a:

2: pi/2<a<pi

=>sin a>0 và cosa<0

tan a=-2

1+tan^2a=1/cos^2a=1+4=5

=>cos^2a=1/5

=>\(cosa=-\dfrac{1}{\sqrt{5}}\)

\(sina=\sqrt{1-\dfrac{1}{5}}=\dfrac{2}{\sqrt{5}}\)

cot a=1/tan a=-1/2

3: pi<a<3/2pi

=>cosa<0; sin a<0

1+cot^2a=1/sin^2a

=>1/sin^2a=1+9=10

=>sin^2a=1/10

=>\(sina=-\dfrac{1}{\sqrt{10}}\)

\(cosa=-\dfrac{3}{\sqrt{10}}\)

tan a=1:cota=1/3

b;

tan x=-2

=>sin x=-2*cosx

\(A=\dfrac{2\cdot sinx+cosx}{cosx-3sinx}\)

\(=\dfrac{-4cosx+cosx}{cosx+6cosx}=\dfrac{-3}{7}\)

2: tan x=-2 

=>sin x=-2*cosx

\(B=\dfrac{-4cosx+3cosx}{-6cosx-2cosx}=\dfrac{1}{8}\)

Ta có:

a) \(\sin \left( {\alpha  + \frac{\pi }{6}} \right) = \sin \alpha \cos \frac{\pi }{6} + \cos \alpha \sin \frac{\pi }{6} = \frac{{\sqrt 6 }}{3}.\frac{{\sqrt 3 }}{2} + \left( { - \frac{1}{{\sqrt 3 }}} \right).\frac{1}{2} = \frac{{ - \sqrt 3  + 3\sqrt 2 }}{6}\)      

b) \(\cos \left( {\alpha  + \frac{\pi }{6}} \right) = \cos \alpha .\cos \frac{\pi }{6} - \sin \alpha \sin \frac{\pi }{6} = \left( { - \frac{1}{{\sqrt 3 }}} \right).\frac{{\sqrt 3 }}{2} - \frac{{\sqrt 6 }}{3}.\frac{1}{2} =  - \frac{{3 + \sqrt 6 }}{6}\)

c) \(\sin \left( {\alpha  - \frac{\pi }{3}} \right) = \sin \alpha \cos \frac{\pi }{3} - \cos \alpha \sin \frac{\pi }{3} = \frac{{\sqrt 6 }}{3}.\frac{1}{2} - \left( { - \frac{1}{{\sqrt 3 }}} \right).\frac{{\sqrt 3 }}{2} = \frac{{3 + \sqrt 6 }}{6}\)

d) \(\cos \left( {\alpha  - \frac{\pi }{6}} \right) = \cos \alpha \cos \frac{\pi }{6} + \sin \alpha \sin \frac{\pi }{6} = \left( { - \frac{1}{{\sqrt 3 }}} \right).\frac{{\sqrt 3 }}{2} + \frac{{\sqrt 6 }}{3}.\frac{1}{2} = \frac{{ - 3 + \sqrt 6 }}{6}\)

Do \(\pi< a< \frac{3\pi}{2}\Rightarrow cosa< 0\)

\(cosa=-\sqrt{1-sin^2a}=-\sqrt{1-0,6^2}=-\frac{4}{5}\)

\(tana=\frac{sina}{cosa}=-\frac{3}{4}\)

\(cota=\frac{1}{tana}=-\frac{4}{3}\)

\(a,\dfrac{1}{tan\alpha+1}+\dfrac{1}{cot\alpha+1}\\ =\dfrac{cot\alpha+1+tan\alpha+1}{\left(tan\alpha+1\right)\left(cot\alpha+1\right)}\\ =\dfrac{tan\alpha+cot\alpha+2}{tan\alpha\cdot cot\alpha+tan\alpha+cot\alpha+1}\\ =\dfrac{tan\alpha+cot\alpha+2}{tan\alpha+cot\alpha+2}\\ =1\)

\(b,cos\left(\dfrac{\pi}{2}-\alpha\right)-sin\left(\pi+\alpha\right)\\ =sin\alpha+sin\alpha\\ =2sin\alpha\)

\(c,sin\left(\alpha-\dfrac{\pi}{2}\right)+cos\left(-\alpha+6\pi\right)-tan\left(\alpha+\pi\right)cot\left(3\pi-\alpha\right)\\ =-sin\left(\dfrac{\pi}{2}-\alpha\right)+cos\left(\alpha\right)-tan\left(\alpha\right)cot\left(\pi-\alpha\right)\\ =-cos\left(\alpha\right)+cos\left(\alpha\right)+tan\left(\alpha\right)\cdot cot\left(\alpha\right)\\ =1\)

a) Vì \(0<\alpha <\frac{\pi }{2} \) nên \(\sin \alpha  > 0\). Mặt khác, từ \({\sin ^2}\alpha  + {\cos ^2}\alpha  = 1\) suy ra

