21.19. Вычислите: a) \(sin(2 arcsin \frac{1}{2} - 3 arccos(\frac{1}{2}))\) b) \(cos(\frac{1}{2} arcsin 1 + arcsin(\frac{\sqrt{2}}{2}))\) c) \(tg(arcsin \frac{\sqrt{3}}{2} + 2 arccos(\frac{\sqrt{2}}{2}))\) d) \(ctg(3 arccos (-1) - arcsin(\frac{1}{2}))\)

a) \(sin(2 arcsin \frac{1}{2} - 3 arccos(\frac{1}{2}))\) * \(arcsin(\frac{1}{2}) = \frac{\pi}{6}\) * \(arccos(\frac{1}{2}) = \frac{\pi}{3}\) * \(sin(2 * \frac{\pi}{6} - 3 * \frac{\pi}{3}) = sin(\frac{\pi}{3} - \pi) = sin(-\frac{2\pi}{3}) = -\frac{\sqrt{3}}{2}\) Ответ: \(-\frac{\sqrt{3}}{2}\) b) \(cos(\frac{1}{2} arcsin 1 + arcsin(\frac{\sqrt{2}}{2}))\) * \(arcsin(1) = \frac{\pi}{2}\) * \(arcsin(\frac{\sqrt{2}}{2}) = \frac{\pi}{4}\) * \(cos(\frac{1}{2} * \frac{\pi}{2} + \frac{\pi}{4}) = cos(\frac{\pi}{4} + \frac{\pi}{4}) = cos(\frac{\pi}{2}) = 0\) Ответ: 0 c) \(tg(arcsin \frac{\sqrt{3}}{2} + 2 arccos(\frac{\sqrt{2}}{2}))\) * \(arcsin(\frac{\sqrt{3}}{2}) = \frac{\pi}{3}\) * \(arccos(\frac{\sqrt{2}}{2}) = \frac{\pi}{4}\) * \(tg(\frac{\pi}{3} + 2 * \frac{\pi}{4}) = tg(\frac{\pi}{3} + \frac{\pi}{2}) = tg(\frac{5\pi}{6}) = -\frac{\sqrt{3}}{3}\) Ответ: \(-\frac{\sqrt{3}}{3}\) d) \(ctg(3 arccos (-1) - arcsin(\frac{1}{2}))\) * \(arccos(-1) = \pi\) * \(arcsin(\frac{1}{2}) = \frac{\pi}{6}\) * \(ctg(3 * \pi - \frac{\pi}{6}) = ctg(\frac{18\pi - \pi}{6}) = ctg(\frac{17\pi}{6}) = ctg(\frac{5\pi}{6}) = -\sqrt{3}\) Ответ: \(-\sqrt{3}\)