DeMoivre's Theorem

Any pair $\left(\vphantom{ a,b}\right.$a, b$\left.\vphantom{ a,b}\right)$ of real numbers can be represented in polar coordinates with a = r cosθ and b = r sinθ where r = $\sqrt{{%
a^{2}+b^{2}}}$ is the distance from the point $\left(\vphantom{ a,b}\right.$a, b$\left.\vphantom{ a,b}\right)$ to the origin and θ is an angle satisfying tanθ = ${\frac{{a}}{{b}}}$. Thus any complex number can be written in the form

z = r$\displaystyle \left(\vphantom{ \cos \theta +i\sin \theta }\right.$cosθ + i sinθ$\displaystyle \left.\vphantom{ \cos \theta +i\sin \theta }\right)$

where r = $\left\vert\vphantom{ \,z}\right.$ z$\left.\vphantom{ \,z}\right\vert$.

DeMoivre's Theorem says that if z = r$\left(\vphantom{ \cos \theta +i\sin \theta
}\right.$cosθ + i sinθ$\left.\vphantom{ \cos \theta +i\sin \theta
}\right)$ and n is a positive integer, then

zn = $\displaystyle \left(\vphantom{ r\left( \cos t+i\sin t\right) }\right.$r$\displaystyle \left(\vphantom{ \cos t+i\sin t}\right.$cos t + i sin t$\displaystyle \left.\vphantom{ \cos t+i\sin t}\right)$$\displaystyle \left.\vphantom{ r\left( \cos t+i\sin t\right) }\right)^{{n}}_{}$ = rn$\displaystyle \left(\vphantom{ \cos
nt+i\sin nt}\right.$cos nt + i sin nt$\displaystyle \left.\vphantom{ \cos
nt+i\sin nt}\right)$

With Scientific Notebook, you can obtain this result for small values of n by the sequence of operations Expand followed by MenuDialogCombine + Trig Functions and then Factor.

$\blacktriangleright$ Expand, Combine + Trig Functions, Factor

$\left(\vphantom{ r\left( \cos t+i\sin t\right) }\right.$r$\left(\vphantom{ \cos t+i\sin t}\right.$cos t + i sin t$\left.\vphantom{ \cos t+i\sin t}\right)$$\left.\vphantom{ r\left( \cos t+i\sin t\right) }\right)^{{3}}_{}$ =   r3cos3t + 3ir3cos2t sin t - 3r3cos t sin2t - ir3sin3t


=  r3cos 3t + ir3sin 3t =  r3$\left(\vphantom{ \cos 3t+i\sin 3t}\right.$cos 3t + i sin 3t$\left.\vphantom{ \cos 3t+i\sin 3t}\right)$

Or, you can use Simplify followed by Combine + Trig Functions and Factor.

$\blacktriangleright$ Simplify, Combine + Trig Functions, Factor

$\left(\vphantom{ r\left( \cos t+i\sin t\right) }\right.$r$\left(\vphantom{ \cos t+i\sin t}\right.$cos t + i sin t$\left.\vphantom{ \cos t+i\sin t}\right)$$\left.\vphantom{ r\left( \cos t+i\sin t\right) }\right)^{{3}}_{}$ =  4r3cos3t + 4ir3cos2t sin t - 3r3cos t - ir3sin t


=  r3cos 3t + ir3sin 3t =  r3$\left(\vphantom{ \cos 3t+i\sin 3t}\right.$cos 3t + i sin 3t$\left.\vphantom{ \cos 3t+i\sin 3t}\right)$ 6pt

You can get the same results in complete generality by working with reit, since

$\displaystyle \left(\vphantom{ re^{it}}\right.$reit$\displaystyle \left.\vphantom{ re^{it}}\right)^{{n}}_{}$ =  rneitn

You can get the identity

reit = r$\displaystyle \left(\vphantom{ \cos t+i\sin t}\right.$cos t + i sin t$\displaystyle \left.\vphantom{ \cos t+i\sin t}\right)$

with Evaluate. (You will find that CTRL +E has no effect on the expression reit. This is one circumstance where Evaluate and CTRL +E produce a different result.)

$\blacktriangleright$ Evaluate, Factor

reit = r cos t + ir sin t = r$\left(\vphantom{ \cos t+i\sin t}\right.$cos t + i sin t$\left.\vphantom{ \cos t+i\sin t}\right)$ 6pt


\begin{example}
The following sequence of operations will change the complex nu...
...ft( \cos .87606+i\sin .87606\right) =5.0+6.0i
\end{displaymath}
\end{example}