Calcule a expressãoa) 2\/3sen + pi\/2 + 3\/5sen . 5pi\/3 - 6\/7 sen. 7pi\/6

Sabemos que:

\[sen(\dfrac{\pi}{2}) = 1\]

\[sen(\dfrac{5\pi}{3}) = -\dfrac{\sqrt{3}}{2}\]

\[sen(\dfrac{7\pi}{6}) = -\dfrac{1}{2}\]

Substituindo tais valores na expressão inicial, temos:

\[(\dfrac{2}{3}\cdot 1) + (\dfrac{3}{5}\cdot -\dfrac{\sqrt{3}}{2})-(\dfrac{6}{7}\cdot -\dfrac{1}{2})\]

\[= \dfrac{2}{3}-\dfrac{3\sqrt3}{10}+\dfrac{6}{14}\]

\[= \dfrac{2}{3}-\dfrac{3\sqrt3}{10}+\dfrac{3}{7}\]

\[= \dfrac{140}{210}-\dfrac{63\sqrt3}{210}+\dfrac{90}{210}\]

\[= \dfrac{210-63\sqrt3}{210}\]

\[= 1 -\dfrac{3\sqrt3}{10}\]