# Thread: Using De Moivre's Theorem

1. ## Using De Moivre's Theorem

Hello,
Doing a practice paper and found this question, not sure what to do:
Write down de Moivre's Theorem for n=5. hence show that for $\sin \theta \neq 0$

$\frac{\sin 5 \theta}{\sin \theta} = A cos^4 \theta + B \cos^2 \theta + C$

where A, B, C are constants to be determined.
Deduce the limiting value of $\frac{\sin 5 \theta}{\sin \theta}$ as $\theta$ tends to zero.
I've written de Moivre for n=5:

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

But don't know where to go from there, at all.

Thanks

2. Originally Posted by alexgeek
Hello,
Doing a practice paper and found this question, not sure what to do:

I've written de Moivre for n=5:

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

But don't know where to go from there, at all.

Thanks
From De'moivre's theorem it follows that

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

Now I suggest you expand the left hand side of this identity and equate the imaginary part of it to $\sin (5 \theta)$.

3. Finally got it, tried expanding it by hand then realised I could do it by binomial. Cheers