Soal yang Akan Dibahas
Jika $ \alpha + \beta = \frac{\pi}{4} $ dan $ \cos \alpha \cos \beta = \frac{3}{4} $ ,
maka $ \cos (\alpha - \beta ) = .... $
A). $ \frac{2}{3} - \frac{\sqrt{2}}{2} \, $ B). $ \frac{1}{2} - \frac{\sqrt{2}}{2} \, $
C). $ \frac{1}{2} - \frac{\sqrt{3}}{2} \, $ D). $ 1 - \frac{\sqrt{3}}{3} \, $
E). $ \frac{3}{4} - \frac{\sqrt{3}}{3} $
A). $ \frac{2}{3} - \frac{\sqrt{2}}{2} \, $ B). $ \frac{1}{2} - \frac{\sqrt{2}}{2} \, $
C). $ \frac{1}{2} - \frac{\sqrt{3}}{2} \, $ D). $ 1 - \frac{\sqrt{3}}{3} \, $
E). $ \frac{3}{4} - \frac{\sqrt{3}}{3} $
$\spadesuit $ Konsep Dasar
*). Rumus jumlah dan selisih sudut trigonometri :
$ \cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta $
$ \cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta $
*). Rumus jumlah dan selisih sudut trigonometri :
$ \cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta $
$ \cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta $
$\clubsuit $ Pembahasan
*). Menentukan nilai $ \sin \alpha \sin \beta $ :
$ \begin{align} \alpha + \beta & = \frac{\pi}{4} \\ \cos ( \alpha + \beta ) & = \cos \frac{\pi}{4} \\ \cos \alpha \cos \beta - \sin \alpha \sin \beta & = \frac{\sqrt{2}}{2} \\ \frac{3}{4} - \sin \alpha \sin \beta & = \frac{\sqrt{2}}{2} \\ \sin \alpha \sin \beta & = \frac{3}{4} - \frac{\sqrt{2}}{2} \end{align} $
*). Menentukan nilai $ \cos ( \alpha - \beta ) $ :
$ \begin{align} \cos (\alpha - \beta) & = \cos \alpha \cos \beta + \sin \alpha \sin \beta \\ & = \frac{3}{4} + \frac{3}{4} - \frac{\sqrt{2}}{2} \\ & = \frac{6}{4} - \frac{\sqrt{2}}{2} \\ & = \frac{3}{2} - \frac{\sqrt{2}}{2} \end{align} $
Jadi, nilai $ \cos (\alpha - \beta) = \frac{3}{2} - \frac{\sqrt{2}}{2} . \, \heartsuit $
(Tidak ada jawaban pada optionnya)
*). Menentukan nilai $ \sin \alpha \sin \beta $ :
$ \begin{align} \alpha + \beta & = \frac{\pi}{4} \\ \cos ( \alpha + \beta ) & = \cos \frac{\pi}{4} \\ \cos \alpha \cos \beta - \sin \alpha \sin \beta & = \frac{\sqrt{2}}{2} \\ \frac{3}{4} - \sin \alpha \sin \beta & = \frac{\sqrt{2}}{2} \\ \sin \alpha \sin \beta & = \frac{3}{4} - \frac{\sqrt{2}}{2} \end{align} $
*). Menentukan nilai $ \cos ( \alpha - \beta ) $ :
$ \begin{align} \cos (\alpha - \beta) & = \cos \alpha \cos \beta + \sin \alpha \sin \beta \\ & = \frac{3}{4} + \frac{3}{4} - \frac{\sqrt{2}}{2} \\ & = \frac{6}{4} - \frac{\sqrt{2}}{2} \\ & = \frac{3}{2} - \frac{\sqrt{2}}{2} \end{align} $
Jadi, nilai $ \cos (\alpha - \beta) = \frac{3}{2} - \frac{\sqrt{2}}{2} . \, \heartsuit $
(Tidak ada jawaban pada optionnya)
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