16. Чему равно значение выражения: 1) 2sin 150° − 4cos 120°; 2) tg 45° sin 120° ctg 150°; 3) sin 90°(tg 150° cos 135° − tg 120° cos 135°)²?

Question

Accepted Answer

\(2\sin150^\circ-4\cos120^\circ=2\cdot\frac12-4\cdot\left(-\frac12\right)=1+2=3\).
\(\operatorname{tg}45^\circ\sin120^\circ\operatorname{ctg}150^\circ=1\cdot\frac{\sqrt3}{2}\cdot(-\sqrt3)=-\frac32\).
\(\sin90^\circ(\operatorname{tg}150^\circ\cos135^\circ-\operatorname{tg}120^\circ\cos135^\circ)^2\)
\(=1\cdot\bigl((\operatorname{tg}150^\circ-\operatorname{tg}120^\circ)\cos135^\circ\bigr)^2\)
\(=\left(\left(-\frac{\sqrt3}{3}+\sqrt3\right)\cdot\left(-\frac{\sqrt2}{2}\right)\right)^2\)
\(=\left(\frac{2\sqrt3}{3}\cdot\left(-\frac{\sqrt2}{2}\right)\right)^2=\left(-\frac{\sqrt6}{3}\right)^2=\frac23\).

Ответ: 1) \(3\); 2) \(-\frac32\); 3) \(\frac23\).