## Complex Numbers, De Moivre

Discussions for the Core part of the syllabus. Algebra, Functions and equations, Circular functions and trigonometry, Vectors, Statistics and probability, Calculus. IB Maths HL Revision Notes

### Complex Numbers, De Moivre

De Moivre’s Theorem - IB Mathematics HL

How can we find the following complex number

$z=(1- \sqrt3 i)^{24}$ using De Moivre’s Theorem?

Thanks
lora

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Joined: Wed Apr 10, 2013 7:36 pm

### Re: Complex Numbers, De Moivre

De Moivre’s Theorem - IB Math HL

The De Moivre’s theorem is given by the following formula

$z^n=( |z|(cos \theta +isin \theta))^n$

$= |z|^n (cos \theta +isin \theta)^n$

$= |z|^n (cos (n \theta) +isin (n \theta))$

Before apply De Moivre’s theorem we must convert the Cartesian form to polar form

So

$z=(1- \sqrt3 i)=$

$z= 2(cos ( -\frac{\pi}{3}) +isin( -\frac{\pi}{3}))$

Therefore,

$(1-\sqrt3 \cdot i)^ {24}=$

$=(2(cos(-\frac{\pi}{3})+isin(-\frac{\pi}{3})))^{24}$

$=2^{24} (cos(-24 \frac{\pi}{3}) +isin(-24 \frac{\pi}{3}))$

$=2^{24} (cos(-8 \pi) +isin(-8 \pi))$

$=2^{24} (1) +0 i$

$=2^{24}$

Hope these help!!

Hope these help!!
lily

Posts: 0
Joined: Mon Jan 28, 2013 8:04 pm

### Re: Complex Numbers, De Moivre

Thanks Lily!!
lora

Posts: 0
Joined: Wed Apr 10, 2013 7:36 pm