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Giải bài 64 trang 31 sách bài tập toán 11 - Cánh diều

Cho \(\tan \alpha = 2\). Giá trị của biểu thức \(A = \frac{{3\sin \alpha + \cos \alpha }}{{\sin \alpha - \cos \alpha }}\) bằng bao nhiêu?

𝐆Tổng 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

Cho \(\tan \alpha  = 2\). Giá trị của biểu thức \(A = \frac{{3\sin \alpha  + \cos \alpha }}{{\sin \alpha  - \cos \alpha }}\) bằng bao nhiêu?

Phương pháp giải - Xem chi tiết

Do \(\tan \alpha \) xác định nên \(\cos \alpha  \ne 0\). Chia cả tử và mẫu của \(A\) cho \(\cos \alpha \), và sử dụng công thức \(\tan \alpha  = \frac{{\sin \alpha }}{{\cos \alpha }}\).

Lời giải chi tiết

Do \(\tan \alpha \) xác định nên \(\cos \alpha  \ne 0\).Chia cả tử và mẫu của \(A\) cho \(\cos \alpha \), ta được:

\(A = \frac{{3\frac{{\sin \alpha }}{{\cos \alpha }} + 1}}{{\frac{{\sin \alpha }}{{\cos \alpha }} - 1}} = \frac{{3\tan \alpha  + 1}}{{\tan \alpha  - 1}} = \frac{{3.2 + 1}}{{2 - 1}} = 7\)

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