Determine o valor de

$p = \sin{\frac{\pi}{24}} \cdot \sin{\frac{5 \pi}{24}} \cdot \sin{\frac{7 \pi}{24}} \cdot \sin{\frac{11 \pi}{24}}$


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ITA IIIT 28/04/2022 14:08
$-$ Conhecida as fórmulas de transformação $produto-soma$, ou simplesmente $\text{Fórmulas de Werner}$, temos: \begin{matrix} p = \sin \underbrace{{\frac{\pi}{24}} \ . \ \sin\overbrace{{\frac{5\pi}{24}} \ . \ \sin}^{produto-soma}{\frac{7\pi}{24}} \ . \ \sin}_{produto-soma}{\frac{11\pi}{24}} \\ \\ p = {\large{( \frac{\cos{\frac{12\pi}{24}} \ - \ \cos{\frac{10\pi}{24}} }{-2} )}} \ . \ {\large{( \frac{\cos{\frac{12\pi}{24}} \ - \ \cos{\frac{2\pi}{24}} }{-2} )}} = \frac{1}{4} \ . \ (\cos{\frac{\pi}{2}} - \cos{\frac{5\pi}{12}}) \ . \ ( \cos{\frac{\pi}{2}} - \cos{\frac{\pi}{12}}) \\ \\ p = \frac{1}{4} \ . \ \underbrace{\cos{\frac{5\pi}{12}} \ . \ \cos{\frac{\pi}{12}}}_{produto-soma} \\ \\ p = \frac{1}{4} \ . \ {\large{( \frac{\cos{\frac{6\pi}{12}} \ + \ \cos{\frac{4\pi}{12}} }{2} )}} = \frac{1}{8} \ . \ (\cos{\frac{\pi}{2}} + \cos{\frac{\pi}{3}}) = \frac{1}{8} \ . \ \cos{\frac{\pi}{3}} \\ \\ \fbox{$p = \large{ \frac{1}{16}}$} \end{matrix}
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