Bài 24 trang 215 SBT đại số 10

Đề bài

Tính giá trị biểu thức \(\cos \dfrac{\pi }{7}\cos \dfrac{{4\pi }}{7}\cos \dfrac{{5\pi }}{7}\)

    A. 1/8          B. -1/8

    C. 1/5          D. 2/5

Lời giải chi tiết

\(\begin{array}{l}
A = \cos \frac{\pi }{7}\cos \frac{{4\pi }}{7}\cos \frac{{5\pi }}{7}\\
\Rightarrow \sin \frac{\pi }{7}A\\
= \sin \frac{\pi }{7}\cos \frac{\pi }{7}\cos \frac{{4\pi }}{7}\cos \frac{{5\pi }}{7}\\
= \frac{1}{2}.2\sin \frac{\pi }{7}\cos \frac{\pi }{7}\cos \frac{{4\pi }}{7}\cos \frac{{5\pi }}{7}\\
= \frac{1}{2}\sin \frac{{2\pi }}{7}\cos \frac{{4\pi }}{7}\cos \frac{{5\pi }}{7}\\
= \frac{1}{2}\sin \frac{{2\pi }}{7}.\frac{1}{2}\left( {\cos \frac{{9\pi }}{7} + \cos \frac{\pi }{7}} \right)\\
= \frac{1}{2}\sin \frac{{2\pi }}{7}.\frac{1}{2}\left( { - \cos \frac{{2\pi }}{7} + \cos \frac{\pi }{7}} \right)\\
= \frac{1}{4}\left( { - \sin \frac{{2\pi }}{7}\cos \frac{{2\pi }}{7} + \sin \frac{{2\pi }}{7}\cos \frac{\pi }{7}} \right)\\
= \frac{1}{4}\left( { - \frac{1}{2}.2\sin \frac{{2\pi }}{7}\cos \frac{{2\pi }}{7} + \sin \frac{{2\pi }}{7}\cos \frac{\pi }{7}} \right)\\
= \frac{1}{4}\left[ { - \frac{1}{2}\sin \frac{{4\pi }}{7} + \frac{1}{2}\left( {\sin \frac{{3\pi }}{7} + \sin \frac{\pi }{7}} \right)} \right]\\
= \frac{1}{8}\left[ { - \sin \frac{{4\pi }}{7} + \sin \frac{{3\pi }}{7} + \sin \frac{\pi }{7}} \right]\\
= \frac{1}{8}\left( { - \sin \frac{{4\pi }}{7} + \sin \frac{{4\pi }}{7} + \sin \frac{\pi }{7}} \right)\\
= \frac{1}{8}\sin \frac{\pi }{7}\\
\Rightarrow A = \frac{{\frac{1}{8}\sin \frac{\pi }{7}}}{{\sin \frac{\pi }{7}}} = \frac{1}{8}
\end{array}\)

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