Cho \(\sin a = \frac{2}{3}\) với \(\frac{\pi }{2} < a < \pi \). Tính:

\(\sin \left( {a + \frac{\pi }{4}} \right),\,\cos \left( {a - \frac{{5\pi }}{6}} \right),\,\tan \left( {a + \frac{{2\pi }}{3}} \right)\);

\(\sin \left( {a + \frac{\pi }{4}} \right)\)\( = \sin a\cos \frac{\pi }{4} + \cos a\sin \frac{\pi }{4}\)\( = \frac{2}{3}.\frac{{\sqrt 2 }}{2} + \left( { - \frac{{\sqrt 5 }}{3}} \right).\frac{{\sqrt 2 }}{2} = \frac{{2\sqrt 2 - \sqrt {10} }}{6}\).

\(\cos \left( {a - \frac{{5\pi }}{6}} \right)\)\( = \cos a\cos \frac{{5\pi }}{6} + \sin a\sin \frac{{5\pi }}{6}\)\( = \left( { - \frac{{\sqrt 5 }}{3}} \right).\left( { - \frac{{\sqrt 3 }}{2}} \right) + \frac{2}{3}.\frac{1}{2} = \frac{{\sqrt {15}  + 2}}{6}\).

\(\tan \left( {a + \frac{{2\pi }}{3}} \right)\)\( = \frac{{\tan a + \tan \frac{{2\pi }}{3}}}{{1 - \tan a\tan \frac{{2\pi }}{3}}} = \frac{{ - \frac{{2\sqrt 5 }}{5} + \left( { - \sqrt 3 } \right)}}{{1 - \left( { - \frac{{2\sqrt 5 }}{5}} \right).\left( { - \sqrt 3 } \right)}} = \frac{{8\sqrt 5 + 9\sqrt 3 }}{7}\).