Pembahasan trigonometri (2)


soal nomor 2
\sin^{2}6^{o}+\sin^{2}42^{o}+\sin^{2}66^{o}+\sin^{2}78^{o}=

\sin6^{o}\cos84^{o}+\sin42^{o}\cos48^{o}+\sin66^{o}\cos24^{o}+\sin78^{o}\cos12^{o}=

\frac{1}{2}(\sin90^{o}-\sin78^{o})+\frac{1}{2}(\sin90^{o}-\sin6^{o})+

\frac{1}{2}(\sin90^{o}+\sin42^{o})+\frac{1}{2}(\sin90^{o}+\sin66^{o})=

\frac{1}{2}-\frac{1}{2}\sin78^{o}+\frac{1}{2}-\frac{1}{2}\sin6^{o}+

\frac{1}{2}+\frac{1}{2}\sin42^{o}+\frac{1}{2}+\frac{1}{2}\sin66^{o}=

2 +\frac{1}{2}(\sin42^{o}-\sin78^{o})+\frac{1}{2}(\sin66^{o}-\sin6^{o})=

2 +\frac{1}{2}.2\cos60^{o}\sin(-18)^{o})+\frac{1}{2}.2\cos36^{o}\sin30^{o}=

2 -\frac{1}{2}\sin18^{o}+\frac{1}{2}\cos36^{o}=

2 +\frac{1}{2}(\cos36^{o}-\sin18^{o})=

2 +\frac{1}{2}(\cos36^{o}-\cos72^{o})=

2 +\frac{1}{2}.-2\sin54^{o}\sin(-18)^{o}=

2 +\sin54^{o}\sin18^{o}=

2 +\sin54^{o}\sin18^{o}\frac{cos18^{o}}{cos18^{o}}=

2 +\frac{1}{2}\sin54^{o}\frac{sin36^{o}}{cos18^{o}}=

2 +\frac{1}{2}\sin54^{o}\frac{cos54^{o}}{cos18^{o}}=

2 +\frac{1}{2}.\frac{1}{2}\frac{\sin108^{o}}{cos18^{o}}=

2 +\frac{1}{4}\frac{\sin72^{o}}{cos18^{o}}=

2 +\frac{1}{4}\frac{\cos18^{o}}{cos18^{o}}=

2 +\frac{1}{4}=\frac{9}{4}

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