Задача 20. Найдите \({\text{ctg}}\,\,\left( {\alpha + \beta } \right)\), если \({\text{tg}}\,\,\left( {\dfrac{{3\pi }}{2} + \alpha } \right) = 3,\,\,\,\,{\text{ctg}}\,\left( {3\pi + \beta } \right) = -7\)
Решение
\({\rm{tg}}\left( {\dfrac{{3\pi }}{2} + \alpha } \right) = 3\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,-{\rm{ctg}}\alpha = 3\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{ctg}}\alpha = -3.\)
\({\rm{ctg}}\left( {3\pi + \beta } \right) = -7\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,{\rm{ctg}}\beta = -7.\)
Воспользуемся формулой котангенса суммы:
\({\rm{ctg}}\left( {\alpha + \beta } \right) = \dfrac{{{\rm{ctg}}\alpha {\rm{ctg}}\beta -1}}{{{\rm{ctg}}\alpha + {\rm{ctg}}\beta }} = \dfrac{{-3 \cdot \left( {-7} \right)-1}}{{-3-7}} = \dfrac{{20}}{{-10}} = -2.\)
Ответ: \(-2.\)