Nếu \(0^\circ var DOMAIN = "https://hoc247.net/"; var ST...

* Hướng dẫn giải

Ta có:

\(1 = {\sin ^2}\alpha  + {\cos ^2}\alpha  \)\(= {\sin ^2}\alpha  + {\left( {\dfrac{1}{2} - \sin \alpha } \right)^2} \)\(= 2{\sin ^2}\alpha  - \sin \alpha  + \dfrac{1}{4}\)

\( \Leftrightarrow 2{\sin ^2}\alpha  - \sin \alpha  - \dfrac{3}{4} = 0 \)

\(\Leftrightarrow \sin \alpha  = \dfrac{{1 \pm \sqrt 7 }}{4}\)

Mà \(0^\circ  < \alpha  < 180^\circ \) nên \(\sin \alpha  > 0\). Chọn \(\sin \alpha  = \dfrac{{1 + \sqrt 7 }}{4}\).

Suy ra \(\cos \alpha  = \dfrac{1}{2} - \dfrac{{1 + \sqrt 7 }}{4} = \dfrac{{1 - \sqrt 7 }}{4}\).

=> \(\tan \alpha  = \dfrac{{\sin \alpha }}{{\cos \alpha }} = \dfrac{{1 + \sqrt 7 }}{{1 - \sqrt 7 }} \)\(\,= \dfrac{{{{\left( {1 + \sqrt 7 } \right)}^2}}}{{1 - 7}} =  - \dfrac{{4 + \sqrt 7 }}{3}\).

Vậy \(\left( {m,n} \right) = \left( {4,7} \right)\).