Задача 24. Найдите \(\cos \left( {\alpha -\beta } \right),\) если \(\sin \alpha = 0,8,\,\,\,\,\,\cos \beta = -0,6\) и \(\dfrac{\pi }{2} < \alpha < \pi ,\,\,\,\,\,\pi < \beta < \dfrac{{3\pi }}{2}\)
Решение
Воспользуемся основным тригонометрическим тождеством:
\({\sin ^2}\alpha + {\cos ^2}\alpha = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{0,8^2} + {\cos ^2}\alpha = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\cos ^2}\alpha = 0,36\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\cos \alpha = 0,6,\,\,\,\,}\\{\cos \alpha = -0,6.}\end{array}} \right.\)
Так как \(\dfrac{\pi }{2} < \alpha < \pi \) (II четверть), то \(\cos \alpha < 0\), то есть \(\cos \alpha = -0,6.\)
\({\sin ^2}\beta + {\cos ^2}\beta = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\sin ^2}\beta + {\left( {-0,6} \right)^2} = 1\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\sin ^2}\beta = 0,64\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\left[ {\begin{array}{*{20}{c}}{\sin \beta = 0,8,\,\,\,}\\{\sin \beta = -0,8.}\end{array}} \right.\)
Так как \(\pi < \beta < \dfrac{{3\pi }}{2}\) (III четверть), то \(\sin \beta < 0\), то есть \(\sin \beta = -0,8.\)
\(\cos \left( {\alpha -\beta } \right) = \cos \alpha \cos \beta + \sin \alpha \sin \beta = -0,6 \cdot \left( {-0,6} \right) + 0,8 \cdot \left( {-0,8} \right) = 0,36-0,64 = -0,28.\)
Ответ: \(-0,28.\)