Soal yang Akan Dibahas
Nilai dari
$ \log (\tan 2^\circ) + \log (\tan 3^\circ) + ... + \log (\tan 88^\circ) = .... $
A). $ 0 \, $ B). $ 1 \, $ C). $ 45 \, $ D). $ 89 \, $ E). $ 90 \, $
$ \log (\tan 2^\circ) + \log (\tan 3^\circ) + ... + \log (\tan 88^\circ) = .... $
A). $ 0 \, $ B). $ 1 \, $ C). $ 45 \, $ D). $ 89 \, $ E). $ 90 \, $
$\spadesuit $ Konsep Dasar
*). Sifat-sifat logaritma :
$ \log a + \log b = \log ab \, $ dan $ \log 1 = 0 $.
*). Sudut komplemen pada trigonometri :
$ \tan x = \cot (90^\circ - x ) = \frac{1}{\tan (90^\circ - x ) } $
*). Sifat-sifat logaritma :
$ \log a + \log b = \log ab \, $ dan $ \log 1 = 0 $.
*). Sudut komplemen pada trigonometri :
$ \tan x = \cot (90^\circ - x ) = \frac{1}{\tan (90^\circ - x ) } $
$\clubsuit $ Pembahasan
*). Menyelesaikan soal :
$\begin{align} & \log (\tan 2^\circ) + \log (\tan 3^\circ) + ... + \log (\tan 88^\circ) \\ & = \log (\tan 2^\circ. \tan 3^\circ ... \tan 45^\circ ... \tan 87^\circ \tan 88^\circ) \\ & = \log \left( \frac{1}{\tan (90^\circ -2^\circ)}. \frac{1}{\tan (90^\circ -3^\circ)} ... \tan 45^\circ ... \tan 87^\circ .\tan 88^\circ \right) \\ & = \log \left( \frac{1}{\tan 88^\circ }. \frac{1}{\tan 87^\circ } ... \tan 45^\circ ... \tan 87^\circ . \tan 88^\circ \right) \\ & = \log \left( \frac{1}{\tan 88^\circ }.\tan 88^\circ . \frac{1}{\tan 87^\circ } . \tan 87^\circ ... \tan 45^\circ \right) \\ & = \log \left( \underbrace{1.1.1...1.1}_{45 \text{kali}} . \tan 45^\circ \right) \\ & = \log \left( \underbrace{1.1.1...1.1}_{45 \text{kali}} . 1 \right) \\ & = \log 1 = 0 \end{align} $
Jadi, hasilnya adalah $ 0 . \, \heartsuit $
*). Menyelesaikan soal :
$\begin{align} & \log (\tan 2^\circ) + \log (\tan 3^\circ) + ... + \log (\tan 88^\circ) \\ & = \log (\tan 2^\circ. \tan 3^\circ ... \tan 45^\circ ... \tan 87^\circ \tan 88^\circ) \\ & = \log \left( \frac{1}{\tan (90^\circ -2^\circ)}. \frac{1}{\tan (90^\circ -3^\circ)} ... \tan 45^\circ ... \tan 87^\circ .\tan 88^\circ \right) \\ & = \log \left( \frac{1}{\tan 88^\circ }. \frac{1}{\tan 87^\circ } ... \tan 45^\circ ... \tan 87^\circ . \tan 88^\circ \right) \\ & = \log \left( \frac{1}{\tan 88^\circ }.\tan 88^\circ . \frac{1}{\tan 87^\circ } . \tan 87^\circ ... \tan 45^\circ \right) \\ & = \log \left( \underbrace{1.1.1...1.1}_{45 \text{kali}} . \tan 45^\circ \right) \\ & = \log \left( \underbrace{1.1.1...1.1}_{45 \text{kali}} . 1 \right) \\ & = \log 1 = 0 \end{align} $
Jadi, hasilnya adalah $ 0 . \, \heartsuit $