Cho \(\tan \alpha - 3\cot \alpha = 6\) và \(\pi < \alpha < {{3\pi } \over 2}\). Tính
a) \(\sin \alpha + \cos \alpha \)
b) \({{2\sin \alpha - \tan \alpha } \over {{\rm{cos}}\alpha {\rm{ + cot}}\alpha }}\)
Gợi ý làm bài
Vì \(\pi < \alpha < {{3\pi } \over 2}\)
Nên \(\cos \alpha < 0,\sin \alpha < 0\) và \(\tan \alpha > 0\)
Ta có: \(\tan \alpha - 3\cot \alpha = 6 \Leftrightarrow \tan \alpha - {3 \over {\tan \alpha }} - 6 = 0\)
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\( \Leftrightarrow {\tan ^2}\alpha - 6\tan \alpha - 3 = 0\)
Vì \(\tan \alpha > 0\) nên \(\tan \alpha = 3 + 2\sqrt 3\)
a) \({\rm{co}}{{\rm{s}}^2}\alpha = {1 \over {1 + {{\tan }^2}\alpha }} = {1 \over {22 + 12\sqrt 3 }}\)
Suy ra \({\rm{cos}}\alpha {\rm{ = - }}{1 \over {\sqrt {22 + 12\sqrt 3 } }},\sin \alpha = - {{3 + 2\sqrt 3 } \over {\sqrt {22 + 12\sqrt 3 } }}.\)
Vậy \(\sin \alpha + c{\rm{os}}\alpha {\rm{ = - }}{{4 + 2\sqrt 3 } \over {\sqrt {22 + 12\sqrt 3 } }}\)
\(\eqalign{
& {{2\sin \alpha - \tan \alpha } \over {{\rm{cos}}\alpha {\rm{ + cot}}\alpha }} = {{\sin \alpha (2 - {1 \over {{\rm{cos}}\alpha }})} \over {{\rm{cos(1 + }}{1 \over {\sin \alpha }})}} \cr
& = \tan \alpha .{{2\cos \alpha - 1} \over {{\rm{cos}}\alpha }}.{{\sin \alpha } \over {\sin \alpha + 1}} = {\tan ^2}\alpha .{{2\cos \alpha - 1} \over {\sin \alpha + 1}} \cr} \)
\(\eqalign{
& {(3 + 2\sqrt 3 )^2}.{{ - {2 \over {\sqrt {22 + 12\sqrt 3 } }}} \over { - {{3 + 2\sqrt 3 } \over {\sqrt {22 + 12\sqrt 3 } }} + 1}} \cr
& = (21 + 12\sqrt 3 ).{{2 + \sqrt {22 + 12\sqrt 3 } } \over {3 + 2\sqrt 3 - \sqrt {22 + 12\sqrt 3 } }} \cr} \)