Denote by V(x,y) the vertex at ordered pair (x,y) in the Cartesian coordinate system. Denote...

veceriraby

veceriraby

Answered

2022-01-29

Denote by V(x,y) the vertex at ordered pair (x,y) in the Cartesian coordinate system. Denote by A(x,y) the measure of angle V(x,y)V(0,0)V(1,0). Let
θ=0i,j5  (i,j)(0,0)A(i,j)
Find θ (mod 2π)
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
0i,j5  (i,j)(0,0)arctanij
But I don't know how to evaluate this.arctan(x+y) formula fails. I think there is a different approach I am not aware of.
Is there a nice way to generalize to
0i,jk  (i,j)(0,0)A(i,j)

Answer & Explanation

Jason Duke

Jason Duke

Expert

2022-01-30Added 11 answers

I got it now, Note that if we represent Am,n+An,m in terms of complex numbers we get arg(m+ni)(n+mi)=π2. There are k(k+1) of these numbers (ignnoring diagonals so we get (kk+12)(π2). Adding back the diagonal (ignoring (0,0)) we get kπ4
(kk+12)(π2)+kπ4=k(k+2)π4
Which is the general solution. Substituting k=5 gives 35π43π4(mod 2π)

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