Tuesday, 22 November 2016

X + Y + Z = XYZ

Since I have been sick the past few days and missed a math conference, the math worm in me wants me to write something mathematical today.

I came across an article lately, which has continued to amaze me.  The simple equation

X + Y + Z = XYZ

has many solutions such as (X,Y,Z) = (1,2,3) or  (0,3,-3,).  But to find ALL real solutions, trigonometry surprisingly gives a perfect answer.  All we need is the identity

Consider any solution (X,Y,Z).  Since the range of the tangent function is the set of all real numbers, we can always find real numbers A, B and C such that

X = tan A, Y = tan B and Z = tan C.

So the equation

+ Y + Z = XYZ

is equivalent to

tan A + tan B + tan C = tan A tan B tan C,

which is equivalent to

tan (A + B + C) = 0,

yielding the solutions

A + B + C = nπ

for any integer n.

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