Tính giá trị biểu thức: $B=sin^2(\frac{\pi}{7})+sin^2(\frac{2\pi}{7})+sin^2(\frac{3\pi}{7})$
Tính giá trị biểu thức: $B=sin^2(\frac{\pi}{7})+sin^2(\frac{2\pi}{7})+sin^2(\frac{3\pi}{7})$
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Ta có:
$B = \sin^2{\dfrac{\pi}{7}} + \sin^2{\dfrac{2\pi}{7}} + \sin^2{\dfrac{3\pi}{7}}$
$B = \dfrac{3}{2} - \dfrac{\cos{\dfrac{2\pi}{7}} + \cos{\dfrac{4\pi}{7}} +\cos{\dfrac{6\pi}{7}}}{2}$
$B = \dfrac{3}{2} - \dfrac{2(\sin{\dfrac{4\pi}{7}}.\cos{\dfrac{2\pi}{7}} + \sin{\dfrac{4\pi}{7}}.\cos{\dfrac{4\pi}{7}} + \sin{\dfrac{4\pi}{7}}.\cos{\dfrac{6\pi}{7}})}{4\sin{\dfrac{4\pi}{7}}}$
$= \dfrac{3}{2} - \dfrac{\sin{\dfrac{6\pi}{7}} + \sin{\dfrac{2\pi}{7}} + \sin{\dfrac{8\pi}{7}} + \sin{\dfrac{10\pi}{7}} - \sin{\dfrac{2\pi}{7}}}{4\sin{\dfrac{4\pi}{7}}}$
Do $\sin{x} + \sin{(2\pi - x)} = 0$ nên:
$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \sin{\dfrac{6\pi}{7}} + \sin{\dfrac{8\pi}{7}} = 0$ và $\sin{\dfrac{10\pi}{7}} = - \sin{\dfrac{4\pi}{7}}$
Vậy: $B = \dfrac{3}{2} + \dfrac{1}{4} = \dfrac{7}{4}$
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