b: \(-\dfrac{\Omega}{2}< \alpha< 0\)
=>\(sin\alpha< 0\)
\(sin^2\alpha+cos^2\alpha=1\)
=>\(sin^2\alpha=1-cos^2\alpha=1-\left(\dfrac{2}{\sqrt{5}}\right)^2=1-\dfrac{4}{5}=\dfrac{1}{5}\)
mà \(sin\alpha< 0\)
nên \(sin\alpha=\dfrac{-1}{\sqrt{5}}\)
\(tan\alpha=\dfrac{sin\alpha}{cos\alpha}=\dfrac{-1}{\sqrt{5}}:\dfrac{-2}{\sqrt{5}}=\dfrac{-1}{-2}=\dfrac{1}{2}\)
\(cot\alpha=\dfrac{1}{tan\alpha}=1:\dfrac{1}{2}=2\)
d: \(180^0< \alpha< 270^0\)
=>\(cos\alpha< 0\)
\(sin^2\alpha+cos^2\alpha=1\)
=>\(cos^2\alpha=1-\left(-\dfrac{1}{3}\right)^2=1-\dfrac{1}{9}=\dfrac{8}{9}\)
mà \(cos\alpha< 0\)
nên \(cos\alpha=-\sqrt{\dfrac{8}{9}}=-\dfrac{2\sqrt{2}}{3}\)
\(tan\alpha=\dfrac{sin\alpha}{cos\alpha}=\dfrac{-1}{3}:\dfrac{-2\sqrt{2}}{3}=\dfrac{1}{2\sqrt{2}}=\dfrac{\sqrt{2}}{4}\)
\(cot\alpha=\dfrac{1}{tan\alpha}=1:\dfrac{\sqrt{2}}{4}=\dfrac{4}{\sqrt{2}}=2\sqrt{2}\)
f: \(\dfrac{\Omega}{2}< \alpha< \Omega\)
=>\(sin\alpha>0;cos\alpha< 0\)
\(1+tan^2\alpha=\dfrac{1}{cos^2\alpha}\)
=>\(\dfrac{1}{cos^2\alpha}=1+\left(-2\right)^2=5\)
=>\(cos^2\alpha=\dfrac{1}{5}\)
mà \(cos\alpha< 0\)
nên \(cos\alpha=-\dfrac{1}{\sqrt{5}}\)
\(tan\alpha=\dfrac{sin\alpha}{cos\alpha}\)
=>\(sin\alpha=cos\alpha\cdot tan\alpha=\left(-2\right)\cdot\dfrac{-1}{\sqrt{5}}=\dfrac{2}{\sqrt{5}}\)
\(cot\alpha=\dfrac{1}{tan\alpha}=\dfrac{1}{-2}=-\dfrac{1}{2}\)
h: \(\Omega< \alpha< \dfrac{3}{2}\Omega\)
=>\(sin\alpha< 0;cos\alpha< 0\)
\(tan\alpha=\dfrac{1}{cot\alpha}=\dfrac{1}{3}\)
\(1+cot^2\alpha=\dfrac{1}{sin^2\alpha}\)
=>\(\dfrac{1}{sin^2\alpha}=1+9=10\)
=>\(sin^2\alpha=\dfrac{1}{10}\)
mà \(sin\alpha< 0\)
nên \(sin\alpha=-\dfrac{1}{\sqrt{10}}\)
\(cos\alpha=\dfrac{sin\alpha}{tan\alpha}=\dfrac{-1}{\sqrt{10}}:\dfrac{1}{3}=\dfrac{-3}{\sqrt{10}}\)
