Complex numbers problem

**rednest** · April 18th 2008, 12:00 AM

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

**CaptainBlack** · April 18th 2008, 01:48 AM

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

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