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.
Solution
c = sin α cos β + cos α sin β = a√(1-b2) + b√(1-a2). Squaring, c2 = a2(1-b2) + b2(1-a2) +2ab√(1-a2)√(1-b2). Leaving the remaining radical on one side and squaring again, we get c4 - 2(a2(1-b2)+b2(1-a2))c2 + (a2(1-b2)+b2(1-a2))2 - 4a2b2(1-a2)(1-b2) = 0, or c4 - 2(a2-2a2b2+b2)c2 + (a2-b2)2 = 0.
We know that one root of this quartic in c is c1 = a√(1-b2) + b√(1-a2). Since a and b only appear in the quartic as squares, the other roots must be c2 = -a√(1-b2) + b√(1-a2), c3 = a√(1-b2) - b√(1-a2), and c4 = -a√(1-b2) - b√(1-a2). Now c1 = c2 iff a√(1-b2) = 0 ie a = 0 or b = ±1. Similarly c1 = c3 iff a = ±1 or b = 0. We find c1 = c4 iff a√(1-b2) = -b√(1-a2) or a = -b. Similarly, we find c2 = c3 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.
© John Scholes
jscholes@kalva.demon.co.uk
29 Oct 2003
Last corrected/updated 29 Oct 03