# Complex numbers problem

• Apr 18th 2008, 12:00 AM
rednest
Complex numbers problem
Using De Moivre's Theorm, express $tan 5\theta$ in terms of $tan \theta$.
• Apr 18th 2008, 01:48 AM
CaptainBlack
Quote:

Originally Posted by rednest
Using De Moivre's Theorm, express $tan 5\theta$ in terms of $tan \theta$.

De Moivre's theorem tells us that:

$(\cos(\theta)+i \sin(\theta))^5=\cos(5\theta)+i \sin(5\theta)$

Now I will write $s$ for $\sin(\theta)$ , $c$ for $\cos(\theta)$ and $t$ for $\tan(\theta)$. Then expanding the left hand side and equating real and imaginary parts we get:

$
\cos(5\theta)=c^5-10c^3s^2+5c~s^4
$

and:

$
\sin(5\theta)=5c^4s-10c^2s^3+s^5
$
.

Therefore:

$
\tan(\theta)=\frac{5c^4s-10c^2s^3+s^5}{c^5-10c^3s^2+5c~s^4}
$
.

Now on the right divide top and bottom through by $c^5$ and you are done

RonL