\(\sin \alpha  = \sqrt {1 - {{\cos }^2}a}  = \sqrt {1 - \frac{1}{{25}}}  = \frac{{2\sqrt 6 }}{5}\)

Do đó, \(\tan \alpha  = \frac{{\sin \alpha }}{{\cos \alpha }} = \frac{{\frac{{2\sqrt 6 }}{5}}}{{\frac{1}{5}}} = 2\sqrt 6 \) và \(\cot \alpha  = \frac{{\cos \alpha }}{{\sin \alpha }} = \frac{{\frac{1}{5}}}{{\frac{{2\sqrt 6 }}{5}}} = \frac{{\sqrt 6 }}{{12}}\)

b) Vì \(\frac{\pi }{2} < \alpha  < \pi\) nên \(\cos \alpha  < 0\). Mặt khác, từ \({\sin ^2}\alpha  + {\cos ^2}\alpha  = 1\) suy ra

       \(\cos \alpha  = \sqrt {1 - {{\sin }^2}a}  = \sqrt {1 - \frac{4}{9}}  = -\frac{{\sqrt 5 }}{3}\)

Do đó, \(\tan \alpha  = \frac{{\sin \alpha }}{{\cos \alpha }} = \frac{{\frac{2}{3}}}{{-\frac{{\sqrt 5 }}{3}}} = -\frac{{2\sqrt 5 }}{5}\) và \(\cot \alpha  = \frac{{\cos \alpha }}{{\sin \alpha }} = \frac{{-\frac{{\sqrt 5 }}{3}}}{{\frac{2}{3}}} = -\frac{{\sqrt 5 }}{2}\)

c) Ta có: \(\cot \alpha  = \frac{1}{{\tan \alpha }} = \frac{1}{{\sqrt 5 }}\)

Ta có: \({\tan ^2}\alpha  + 1 = \frac{1}{{{{\cos }^2}\alpha }} \Rightarrow {\cos ^2}\alpha  = \frac{1}{{{{\tan }^2}\alpha  + 1}} = \frac{1}{6} \Rightarrow \cos \alpha  =  \pm \frac{1}{{\sqrt 6 }}\)

Vì \(\pi  < \alpha  < \frac{{3\pi }}{2} \Rightarrow \sin \alpha  < 0\;\) và \(\,\,\cos \alpha  < 0 \Rightarrow \cos \alpha  = -\frac{1}{{\sqrt 6 }}\)

Ta có: \(\tan \alpha  = \frac{{\sin \alpha }}{{\cos \alpha }} \Rightarrow \sin \alpha  = \tan \alpha .\cos \alpha  = \sqrt 5 .(-\frac{1}{{\sqrt 6 }}) = -\sqrt {\frac{5}{6}} \)

d) Vì \(\cot \alpha  =  - \frac{1}{{\sqrt 2 }}\;\,\) nên \(\,\,\tan \alpha  = \frac{1}{{\cot \alpha }} =  - \sqrt 2 \)

Ta có: \({\cot ^2}\alpha  + 1 = \frac{1}{{{{\sin }^2}\alpha }} \Rightarrow {\sin ^2}\alpha  = \frac{1}{{{{\cot }^2}\alpha  + 1}} = \frac{2}{3} \Rightarrow \sin \alpha  =  \pm \sqrt {\frac{2}{3}} \)

Vì \(\frac{{3\pi }}{2} < \alpha  < 2\pi  \Rightarrow \sin \alpha  < 0 \Rightarrow \sin \alpha  =  - \sqrt {\frac{2}{3}} \)

Ta có: \(\cot \alpha  = \frac{{\cos \alpha }}{{\sin \alpha }} \Rightarrow \cos \alpha  = \cot \alpha .\sin \alpha  = \left( { - \frac{1}{{\sqrt 2 }}} \right).\left( { - \sqrt {\frac{2}{3}} } \right) = \frac{{\sqrt 3 }}{3}\)

Ý bạn là \(\pi< a< \frac{3\pi}{2}\) và tìm \(cosa,tana,cota\)?

Khi đó \(cosa< 0\) \(\Rightarrow cosa=-\sqrt{1-sin^2a}=-\frac{12}{13}\)

\(tana=\frac{sina}{cosa}=\frac{5}{12}\)

\(cota=\frac{1}{tana}=\frac{12}{5}\)