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Giải bài 8 trang 11 sách bài tập toán 11 - Cánh diều

Cho \(\cot x = - 3\), \(\frac{\pi }{2} < x < \pi \). Tính \(\sin x\), \(\cos x\), \(\tan x\).

﷽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

Cho \(\cot x =  - 3\), \(\frac{\pi }{2} < x < \pi \). Tính \(\sin x\), \(\cos x\), \(\tan x\).

Phương pháp giải - Xem chi tiết

Sử dụng công thức \(\tan x = \frac{1}{{\cot x}}\) để tính \(\tan x\). Sử dụng công thức \(1 + {\cot ^2}x = \frac{1}{{{{\sin }^2}x}}\)  và điều kiện \(\frac{\pi }{2} < x < \pi \) để tính \(\sin x\). Sử dụng công thức \(\cot x = \frac{{\cos x}}{{\sin x}}\) để tính \(\cos x\) theo \(\sin x\) và \(\cot x\).

Lời giải chi tiết

Ta có \(\tan x = \frac{1}{{\cot x}} = 1:\left( { - 3} \right) =  - \frac{1}{3}\).Do \(1 + {\cot ^2}x = \frac{1}{{{{\sin }^2}x}} \Rightarrow {\sin ^2}x = \frac{1}{{1 + {{\cot }^2}x}} = \frac{1}{{1 + {{\left( { - 3} \right)}^2}}} = \frac{1}{{10}} \Rightarrow \sin x =  \pm \frac{{\sqrt {10} }}{{10}}\)

Vì \(\frac{\pi }{2} < x < \pi  \Rightarrow \sin x > 0 \Rightarrow \sin x = \frac{{\sqrt {10} }}{{10}}\).

Vì \(\cot x = \frac{{\cos x}}{{\sin x}} \Rightarrow \cos x = \cot x.\sin x =  - 3.\frac{{\sqrt {10} }}{{10}} =  - \frac{{3\sqrt {10} }}{{10}}\).

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