$\begin{array}{l} {0^o} < x < {180^o} \Rightarrow \sin x > 0\\ \left( {\sin x + \cos x} \right) = \dfrac{1}{2}\\ \Leftrightarrow {\left( {\sin x + \cos x} \right)^2} = \dfrac{1}{4} \Leftrightarrow 1 + 2\sin x\cos x = \dfrac{1}{4}\\ \Leftrightarrow \sin x\cos x = \dfrac{{ - 3}}{8}\\ \sin x,\cos x\,là \,nghiệm\,của\,phương\,trình\,\\ {a^2} - \dfrac{1}{2}a - \dfrac{3}{8} = 0 \Leftrightarrow \left[ \begin{array}{l} a = \dfrac{{1 + \sqrt 7 }}{4}\\ a = \dfrac{{1 - \sqrt 7 }}{4} \end{array} \right.\\ \Rightarrow \sin x = \dfrac{{1 + \sqrt 7 }}{4},\cos x = \dfrac{{1 - \sqrt 7 }}{4}\\ \tan x = \dfrac{{\sin x}}{{\cos x}} = \dfrac{{1 + \sqrt 7 }}{4}:\dfrac{{1 - \sqrt 7 }}{4} = - \dfrac{{4 + \sqrt 7 }}{3}\\ \Rightarrow p = 4,q = 7\\ \Rightarrow \left( {p;q} \right) = \left( {4;7} \right) \end{array}$
