Let $\alpha =\cos 2\pi /7+i\sin 2\pi /7$. Show that $\alpha$ is a root of $z^7-1=0$ and find the other roots in terms of $\alpha$. The number $\alpha +{\alpha }^2+{\alpha }^4$ is a root of the quadratic equation $z^2+Az+B=0$ where $A$ and $B$ are real. Find the numerical values of $A$ and $B$.

For part one, I thought that ${\alpha }^n$, where $n$ is an integer, would suffice. As for part two, I'm completely stuck of a nice solution. Thanks,

Chris

PS. Mark, I agree with you

$A=-2(\cos (2\pi /7)+\cos (4\pi /7)+\cos (8\pi /7)$

Which from my calculator = 1, but I'm having difficulty proving it.

If you say $A=-u-v$ (which was you had to show in the 1st part which Chris didn't post)

then $A=-(\alpha +{\alpha }^2+{\alpha }^4)-({\alpha }^6+{\alpha }^5+{\alpha }^3)$ (as the 2nd root is the conjugate of the 1st - by looking at the roots on the Argand Diagram you can see what the conjugate is)

$A=-(\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4+{\alpha }^5+{\alpha }^6)$

But the sum of the seven roots found above =0 so $1+\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4+{\alpha }^5+{\alpha }^6=0$

Therefore, $A=1$

For $B$, say $B=uv$

Therefore, $B=(\alpha +{\alpha }^2+{\alpha }^4)({\alpha }^6+{\alpha }^5+{\alpha }^3)$

Multiplying out and simplifying the powers of $\alpha$ that are > 6 gives:

$B=3+\alpha +{\alpha }^2+{\alpha }^3+{\alpha }^4+{\alpha }^5+{\alpha }^6$

$B=2+0$ (as the sum of the roots is 0)

$B=2$

The final part of the question was:

Show that $\cos (2\pi /7)+\cos (4\pi /7)+\cos (8\pi /7)=-1/2$ and find the value of $\sin (2\pi /7)+\sin (4\pi /7)+\sin (8\pi /7)$

I did this by comparing my answers for $A$ and $B$ with their values in sin and cos form (like Alice found). I got $\sin (2\pi /7)+\sin (4\pi /7)+\sin (8\pi /7)=\sqrt{7}/2$.

The question asked you to show carefully how you obtained the sign of your answer (since it was obtained by square rooting something) - I did this graphically by showing that $\sin (2\pi /7)+\sin (4\pi /7)$ was more positive than $\sin (8\pi /7)$ is negative - does anyone know if this would be considered sufficient in a STEP exam?