# COmplex no

• Apr 14th 2013, 09:46 AM
Tutu
COmplex no
Hi this is about complex numbers and I apologize if it doesn't fall under this thread!

I really need help with this one, I really can't figure it out. Its just one question but there's a lot to refer to from previous parts, I know it taxing and I would appreciate your help so so much!

Use the results of ___________ to prove
$\sin^3\theta\cos^3\theta= \frac{1}{32}(3\sin2\theta-\sin6\theta)$

Now
_______ will refer to the previous parts of this question, ( I don't know which part will be helpful in solving the question so I'm going to put all the parts down..)

a.) If z = $cis\theta$ prove that $z^{n}+\frac{1}{z^{n}} = 2\cosn\theta$
b.) Use the binomial theorem to expand $(z+\frac{1}{z})^{3}$ and simplify your result.
c.)By using a and b above, show that
$\cos^3\theta=\frac{1}{4}\cos3\theta+\frac{3}{4}\co s\theta$
d.) Hence show the exact value of $\cos^3\frac{13\pi}{12}$ is $\frac{-5sqrt(2)-3sqrt(6)}{16}$

e.) Show that if z = $cis\theta$ then $z^{n}-\frac{1}{z^{n}} = 2i\sinn\theta$ and hence that $\sin^3\theta=\frac{3}{4}\sin\theta-\frac{1}{4}\sin3\theta$

I know how to solve parts a to e, I'm just stuck at the very last part f.) which question is at the start of this post, please help me, thank you so so so much!