# Trigo + Complex

Let $A_1,A_2,...A_n$ be a regular polygon of $n$ sides whose centre is origin $O$.Llet the complex numbers representing the vertices $A_1,A_2.....A_n$be$z_1,z_2....z_n$ respectively.The radius is of length unity.Then, $\prod_{n=2}^{n}|A_1A_n|=\mathbb{?}$

4 years, 2 months ago

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Let the circumradius be $R$. Now we consider all $z$ that satisfy the circle and write a certain expansion:

$z^n-R^n=(z-R)(z^{n-1} + Rz^{n-2}+Rz^{n-3} + \dots + zR^{n-2}+R^{n-1})$

Then we have $z_1=R$ and $,z_2,z_3\dots z_n$ as roots of $(z^{n-1} + Rz^{n-2}+Rz^{n-3} + \dots + zR^{n-2}+R^{n-1})$ , So we can write :

$(z-z_2)(z-z_3) \dots (z-z_n)=z^{n-1} + Rz^{n-2}+Rz^{n-3} + \dots + zR^{n-2}+R^{n-1})$

Since $z_1=R$ is one of the $z$'s , we substitute $z=R$ in RHS . Thus the equation becomes:

$|(z-z_1)||(z-z_2)||(z-z_3)| \dots |(z-z_n)|=nR^{n-1} \dots (1)$

To calculate $R$ we use cosine rule and get $R=\dfrac{1}{2\sin\left(\dfrac{\pi}{n}\right)}$. So finally

$\displaystyle\prod_{i=2}^n |A_1A_n|=\dfrac{n}{2\sin^{n-1}\left(\dfrac{\pi}{n}\right)}$

Edit: The problem is edited and the circumradius is unity and not the side. Hence substituting $R=1$ in $(1)$ we get:

$\displaystyle\prod_{i=2}^n |A_1A_n|=n\times 1^{n-1} = \boxed{n}$

- 4 years, 2 months ago

Answer is $n$.

- 4 years, 2 months ago

Are you sure that the side equals unity? Or the circumradius?

- 4 years, 2 months ago

Oh!Sorry!R=1!

- 4 years, 2 months ago

Yes , now the answer is $n$.

- 4 years, 2 months ago

I asked this question a while ago and Alan Yan wrote a nice solution

- 4 years, 2 months ago

- 4 years, 2 months ago

Thanks a lot sir!

- 4 years, 2 months ago