Problem
ALG-B1-M06-P023 Rational trigonometric sums
A number \(t\) is such that both sums \(S=\sin 48t+\sin 49t\) and \(C=\cos 48t+\cos 49t\) are rational. Prove that both terms in the sum \(C\) are rational.
Find \(S^2+C^2\). Then recall that \(\cos nt\) is a polynomial in \(\cos t\).
Since \(S\) and \(C\) are rational, \(S^2+C^2\) is rational. Expanding, \[S^2+C^2=2+2(\sin48t\sin49t+\cos48t\cos49t)=2+2\cos t.\] Hence \(\cos t\) is rational.
For every positive integer \(n\), \(\cos nt\) is a polynomial with integer coefficients in \(\cos t\). Therefore \(\cos48t\) is rational. Since \(C=\cos48t+\cos49t\) is rational, \(\cos49t=C-\cos48t\) is rational as well.
Source analysis: the objects are two rational sums of neighboring sine and cosine terms; the first approach is to try to split the sum; the hidden observation is that the squared sums give \(\cos t\); the key ideas are the trigonometric dot product and the Chebyshev polynomial idea. The numbers are changed and the statement is rewritten.