Question

"Find the cube roots of 27(cos 330 + i sin 330)"
Please help? I am completely clueless on what it is asking or how to go about solving it.


(I'm asking this for a friend)

Answer 1

are you familiar with cosx + isinx = e^ix ?

Answer 2

divide by 3

Answer 3

idk but this was a given
http://prntscr.com/6213hi

Answer 4

well also take the cube root of 27, which is 3 since \(3^3=27\)
i meant "divide the angle by 3"

Answer 5

that is a bunch of math teacherse designed to confuse something that is very very (very) easy

Answer 6

divide the angle by 3, you can do it in your head

Answer 7

110 degrees?

Answer 8

bingo

Answer 9

that's De'Moivres theorem which satellite is trying to explain to you

Answer 10

so one answer is
\[3(\cos(110^\circ)+i\sin(110^\circ))\]

Answer 11

now you have a choice of ways to find the other two answers

Answer 12

one is to divide the unit circle in two three equal parts with \(110^\circ\) as one of the parts

Answer 13

I have to show work, what is this De'Moivres theorem?

Answer 14

the other is to add 360 and divide by 3 again

Answer 15

show work: \(\sqrt[3]{27}=3\) and \(330\div 3=110\) is the first part of the "work"

Answer 16

haha ok :)

Answer 17

ok now here is an interesting question before we continue

Answer 18

oh! the weird hieroglyphics are starting to make sense!

Answer 19

most adults do this using radians (numbers) and not degrees
in fact the formula you gave as a screen shot is done in radians
but the question is asked in degrees

Answer 20

I realized that xD hehe what's the k in that equation?

Answer 21

well degrees we have, so degrees we work with
for the next bit of work, take \(330+360\) and divide it by \(3\)

Answer 22

330 +360 = 690
/3 = 230

Answer 23

the \(k\) is \(0,1,2,3...\) however many you need for the roots
we will use 0,1, 2

Answer 24

is that the new angles we put in there?
would we still keep it as 3 on the outside part?

Answer 25

yes

Answer 26

the real cube root of 27 does not change

Answer 27

so 3(cos(230) + isin(230)) ?

Answer 28

bingo
one more

Answer 29

add 360 again?

Answer 30

yup

Answer 31

to the 230 or the 690?

Answer 32

to the 690

Answer 33

in other words \(330+2\times 360\) this time \(k=2\)

Answer 34

ok :)
690 + 360 = 1050
1050 / 3 = 350

Answer 35

you found the sound

Answer 36

I think I'm getting the hang of this now! :D

Answer 37

let me draw you a picture

Answer 38

so the final one is 3 (cos(350) + isin(350))

Answer 39

yes

Answer 40

we divided the unit circle in to three parts, with one at \(110^\circ\)

Answer 41

i should say "three equal parts"

Answer 42

unit circle divided into thirds to find the cubes?

Answer 43

yes, so long as you know one of them to start you can do it that way

Answer 44

we had to find the 110 first, but that was an easy matter of division

Answer 45

no prized for guessing how many part for fourth roots etc

Answer 46

*prizes

Answer 47

ooh i should correct what you said
divide in to three parts for cubed roots, not cubes

Answer 48

oh right ^_^

Answer 49

now go show your friend how easy this was

Answer 50

I will! :D Thank you so much for your help!

Answer 51

it is the whole reason to write a complex number in this form
at least one main reason

Answer 52

yw