but it should be clear that all the angles will be
\[\frac{2\pi}{7}, \frac{4\pi}{7},\frac{6\pi}{7},\frac{8\pi}{7},\frac{10\pi}{7},\frac{12\pi}{7}\] and of course \(0\)

Answer 13

oh ok thanks so much! :)

Answer 14

as for #3 you need to write
\[5-5\sqrt3 i\] is trig form
do you know how to do that?

Answer 15

I got r=10 and theta=pi/3

Answer 16

\[r=\sqrt{5^2+(5\sqrt{3})^2}\] yeah 10 looks good

Answer 17

\(\frac{\pi}{3}\) is not right however

Answer 18

negative pi/3?

Answer 19

yeah you can use that one

Answer 20

yay thanks

Answer 21

yw