Question

find 1/z by first putting the complex number in polar form. z= 56(cis(11pi/6))

Answer 1

z = 28(sqrt(3))-28(i) in rectangular form.

Answer 2

\[\Large z=28\sqrt3 - 28i \] ?

Answer 3

correct that is what z is in rectangular form

Answer 4

So, the first most general I always ask myself, in which quadrant is this complex number?

Answer 5

\[z=x+iy \]

Answer 6

its in the 4th quadrant, were on the same page then cause thats what i did first :)

Answer 7

Exactly, I always try keeping that in mind for the phase later on.

So an argument is given by:
\[\Large |z|= \sqrt{x^2+y^2}= \sqrt{3136}=56 \]

Answer 8

And a phase is given by \[\Large \tan \varphi = \frac{y}{x}=\frac{-28}{28\sqrt{3}}=\frac{-1}{\sqrt{3}} \]

Answer 9

i got 11pi/6

Answer 10

\[\Large \varphi =-30 \\
\\ \Large\varphi'=330=\frac{11\pi}{6} \]

Answer 11

yes so we agree. Did you just want to check on that?

Answer 12

i have to find 1/z

Answer 13

it has something to do with subtracting angle measures

Answer 14

ohh, do you know the laws of DeMoivre?

Answer 15

\[\Large \frac{1}{z}= \frac{1}{56\text{cis}(\frac{11\pi}{6})}= \left( 56\text{cis}\left(\frac{11\pi}{6}\right)\right)^{-1} \]

Answer 16

And now by the Law of De Moivre
\[\Large \frac{1}{56}\text{cis}\left(-\frac{11\pi}{6}\right) \]

Answer 17

Maybe that's why they want you to get it into polar form, otherwise I wouldn't see a reason to do that, I would rather get the complex number out of the denominator in the old fashioned way - by the complex conjugate - and then deal with this argument instead.

But De Moivre is a more elegant way.

Answer 18

thats what my tutor did, seemed as though she only knew the old way. Dont reinvent the wheel thats my moto, either way, your insight into writing it with an exponent has shown me the solution, thank you!

Answer 19

you're very welcome.