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I've been trying to help a friend with something that I learnt in class but I'm now finding it hard to solve myself. The problem goes as follows:

Use geometry to show that |z^{3}-z^{-3}| = 2sin3θ

For z=cisθ, 0<θ<∏/6

Now, I chose ∏/12 as my angle and plotted all this on an Argand diagram, but I don't know how to prove that |z^{3}-z^{-3}| = 2sin3θ in such general form. I know that it is right, but I don't know how to get there. Right now I have a isosceles triangle with two lengths known and two angles known (both 3θ). I know the triangle is right-angled but I don't know if I can use this information since that is specific to my chosen angle, right?

I've spent quite some time on it and I'd really like to understand it. So far I've got xsin3θ=1 where x = |z^{3}-z^{-3}|.

Please help!

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# Modulus of the difference of two complex numbers

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