a.
\(0< a< \dfrac{\pi}{2}\Rightarrow sina>0\Rightarrow sina=\sqrt{1-cos^2a}=\dfrac{3}{\sqrt{5}}\)
\(tana=\dfrac{sina}{cosa}=\dfrac{3}{2}\); \(cota=\dfrac{1}{tana}=\dfrac{2}{3}\)
b.
\(\pi< a< \dfrac{3\pi}{2}\Rightarrow cosa< 0\Rightarrow cosa=-\sqrt{1-sin^2a}=-\dfrac{\sqrt{51}}{10}\)
\(tana=\dfrac{sina}{cosa}=-\dfrac{7\sqrt{51}}{51}\) ; \(cota=\dfrac{1}{tana}=-\dfrac{\sqrt{51}}{7}\)
c.
\(\dfrac{\pi}{2}< a< \pi\Rightarrow sina>0\Rightarrow sina=\sqrt{1-cos^2a}=\dfrac{4\sqrt{3}}{7}\)
\(cos2a=2cos^2a-1=-\dfrac{47}{49}\) ;
\(cos\left(\dfrac{\pi}{3}+a\right)=cos\left(\dfrac{\pi}{3}\right)cosa-sin\left(\dfrac{\pi}{3}\right).sina=\dfrac{1}{2}.\left(-\dfrac{1}{7}\right)-\dfrac{\sqrt{3}}{2}.\dfrac{4\sqrt{3}}{7}=...\)
d.
\(\dfrac{\pi}{2}< a< \pi\Rightarrow cosa< 0\Rightarrow cosa=-\sqrt{1-sin^2a}=-\dfrac{2\sqrt{11}}{7}\)
\(sin2a=2sina.cosa=-\dfrac{4\sqrt{55}}{49}\)
\(sin\left(\dfrac{\pi}{6}-a\right)=sin\left(\dfrac{\pi}{6}\right).cosa-cos\left(\dfrac{\pi}{6}\right).sina\)
\(=\dfrac{1}{2}.\left(-\dfrac{2\sqrt{11}}{7}\right)-\dfrac{\sqrt{3}}{2}.\dfrac{\sqrt{5}}{7}=-\dfrac{2\sqrt{11}+\sqrt{15}}{14}\)
d.
\(-\dfrac{\pi}{2}< a< 0\Rightarrow\left\{{}\begin{matrix}sina< 0\\cosa>0\end{matrix}\right.\)
\(\dfrac{1}{cos^2a}=1+tan^2a\Rightarrow cosa=\dfrac{1}{\sqrt{1+tan^2a}}=\dfrac{\sqrt{5}}{5}\)
\(sina=cosa.tana=-\dfrac{2\sqrt{5}}{5}\)
\(cota=\dfrac{1}{tana}=-\dfrac{1}{2}\)
