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Trigonometry with natural parameter

Source: Moldova TST 2020

March 7, 2020
trigonometryalgebrasystem of equationsparameterization

Problem Statement

Let nn be a natural number. Find all solutions xx of the system of equations {sinx+cosx=n2tgx2=n−23\left\{\begin{matrix} sinx+cosx=\frac{\sqrt{n}}{2}\\tg\frac{x}{2}=\frac{\sqrt{n}-2}{3}\end{matrix}\right. On interval [0,π4).\left[0,\frac{\pi}{4}\right).
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