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Giải bài 3.25 trang 41 sách bài tập toán 10 - Kết nối tri thức với cuộc sống

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🦄Tổng hợp đề thi học kì 2 lớp 10 tất cả các môn - Kết nối tri thức

Toán - Văn - Anh - Lí - Hóa - Sinh - Sử - Địa...
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Đề bài

Cho \(\cos \alpha  = \frac{1}{4}.\) Giá trị của \(P = \frac{{\tan \alpha  + 2\cot \alpha }}{{2\tan \alpha  + 3\cot \alpha }}\) là: A. \( - \frac{{17}}{{33}}.\) B. \(\frac{{17}}{{33}}.\) C. \(\frac{1}{2}.\) D. \(\frac{{16}}{{33}}.\)

Phương pháp giải - Xem chi tiết

- Tính \({\tan ^2}\alpha \) - Biến đổi \(P = \frac{{\tan \alpha  + 2\cot \alpha }}{{2\tan \alpha  + 3\cot \alpha }} = \frac{{{{\tan }^2}\alpha  + 2}}{{2{{\tan }^2}\alpha  + 3}}\)

Lời giải chi tiết

Ta có: \(1 + {\tan ^2}\alpha  = \frac{1}{{{{\cos }^2}\alpha }}\,\, \Rightarrow {\tan ^2}\alpha  = \frac{1}{{{{\cos }^2}\alpha }} - 1 = 15.\)Ta có: \(P = \frac{{\tan \alpha  + 2\cot \alpha }}{{2\tan \alpha  + 3\cot \alpha }} = \frac{{\frac{{\tan \alpha }}{{\cot \alpha }} + 2}}{{\frac{{2\tan \alpha }}{{\cot \alpha }} + 3}} = \frac{{{{\tan }^2}\alpha  + 2}}{{2{{\tan }^2}\alpha  + 3}} = \frac{{15 + 2}}{{2.15 + 3}} = \frac{{17}}{{33}}.\)

Chọn B.

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