434 Найти значение выражения: 1) 3 sinπ/6 + 2 cos π/6 - tg π/3; 2) 5 sin π/4 + 3 tg π/4 - 5 cos π/4 - 10 ctg π/4; 3) (2 tg π/6 - tg π/3) : cos π/6; 4) sin π/3 · cos π/6 - tg π/4.

Ответ: Решение ниже

1) \( 3 \sin(\frac{\pi}{6}) + 2 \cos(\frac{\pi}{6}) - \tan(\frac{\pi}{3}) \)

\( \sin(\frac{\pi}{6}) = \frac{1}{2} \)

\( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \)

\( \tan(\frac{\pi}{3}) = \sqrt{3} \)

\( 3 \cdot \frac{1}{2} + 2 \cdot \frac{\sqrt{3}}{2} - \sqrt{3} = \frac{3}{2} + \sqrt{3} - \sqrt{3} = \frac{3}{2} = 1.5 \)

2) \( 5 \sin(\frac{\pi}{4}) + 3 \tan(\frac{\pi}{4}) - 5 \cos(\frac{\pi}{4}) - 10 \cot(\frac{\pi}{4}) \)

\( \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)

\( \tan(\frac{\pi}{4}) = 1 \)

\( \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2} \)

\( \cot(\frac{\pi}{4}) = 1 \)

\( 5 \cdot \frac{\sqrt{2}}{2} + 3 \cdot 1 - 5 \cdot \frac{\sqrt{2}}{2} - 10 \cdot 1 = \frac{5\sqrt{2}}{2} + 3 - \frac{5\sqrt{2}}{2} - 10 = 3 - 10 = -7 \)

3) \( (2 \tan(\frac{\pi}{6}) - \tan(\frac{\pi}{3})) : \cos(\frac{\pi}{6}) \)

\( \tan(\frac{\pi}{6}) = \frac{\sqrt{3}}{3} \)

\( \tan(\frac{\pi}{3}) = \sqrt{3} \)

\( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \)

\( (2 \cdot \frac{\sqrt{3}}{3} - \sqrt{3}) : \frac{\sqrt{3}}{2} = (\frac{2\sqrt{3}}{3} - \frac{3\sqrt{3}}{3}) : \frac{\sqrt{3}}{2} = \frac{-\sqrt{3}}{3} : \frac{\sqrt{3}}{2} = \frac{-\sqrt{3}}{3} \cdot \frac{2}{\sqrt{3}} = -\frac{2}{3} \)

4) \( \sin(\frac{\pi}{3}) \cdot \cos(\frac{\pi}{6}) - \tan(\frac{\pi}{4}) \)

\( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \)

\( \cos(\frac{\pi}{6}) = \frac{\sqrt{3}}{2} \)

\( \tan(\frac{\pi}{4}) = 1 \)

\( \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} - 1 = \frac{3}{4} - 1 = -\frac{1}{4} = -0.25 \)

Ответ: Решение ниже