3. Вычислите cos(20π/3) - sin(17π/6) : sin(-27.5π) : cos(31π/4)

Сначала упростим каждый тригонометрический аргумент: \(\cos(\frac{20\pi}{3}) = \cos(\frac{20\pi}{3} - 6\pi) = \cos(\frac{20\pi - 18\pi}{3}) = \cos(\frac{2\pi}{3}) = -\frac{1}{2}\) \(\sin(\frac{17\pi}{6}) = \sin(\frac{17\pi}{6} - 2\pi) = \sin(\frac{17\pi - 12\pi}{6}) = \sin(\frac{5\pi}{6}) = \frac{1}{2}\) \(\sin(-27.5\pi) = \sin(-27.5\pi + 28\pi) = \sin(0.5\pi) = \sin(\frac{\pi}{2}) = 1\) \(\cos(\frac{31\pi}{4}) = \cos(\frac{31\pi}{4} - 6\pi) = \cos(\frac{31\pi - 24\pi}{4}) = \cos(\frac{7\pi}{4}) = \frac{\sqrt{2}}{2}\) Теперь подставим значения в исходное выражение: \(-\frac{1}{2} - \frac{1}{2} : 1 : \frac{\sqrt{2}}{2} = -\frac{1}{2} - \frac{1}{2} : \frac{\sqrt{2}}{2} = -\frac{1}{2} - \frac{1}{2} \cdot \frac{2}{\sqrt{2}} = -\frac{1}{2} - \frac{1}{\sqrt{2}} = -\frac{1}{2} - \frac{\sqrt{2}}{2} = \frac{-1 - \sqrt{2}}{2}\) Ответ: \(\frac{-1 - \sqrt{2}}{2}\)