Question

Precalc !

Answer 1

z1 = 1 + sqrt(3) * i

Part I: Write z1 in polar form.
Part II: Apply De Moivre's theorem to find the modulus and angles of the roots of z1.
Part III: What are the fourth roots of z1 = sqrt(3) + i ?

Answer 2

So, for part I I have:
\[2(\cos(\frac{ \pi }{ 3 } )+ isin(\frac{ \pi }{ 3 }))\]

Answer 3

Ya?

Answer 4

Yes

Answer 5

Kay. For part II, I got:
\[\sqrt[4]{2}(\cos(\frac{ \pi }{ 12 })+isin \frac{ \pi }{ 12 }))\]
and three more angles, the same thing except instead of pi/12, I have 7pi/12, 13pi/12 and 19pi/12

Answer 6

Part 2 is missing something in your original post,
oh so it's 4th roots?

Answer 7

For the modulus, would it be \[\sqrt[4]{2}\]
?
And for the angles, would I list the whole thing like above? or would I just put pi/12, 7pi/12, etc.?

Answer 8

Oh yes, sorry! At the beginning it says "Find the fourth roots of the complex number z1 = 1 + sqrt3 * i

Answer 9

Probably list them individually like that, not the whole thing.

Answer 10

Those values all look correct.

Answer 11

Cool! Now for Part III, I'm confused as to how it relates to the original question.. .

Answer 12

Hehe, fourth roots = the whole thing :)

Answer 13

Ahha I see xD I was confused because the order was different, so I thought it was a different equation...
At first they listed 1 + sqrt3 * i
then they put sqrt3 + i

Answer 14

So for part III, it would be this?
2√4(cos(π12)+isinπ12))

Answer 15

All four of them though?

Answer 16

Oh no no you're right, I didn't notice it was different :OO
I have been trickeded!

Answer 17

\[\large\rm \sqrt{3}+i\quad=\quad 2\left(\frac{\sqrt3}{2}+\frac12i\right)\]This refers to a different angle.

Answer 18

Wait so it IS different? :O

Answer 19

shoots

Answer 20

So for Part II, do you think I should put the whole thing? Or still just the pi/12, 7pi/12, things

Answer 21

Based on the wording, it seems like the second step only wants the parts listed. Whereas in the third step we'll list "the whole thing", all 4 of them.

Answer 22

But for the new angles.

Answer 23

for part III,\[\large z^{1/n}= r^{1/n}(\cos \frac{ \theta + 2k \pi }{ n }+ i \sin \frac{ \theta + 2k \pi }{ n })\]for fourth roots means you need to plug in k=0, then k=1, k=2, and k=3.

Answer 24

So basically the same thing I did for part II?
How is part III related to the original equation?

Answer 25

It doesn't relate... which is weird that they're calling this new complex value z1 still.

Answer 26

Oh nuts. (Cashews!!! <3) That is weird... hm

Answer 27

And it's like a subset thingy under the same problem. Why would they call it "part 3"?

Answer 28

hmm weirdddd

Answer 29

Typo is my guess.

Answer 30

Cashews are good, esp. salted

Answer 31

This whole course is wacky

Answer 32

And Agent, I prefer plain but I do love some curried cashews every now and then

Answer 33

Wacky tabbacky.

Cashews? Nawwww.. I aint bout that life.

Answer 34

Idk if i've tried curried but sounds good.

Answer 35

ZEP! Your tastebuds are what's wacked.
Agent - real good.

Answer 36

I do like curry. Thai/Indian.

Answer 37

YES. Mmmm.

\[\sqrt[4]{2}(\cos(\frac{ \pi }{ 24 })+isin \frac{ \pi }{ 24 })\]
For part III?

Answer 38

For the very first of your four roots?
Yes looks good.

Answer 39

Cool cool. Cashew hater -_- (do you at least like pistachios?)
13pi/24, 26pi/24 and 39pi/24 for the other three angles?

Answer 40

+12 in the numerator each time?

1, 13, 25, 37, right?

Answer 41

I've had foods with pistachios.. never tried them on their own though.
This bakery near me used to make these amazing green muffins, pistachios in them

Answer 42

You. Have. never. tried. pistachios. BEFORE??!?!?!
What world do you live in?

Answer 43

lol :P

Answer 44

pi/24, 13pi/24, 25pi/24, 37pi/24 look better? nice catch

Answer 45

The normal world.. where we understand that spinach is not a beverage LOL

Answer 46

Oh my gosh zep. You are missing out.

Answer 47

Haha xD

Answer 48

ya those angles look better

Answer 49

Cool! Well, thanks to my two favorite open studiers :3
I'll convert you to the spinach-eating, almond butter-loving side soon enough! We have figs over here, too. Cashews are optional.

Answer 50

:3

Answer 51

Spinach flavoured beverages? Pukeatronic.

Answer 52

Ya she puts SPINACH in a blender... :[
Makes ... ah sorry I threw up a little just thinking about it...
smoothies <:O

Answer 53

Ew. I repeat my earlier statement. Pukeatronic.

Answer 54

XD

Answer 55

Btw @abbles a while ago, at the students house (the Indian kids who feed me every week) they had some cashews that they'd cooked in something, but i forget what it was.

They were tasty, but i don't think it was curry, ugh i can't remember though.

Answer 56

Ooh that sounds good Agent.

And spinach puree is not as gross as McDonald's sludge crap. -_-

Answer 57

@abbles... so... you're pointing out how non-disgusting spinach puree is, by comparing it... to McDonald's?