Tan 20. Tan 40. Tan 80
Tan 20. Tan 40. Tan 80 = ....
PEMBAHASAN:
$\begin{align} \tan 20. \tan 40. \tan 80 &= \tan 80. \tan 40. \tan 20 \\ &= \frac{\sin 80}{\cos 80}.\frac{\sin 40}{\cos 40}.\frac{\sin 20}{\cos 20}\\ &= \frac{2.\sin 80.\sin 40.\sin 20}{2.\cos 80.\cos 40.\cos 20}\\ &= \frac{(-\cos (80+40) + \cos (80-40)).\sin 20}{(\cos (80+40) + \cos (80-40)).\cos 20}\\ &= \frac{(-\cos 120 + \cos 40).\sin20}{(\cos 120 + \cos 40).\cos 20}\\ &=\frac{(\frac{1}{2} + \cos 40).\sin 20}{(\frac{-1}{2}+\cos 40).\cos 20}\\ &= \frac{(2\cos 40 + 1).\sin 20}{(2\cos 40 - 1).\cos 20}\\ &= \frac{2\cos 40.\sin 20 + \sin 20}{2\cos 40.\cos 20 - \cos 20}\\ &= \frac{\sin 60 - \sin 20 + \sin 20}{\cos 60 + \cos 20 - \cos 20}\\ &= \frac{\sin 60}{\cos 60}\\ &= \tan 60\\ &= \sqrt{3} \end{align}$
PEMBAHASAN:
$\begin{align} \tan 20. \tan 40. \tan 80 &= \tan 80. \tan 40. \tan 20 \\ &= \frac{\sin 80}{\cos 80}.\frac{\sin 40}{\cos 40}.\frac{\sin 20}{\cos 20}\\ &= \frac{2.\sin 80.\sin 40.\sin 20}{2.\cos 80.\cos 40.\cos 20}\\ &= \frac{(-\cos (80+40) + \cos (80-40)).\sin 20}{(\cos (80+40) + \cos (80-40)).\cos 20}\\ &= \frac{(-\cos 120 + \cos 40).\sin20}{(\cos 120 + \cos 40).\cos 20}\\ &=\frac{(\frac{1}{2} + \cos 40).\sin 20}{(\frac{-1}{2}+\cos 40).\cos 20}\\ &= \frac{(2\cos 40 + 1).\sin 20}{(2\cos 40 - 1).\cos 20}\\ &= \frac{2\cos 40.\sin 20 + \sin 20}{2\cos 40.\cos 20 - \cos 20}\\ &= \frac{\sin 60 - \sin 20 + \sin 20}{\cos 60 + \cos 20 - \cos 20}\\ &= \frac{\sin 60}{\cos 60}\\ &= \tan 60\\ &= \sqrt{3} \end{align}$
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