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Let $A_1,A_2,...,A_n$ be a regular polygon of $n$ sides whose center is origin $O$. Let the complex numbers representing vertices $A_1,A_2,...,A_n$ be $z_1,z_2,z_3,...,z_n$ respectively. Let $OA_1=OA_2=...=OA_n=1$. Find the value of $|A_1A_2||A_1A_3|...|A_1A_n|$

My Attempt:

In the triangle $A_1OA_2$, the angle $O$ is $\dfrac{2\pi}n$, so, $$\cos\frac{2\pi}n=\frac{1^2+1^2-(A_1A_2)^2}{2\times1\times1}\\\implies(A_1A_2)^2=2-2\cos\frac{2\pi}n=4\sin^2\frac\pi n\\\implies|A_1A_2|=2\sin\frac\pi n$$

Therefore, required product is $\left(2\sin\dfrac\pi n\right)\left(2\sin\dfrac{2\pi}n\right)...\left(2\sin\dfrac{(n-1)\pi}n\right)$

How to proceed next?

aarbee
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  • Let $f(x)=(x-z_1)\times\cdots\times(x-z_n)=x^n-1$ (if we take $z_1=1$). Then $|f(x)/(x-z_1)|$ evaluated at $x=z_1$ gives $|(z_1-z_2)\times\cdots\times(z_1-z_n)|$. – Gerry Myerson Feb 06 '22 at 12:22

0 Answers0