\(\begin{array}{l}\cos \dfrac{{11\pi }}{{12}} = \cos \left( {\dfrac{\pi }{2} + \dfrac{{5\pi }}{{12}}} \right) = - \sin \dfrac{{5\pi }}{{12}},\\\cos \dfrac{{9\pi }}{{12}} = \cos \left( {\dfrac{\pi }{2} + \dfrac{{3\pi }}{{12}}} \right) = - \sin \dfrac{{3\pi }}{{12}},\\\cos \dfrac{{7\pi }}{{12}} = \cos \left( {\dfrac{\pi }{2} + \dfrac{\pi }{{12}}} \right) = - \sin \dfrac{\pi }{{12}}\end{array}\)
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Tính LG a \({\sin ^2}{15^0} + {\sin ^2}{35^0} + {\sin ^2}{55^0} + {\sin ^2}{75^0};\) Lời giải chi tiết: ) Vì \(\sin {75^0} = \cos {15^0},\sin {55^0} = \cos {35^0}\) nên \({\sin ^2}{15^0} + {\sin ^2}{35^0} + {\sin ^2}{55^0} + {\sin ^2}{75^0} = 2.\) LG b \({\sin ^2}\dfrac{\pi }{8} + {\sin ^2}\dfrac{{3\pi }}{8} + {\sin ^2}\dfrac{{5\pi }}{8} + {\sin ^2}\dfrac{{7\pi }}{8};\) Lời giải chi tiết: Vì \(\begin{array}{l}\sin \dfrac{{7\pi }}{8} = \sin \left( {\dfrac{{3\pi }}{8} + \dfrac{\pi }{2}} \right) = \cos \dfrac{{3\pi }}{8};\\\sin \dfrac{{5\pi }}{8} = \sin \left( {\dfrac{\pi }{8} + \dfrac{\pi }{2}} \right) = \cos \dfrac{\pi }{8}\end{array}\) nên \({\sin ^2}\dfrac{\pi }{8} + {\sin ^2}\dfrac{{3\pi }}{8} + {\sin ^2}\dfrac{{5\pi }}{8} + {\sin ^2}\dfrac{{7\pi }}{8} = 2.\) LG c \({\cos ^2}\dfrac{\pi }{{12}} + {\cos ^2}\dfrac{{3\pi }}{{12}} + {\cos ^2}\dfrac{{5\pi }}{{12}} + {\cos ^2}\dfrac{{7\pi }}{{12}} + {\cos ^2}\dfrac{{9\pi }}{{12}} + {\cos ^2}\dfrac{{11\pi }}{{12}}.\) Lời giải chi tiết: Tương tự \(\begin{array}{l}\cos \dfrac{{11\pi }}{{12}} = \cos \left( {\dfrac{\pi }{2} + \dfrac{{5\pi }}{{12}}} \right) = - \sin \dfrac{{5\pi }}{{12}},\\\cos \dfrac{{9\pi }}{{12}} = \cos \left( {\dfrac{\pi }{2} + \dfrac{{3\pi }}{{12}}} \right) = - \sin \dfrac{{3\pi }}{{12}},\\\cos \dfrac{{7\pi }}{{12}} = \cos \left( {\dfrac{\pi }{2} + \dfrac{\pi }{{12}}} \right) = - \sin \dfrac{\pi }{{12}}\end{array}\) nên ta có: \({\cos ^2}\dfrac{\pi }{{12}} + {\cos ^2}\dfrac{{3\pi }}{{12}} + {\cos ^2}\dfrac{{5\pi }}{{12}} + {\cos ^2}\dfrac{{7\pi }}{{12}} + {\cos ^2}\dfrac{{9\pi }}{{12}} + {\cos ^2}\dfrac{{11\pi }}{{12}} = 3\)
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