Cho \(\cot x = - \sqrt 3 ,\dfrac{{3\pi }}{2} < x < 2\pi \). Các mệnh đề sau đúng hay
Cho \(\cot x = - \sqrt 3 ,\dfrac{{3\pi }}{2} < x < 2\pi \). Các mệnh đề sau đúng hay sai?
Đúng | Sai | |
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1) a) \(\sin x = - \dfrac{{\sqrt {10} }}{{10}}\) |
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2) b) \(\cos x = \dfrac{{\sqrt 3 }}{{10}}\) |
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3) c) \(\sin \left( {\dfrac{{4\pi }}{3} - x} \right) = \dfrac{{ - \sqrt {10} }}{5}\) |
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4) d) \(\tan \left( {x + \dfrac{\pi }{3}} \right) = \dfrac{{\sqrt 3 }}{3}\) |
Đáp án đúng là: 1Đ, 2S, 3Đ, 4Đ
\(\cot x = - \sqrt 3 ,\dfrac{{3\pi }}{2} < x < 2\pi \)
\(\dfrac{1}{{{{\sin }^2}x}} = 1 + {\cot ^2}x = 1 + {( - \sqrt 3 )^2} = 10 \Rightarrow {\sin ^2}x = \dfrac{1}{{10}} \Rightarrow \sin x = \pm \dfrac{{\sqrt {10} }}{{10}}\)
Vì \(\dfrac{{3\pi }}{2} < x < 2\pi \) nên \(\sin x = - \dfrac{{\sqrt {10} }}{{10}}\)
\(\cos x = \cot x.\sin x = - \sqrt 3 \cdot \left( { - \dfrac{{\sqrt {10} }}{{10}}} \right) = \dfrac{{\sqrt {30} }}{{10}}.{\rm{ }}\)
\(\sin \left( {\dfrac{{4\pi }}{3} - x} \right) = \sin \dfrac{{4\pi }}{3}\cos x - \cos \dfrac{{4\pi }}{3}\sin x\)
\( = - \dfrac{{\sqrt 3 }}{2} \cdot \left( {\dfrac{{\sqrt {30} }}{{10}}} \right) - \dfrac{{ - 1}}{2} \cdot \dfrac{{ - \sqrt {10} }}{{10}} = \dfrac{{ - \sqrt {10} }}{5}{\rm{ }}\)
\(\tan x = \dfrac{1}{{\cot x}} = - \dfrac{{\sqrt 3 }}{3}\)
\(\tan \left( {x + \dfrac{\pi }{3}} \right) = \dfrac{{\tan x + \tan \dfrac{\pi }{3}}}{{1 - \tan x \cdot \tan \dfrac{\pi }{3}}} = \dfrac{{\dfrac{{ - \sqrt 3 }}{3} + \sqrt 3 }}{{1 - \dfrac{{ - \sqrt 3 }}{3} \cdot \sqrt 3 }} = \dfrac{{\sqrt 3 }}{3}.\)
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