Задание 5. Найдите значение выражения: 1) sin²(π/6) + cos²(π/3) − tg(π/4); 2) 2 sin²(π/4) + 3 ctg²(π/6) − cos(π/3); 3) (tg(π/3) − ctg(π/6))(sin(π/4) + cos(π/4)); 4) 4 cos²(π/6) − sin²(π/3) + tg(π/3) · ctg(π/6); 5) sin(π/4) cos(π/3) + ctg(π/4) sin(π/6).

Question

Вопрос:

Accepted Answer

Подставим табличные значения функций и выполним вычисления.

\(\sin^2\frac{\pi}{6}+\cos^2\frac{\pi}{3}- g\frac{\pi}{4}=\left(\frac12\right)^2+\left(\frac12\right)^2-1=-\frac12\).
\(2\sin^2\frac{\pi}{4}+3\ctg^2\frac{\pi}{6}-\cos\frac{\pi}{3}=2\left(\frac{\sqrt2}{2}\right)^2+3(\sqrt3)^2-\frac12=1+9-\frac12=\frac{19}{2}\).
\(\left( g\frac{\pi}{3}-\ctg\frac{\pi}{6}\right)\left(\sin\frac{\pi}{4}+\cos\frac{\pi}{4}\right)=(\sqrt3-\sqrt3)\left(\frac{\sqrt2}{2}+\frac{\sqrt2}{2}\right)=0\).
\(4\cos^2\frac{\pi}{6}-\sin^2\frac{\pi}{3}+ g\frac{\pi}{3}\cdot\ctg\frac{\pi}{6}=4\cdot\frac34-\frac34+\sqrt3\cdot\sqrt3=\frac{21}{4}\).
\(\sin\frac{\pi}{4}\cos\frac{\pi}{3}+\ctg\frac{\pi}{4}\sin\frac{\pi}{6}=\frac{\sqrt2}{2}\cdot\frac12+1\cdot\frac12=\frac{\sqrt2+2}{4}\).

Ответ: 1) \(-\frac12\); 2) \(\frac{19}{2}\); 3) 0; 4) \(\frac{21}{4}\); 5) \(\frac{\sqrt2+2}{4}\).