1996 AIME Problem 10

Question

Find the smallest positive integer solution to

\tan(19x^\circ)=\frac{\cos96^\circ+\sin96^\circ}{\cos96^\circ-\sin96^\circ}.

Answer

Recognize the right-hand side using the tangent addition formula

Solve the resulting congruence modulo

180

The tangent addition formula gives

\begin{gathered}\frac{\cos96^\circ+\sin96^\circ}{\cos96^\circ-\sin96^\circ}\\=\tan(45^\circ+96^\circ)\\=\tan141^\circ.\end{gathered}

Hence

19x\equiv141\pmod {180}.

Since

19^2=361\equiv1\pmod {180},

multiplying by

19

gives

x\equiv19\cdot141\equiv159\pmod {180}.

The smallest positive solution is

159.

Problem 10 in other years