Pembahasan trigonometri (1)


Pembahasan trigonometri problem

Soal nomor 1
\cos^{2}18^{o} + \cos^{2}54^{o}=

\cos18^{o}\cos 18^{o} + \cos54^{o}cos 54^{o}=

\sin 72^{o}\cos 18^{o} + \sin 36^{o}cos 54^{o}=

\frac{1}{2}(\sin(72+18)^{o} + \sin(72-18)^{o})+\frac{1}{2}(\sin(36+54)^{o}+\sin(36-54)^{o})=

\frac{1}{2}(\sin (90)^{o} + \sin(54)^{o})+\frac{1}{2}(\sin (90)^{o}+\sin (-18)^{o})=

\frac{1}{2}(1 + \sin(54)^{o})+\frac{1}{2}(1- \sin (18)^{o})=

1 + \frac{1}{2}(\sin(54)^{o}- \sin (18)^{o})=

1 + \frac{1}{2}(2 \cos36 ^{o}\sin(18)^{o})=

1 +  \cos36 ^{o}\sin(18)^{o}=

1 +  \cos36 ^{o}\sin(18)^{o} \frac{\cos 18^{o}}{\cos 18^{o}}=

1 +  \frac{1}{2} \frac {\cos(36) ^{o}\sin(36)^{o}}{\cos 18^{o}}=

1 +  \frac{1}{4} \frac {\sin(72)^{o}}{\cos 18^{o}}=

1 +  \frac{1}{4} \frac {\cos(18)^{o}}{\cos 18^{o}}=
\mathbf{\frac{5}{4}}

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