Trigonometric product Calculate value of $sin(\frac{\pi}{9})\times sin(\frac{2\pi}{9})\times sin(\frac{4\pi}{9})= $ |
Lemma: For all real numbers $x $, we have $\sin(\frac{\pi}{3} - x) \cdot \sin(\frac{\pi}{3} + x) \cdot \sin(x) = \frac{\sin(3x)}{4} $. Apply the Lemma, let $x = \frac{\pi}{9} $, we have: $\sin(\frac{\pi}{9}) \cdot \sin(\frac{2\pi}{9}) \cdot \sin(\frac{4\pi}{9}) = \frac{\sin(\frac{\pi}{3})}{4} = \frac{\sqrt{3}}{8} $. |
Trích:
$Put:A = sin\frac{\pi }{9}.sin\frac{{2\pi }}{9}.sin\frac{{4\pi }}{9}\\ 4A = 2[c{\rm{os}}\frac{{ - \pi }}{3} - c{\rm{os}}\frac{{5\pi }}{9}]\sin \frac{{2\pi }}{9} = \sin \frac{{2\pi }}{9} - 2\sin \frac{{2\pi }}{9}c{\rm{os}}\frac{{5\pi }}{9}\\ $ $= \sin \frac{{2\pi }}{9} - \sin \frac{{7\pi }}{9} + \frac{{\sqrt 3 }}{2} = \sin \frac{{2\pi }}{9} - \sin \frac{{2\pi }}{9} + \frac{{\sqrt 3 }}{2} = \frac{{\sqrt 3 }}{2}\\ $ $\Rightarrow A = \frac{{\sqrt 3 }}{8} $ |
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