 veceriraby

2022-01-29

Denote by ${V}_{\left(x,y\right)}$ the vertex at ordered pair (x,y) in the Cartesian coordinate system. Denote by ${A}_{\left(x,y\right)}$ the measure of angle $\mathrm{\angle }{V}_{\left(x,y\right)}{V}_{\left(0,0\right)}{V}_{\left(1,0\right)}$. Let

Find
I understand the question and everything, but I am slightly overwhelmed about how to find an organized approach to computing the sum of angles. I got the value down to the sum

But I don't know how to evaluate this.$\mathrm{arctan}\left(x+y\right)$ formula fails. I think there is a different approach I am not aware of.
Is there a nice way to generalize to Jason Duke

Expert

I got it now, Note that if we represent ${A}_{m,n}+{A}_{n,m}$ in terms of complex numbers we get $arg\left(m+ni\right)\left(n+mi\right)=\frac{\pi }{2}$. There are $k\left(k+1\right)$ of these numbers (ignnoring diagonals so we get $\left(k\frac{k+1}{2}\right)\left(\frac{\pi }{2}\right)$. Adding back the diagonal (ignoring (0,0)) we get $k\frac{\pi }{4}$
$\left(k\frac{k+1}{2}\right)\left(\frac{\pi }{2}\right)+k\frac{\pi }{4}=\frac{k\left(k+2\right)\pi }{4}$
Which is the general solution. Substituting k=5 gives

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