Expressing a complex number in cartesian form

**FelixHelix** · March 14th 2012, 04:54 AM

Im stuck on a textbook question that has no previous examples. Express $\left(\frac{\sqrt{3}-i}{2}\right)^{101}$ in cartesian form.

OK. So i know De moivre's Theorem but not sure if i can use it here (and thats going away from Cartesian form too)

But could I say that here the form r(cos x + isin x) can be deduced because $\frac{\sqrt{3}}{2}$ is $cos\frac{\pi}{6}$ and then deduce sin.....

But not sure where to go from here!...

**princeps** · March 14th 2012, 05:30 AM

Originally Posted by FelixHelix

Im stuck on a textbook question that has no previous examples. Express $\left(\frac{\sqrt{3}-i}{2}\right)^{101}$ in cartesian form.

OK. So i know De moivre's Theorem but not sure if i can use it here (and thats going away from Cartesian form too)

But could I say that here the form r(cos x + isin x) can be deduced because $\frac{\sqrt{3}}{2}$ is $cos\frac{\pi}{6}$ and then deduce sin.....

But not sure where to go from here!...

$z = \left(\frac{\sqrt 3}{2}-\frac{1}{2}i\right)^{101}=\left(\cos \frac{11 \pi}{6}+i\sin\frac{11 \pi}{6}\right)^{101}$

$z= \cos \frac{1111 \pi}{6}+i\sin\frac{1111 \pi}{6} = \cos \frac{7 \pi}{6}+i\sin\frac{7 \pi}{6}$

$z = -\frac{\sqrt 3}{2}-\frac{1}{2}i$

**Plato** · March 14th 2012, 05:37 AM

Originally Posted by FelixHelix

Im stuck on a textbook question that has no previous examples. Express $\left(\frac{\sqrt{3}-i}{2}\right)^{101}$ in cartesian form....

$\frac{\sqrt{3}-i}{2}=2\exp\left(\frac{-\pi}{6}\right)$
Now apply De moivre's Theorem.

**FelixHelix** · March 14th 2012, 05:50 AM

I'm confused by the workings here. Could you explain how you get the original argument here?

**FelixHelix** · March 14th 2012, 05:54 AM

HI Plato, Im confused, how come your workings are different to Princeps? and how do you get r = 2 in your polar form here?

**Plato** · March 14th 2012, 08:39 AM

Originally Posted by FelixHelix

HI Plato, Im confused, how come your workings are different to Princeps? and how do you get r = 2 in your polar form here?

They are not different; they are equivalent. You should know that.
I just prefer the principal argument, $-\pi<\theta\le\pi$ .

Thread: Expressing a complex number in cartesian form

LinkBack

Thread Tools

Display

Expressing a complex number in cartesian form

Re: Expressing a complex number in cartesian form

Re: Expressing a complex number in cartesian form

Re: Expressing a complex number in cartesian form

Re: Expressing a complex number in cartesian form

Re: Expressing a complex number in cartesian form

Similar Math Help Forum Discussions

Expressing complex numbers in polar form.

Help with complex numbers in polar and cartesian form

complex number polar form

complex number exponential form

complex num to cartesian form

Search Tags