1. Вычислите: а) −9√6 ctg(5π/3) · sin(5π/4); б) sin 5π + 4 cos(14π/3) + tg(−25π/4); в) (ctg(9π/4) + tg(8π/3))(sin(9π/2) − 2 cos(7π/6)).

Question

Вопрос:

Accepted Answer

\(\operatorname{ctg}\frac{5\pi}{3}=\frac{\cos(5\pi/3)}{\sin(5\pi/3)}=-\frac{1}{\sqrt3}\), а \(\sin\frac{5\pi}{4}=-\frac{\sqrt2}{2}\). Поэтому
\(-9\sqrt6\cdot\left(-\frac1{\sqrt3}\right)\cdot\left(-\frac{\sqrt2}{2}\right)=-9\).
\(\sin 5\pi=0\), \(\cos\frac{14\pi}{3}=\cos\frac{2\pi}{3}=-\frac12\), \(\operatorname{tg}\left(-\frac{25\pi}{4}\right)=\operatorname{tg}\left(-\frac\pi4\right)=-1\). Следовательно, \(0+4\cdot\left(-\frac12\right)-1=-3\).
\(\operatorname{ctg}\frac{9\pi}{4}=1\), \(\operatorname{tg}\frac{8\pi}{3}=-\sqrt3\), \(\sin\frac{9\pi}{2}=1\), \(\cos\frac{7\pi}{6}=-\frac{\sqrt3}{2}\). Тогда
\((1-\sqrt3)\left(1-2\cdot\left(-\frac{\sqrt3}{2}\right)\right)=(1-\sqrt3)(1+\sqrt3)=-2\).

Ответ: а) −9; б) −3; в) −2.