10th Eötvös 1903 Problem 2 α and β are real and a = sin α, b = sin β, c = sin(α+β). Find a polynomial p(x, y, z) with integer coefficients, such that p(a, b, c) = 0. Find all values of (a, b) for which there are less than four distinct values of c.
c = sin α cos β + cos α sin β = a√(1-b 2 ) + b√(1-a 2 ). Squaring, c 2 = a 2 (1-b 2 ) + b 2 (1-a 2 ) +2ab√(1-a 2 )√(1- b 2 ). Leaving the remaining radical on one side and squaring again, we get c 4 - 2(a 2 (1-b 2 )+b 2 (1-a 2 ))c 2 + (a 2 (1-b 2 )+b 2 (1-a 2 )) 2 - 4a 2 b 2 (1-a 2 )(1-b 2 ) = 0, or c 4 - 2(a 2 -2a 2 b 2 +b 2 )c 2 + (a 2 -b 2 ) 2 = 0. We know that one root of this quartic in c is c 1 = a√(1-b 2 ) + b√(1-a 2 ). Since a and b only appear in the quartic as squares, the other roots must be c 2 = -a√(1-b 2 ) + b√(1-a 2 ), c 3 = a√(1-b 2 ) - b√(1-a 2 ), and c 4 = - a√(1-b 2 ) - b√(1-a 2 ). Now c 1 = c 2 iff a√(1-b 2 ) = 0 ie a = 0 or b = ±1. Similarly c 1 = c 3 iff a = ±1 or b = 0. We find c 1 = c 4 iff a√(1-b 2 ) = -b√(1-a 2 ) or a = -b. Similarly, we find c 2 = c 3 iff a = b, whilst c2 = c4 and c3 = c4 do not give any new conditions. Thus we have less than 4 distinct roots if a = 0, ±1, or b = 0, ±1, or a = ±b. 10th Eötvös 1903 © John Scholes jscholes@kalva.demon.co.uk 29 Oct 2003 Last corrected/updated 29 Oct 03