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# FP2 Roots of unity watch

1. By considering the roots of z^5 = 1 , show that cos(2pi/5) + cos(4pi/5) + cos(6pi/5) + cos(8pi/5) = -1

I found the solutions of z^5 = 1 in polar form but what am I supposed to do next?
2. (Original post by thers)
By considering the roots of z^5 = 1 , show that cos(2pi/5) + cos(4pi/5) + cos(6pi/5) + cos(8pi/5) = -1

I found the solutions of z^5 = 1 in polar form but what am I supposed to do next?
The sum of the 5th roots of unity is zero. What is the real part of this sum?
3. Ok thanks I've done it now. I'm also stuck on:

Solve 1 - 2z + 4z^2 + 8z^3 = 0 I don't even know how to start.
4. (Original post by thers)
Ok thanks I've done it now. I'm also stuck on:

Solve 1 - 2z + 4z^2 + 8z^3 = 0 I don't even know how to start.
Is the last term supposed to be instead of '+'? If not, then I expect the solution will be very nasty (and possibly unattainable without some sort of very advanced numerical method).

Assuming you have made the above typo; spot the GP.
5. does the question ask for (assuming it is +8) the real root or all 3 roots?
6. I looked at the answers in the back of the textbook and it was 1/2, +1/2i, -1/2i so the textbook must have made a typo when writing the question. Solved it now.
7. yup - those are the factors for -8z^3....

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