Question

Help checking work with De Moivre's theorem?
Let z=13+7i and w=3[cos(1.43)+isin(1.43)]
A) Convert z to polar form
My work:
r=√13^2+7^2=√218=14.765
Θ=tan^-1 7/13=0.494
(14.765, 0.494)

B) Calculate zw using De Moivre's theorem
My work:
3 cos 1.43=0.421
3 i sin 1.43=2.970 i
Multiply both by (13+7i)
5.473+2.947i+-20.79+38.61i=-15.317+41.557i

C) Calculate z/w using De Moivre's theorem
My work:
(13+7i)/(0.421)=30.879+16.627i
(13+7i)/(2.970i)=2.357-4.377i
33.236+12.25i

Answer 1

it says "using de Moivre" so maybe you should multiply/divide r's and add/subtract \(\theta\)'s before putting into rectangular form

Answer 2

How would you do that? I'm not sure how to utilize de Moivre's here...

Answer 3

multiply \(3\times \sqrt{218}\) for the modulus (absolute value) then add the angles

Answer 4

btw your answer is wrong there

Answer 5

i mean the multiplication
lets assume \[13+7i=\sqrt{218}\cos(.494)+i\sin(.494))\]

Answer 6

then to get the product, you multiply the numbers out front and add the angles
that is demoivre

Answer 7

I know it's a lot to ask, but could you walk me through it? I'm not sure what to multiply and what to add.

Answer 8

oh damn, your angle is wrong too

Answer 9

lets check all

Answer 10

\[13+7i\] in polar form you need \(r=\sqrt{13^2+7^2}=14.765\) rounded so that part is right

Answer 11

\[\theta=\tan^{-1}(\frac{7}{13})=0.494\] ok that part is right too, sorry

Answer 12

so \[13+7i=14.765\left(\cos(.494)+i\sin(.494)\right)\]

Answer 13

then \[14.765\left(\cos(.494)+i\sin(.494)\right)\times 3\left(\cos(1.43)+i\sin(1.43)\right)\] is the last job

Answer 14

it will be \[14.765\times 3\left(\cos(.494+1.43)+i\sin(.494+1.43)\right)\]

Answer 15

multiply the modulus, add the angles

Answer 16

you still have to do the arithmetic ( use a calculator) but that is what you get before doing it

Answer 17

let me know if it is not clear

Answer 18

Wait, so 14.765x3(cos(.494+1.43)+i sin (.494+1.43)) is z*w, right?

Answer 19

yes that is right

Answer 20

still have to do the arithmetic though

Answer 21

What would the equation to find z/w be? Is it 14.765x3(cos(.494-1.43)+i sin (.494-1.43))?

Answer 22

close

Answer 23

subtract the angles is right
but divide the modulus

Answer 24

So 14.765/3(cos(.494-1.43)+i sin (.494-1.43))? Sorry for all the questions!

Answer 25

no problem
and yes

Answer 26

don't forget to do the arithmetic, or your teacher will think you are real lazy

Answer 27

I got 2.9186-3.9629i for z/w and -15.321+41.561i for z*w.

Answer 28

i will take your word for it

Answer 29

Alright, sweet, thank you so much! I've been stuck on this problem for a while.

Answer 30

it didn't say to rewrite in standard form
i would have left it in polar form

Answer 31

yw

Answer 32

Wait is leaving it in polar form just 44.295(cos(1.924)+i sin (1.924)) ?