Cho \(\cot x = - \sqrt 3 ,\frac{{3\pi }}{2} < x < 2\pi \). Khi đó:

a) \(\sin x = - \frac{{\sqrt {10} }}{{10}}\).

b) \(\cos x = \frac{{\sqrt 3 }}{{10}}\).

c) \(\sin \left( {\frac{{4\pi }}{3} - x} \right) = \frac{{ - \sqrt {10} }}{5}\).

d) \(\tan \left( {x + \frac{\pi }{3}} \right) = \frac{{\sqrt 3 }}{3}\).

a) \(\cot x =  - \sqrt 3 ,\frac{{3\pi }}{2} < x < 2\pi \).

\(\frac{1}{{{{\sin }^2}x}} = 1 + {\cot ^2}x = 1 + {( - \sqrt 3 )^2} = 10 \Rightarrow {\sin ^2}x = \frac{1}{{10}} \Rightarrow \sin x =  \pm \frac{{\sqrt {10} }}{{10}}\).

\({\rm{V\`i  }}\frac{{3\pi }}{2} < x < 2\pi {\rm{ n\^e n }}\sin x =  - \frac{{\sqrt {10} }}{{10}}\).

b) \(\cos x = \cot x \cdot \sin x =  - \sqrt 3 .\left( { - \frac{{\sqrt {10} }}{{10}}} \right) = \frac{{\sqrt {30} }}{{10}}.\)

c) \(\sin \left( {\frac{{4\pi }}{3} - x} \right) = \sin \frac{{4\pi }}{3}\cos x - \cos \frac{{4\pi }}{3}\sin x =  - \frac{{\sqrt 3 }}{2} \cdot \left( {\frac{{\sqrt {30} }}{{10}}} \right) - \frac{{ - 1}}{2} \cdot \frac{{ - \sqrt {10} }}{{10}} = \frac{{ - \sqrt {10} }}{5}\).

c) \(\tan x = \frac{1}{{\cot x}} =  - \frac{{\sqrt 3 }}{3}\)

\(\tan \left( {x + \frac{\pi }{3}} \right) = \frac{{\tan x + \tan \frac{\pi }{3}}}{{1 - \tan x \cdot \tan \frac{\pi }{3}}} = \frac{{\frac{{ - \sqrt 3 }}{3} + \sqrt 3 }}{{1 - \frac{{ - \sqrt 3 }}{3} \cdot \sqrt 3 }} = \frac{{\sqrt 3 }}{3}\).

Đáp án: a) Đúng;   b) Sai;   c) Đúng;   d) Đúng.