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Giải bài 67 trang 32 sách bài tập toán 11 - Cánh diều

Nếu \(\cos 2\alpha = \frac{{\sqrt 3 }}{6}\) thì giá trị của biểu thức \(\cos \left( {\alpha + \frac{\pi }{3}} \right)\cos \left( {\alpha - \frac{\pi }{3}} \right)\) bằng:

💙Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Cánh diều

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

 Nếu \(\cos 2\alpha  = \frac{{\sqrt 3 }}{6}\) thì giá trị của biểu thức \(\cos \left( {\alpha  + \frac{\pi }{3}} \right)\cos \left( {\alpha  - \frac{\pi }{3}} \right)\) bằng: A. \(\frac{{\sqrt 3 }}{3}\)                           B. \(\frac{{ - 3 + \sqrt 3 }}{{12}}\)   C. \( - \frac{{\sqrt 3 }}{3}\)                         D. \(\frac{{3 + \sqrt 3 }}{{12}}\)

Phương pháp giải - Xem chi tiết

Sử dụng công thức \(\cos x\cos y = \frac{1}{2}\left[ {\cos \left( {x + y} \right) + \cos \left( {x - y} \right)} \right]\)

Lời giải chi tiết

Ta có: \(\cos \left( {\alpha  + \frac{\pi }{3}} \right)\cos \left( {\alpha  - \frac{\pi }{3}} \right) = \frac{1}{2}\left[ {\cos \left( {\alpha  + \frac{\pi }{3} + \alpha  - \frac{\pi }{3}} \right) + \cos \left( {\alpha  + \frac{\pi }{3} - \alpha  + \frac{\pi }{3}} \right)} \right]\)

\(\frac{1}{2}\left[ {\cos \left( {2\alpha } \right) + \cos \left( {\frac{{2\pi }}{3}} \right)} \right] = \frac{1}{2}\left( {\frac{{\sqrt 3 }}{6} - \frac{1}{2}} \right) = \frac{{ - 3 + \sqrt 3 }}{{12}}\)

Đáp án đúng là B.

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