ВН – высота. \(BC = 6,\,\,\,\,\,\,AD = 12,\,\,\,\,\,\,\sin A = 0,8.\)
\(AH = \frac{{AD — BC}}{2} = \frac{{12 — 6}}{2} = 3.\)
По основному тригонометрическому тождеству:
\({\sin ^2}A + {\cos ^2}A = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\cos ^2}A = 1 — {0,8^2} = 0,36\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\cos A = 0,6.\)
По определению косинуса из треугольника АВН:
\(\cos A = \frac{{AH}}{{AB}}\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,0,6 = \frac{3}{{AB}}\,\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,AB = 5.\)
Ответ: 5.