
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
$\displaystyle \sin^3\theta\cos^3\theta= \frac{1}{32}(3\sin2\theta\sin6\theta)$
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 = $\displaystyle cis\theta$ prove that $\displaystyle z^{n}+\frac{1}{z^{n}} = 2\cosn\theta$
b.) Use the binomial theorem to expand $\displaystyle (z+\frac{1}{z})^{3}$ and simplify your result.
c.)By using a and b above, show that
$\displaystyle \cos^3\theta=\frac{1}{4}\cos3\theta+\frac{3}{4}\co s\theta$
d.) Hence show the exact value of $\displaystyle \cos^3\frac{13\pi}{12}$ is $\displaystyle \frac{5sqrt(2)3sqrt(6)}{16}$
e.) Show that if z = $\displaystyle cis\theta$ then $\displaystyle z^{n}\frac{1}{z^{n}} = 2i\sinn\theta$ and hence that $\displaystyle \sin^3\theta=\frac{3}{4}\sin\theta\frac{1}{4}\sin3\theta$
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!