Самостоятельная работа №3: Перевод градусов в радианы

Самостоятельная работа №3. Задание 1. Переведите из градусной меры в радианную. Для перевода используется формула: \(\alpha_{рад} = \frac{\alpha^\circ \cdot \pi}{180^\circ}\). 1) \(120^\circ = \frac{120\pi}{180} = \frac{2\pi}{3}\) 2) \(210^\circ = \frac{210\pi}{180} = \frac{7\pi}{6}\) 3) \(220^\circ = \frac{220\pi}{180} = \frac{11\pi}{9}\) 4) \(150^\circ = \frac{150\pi}{180} = \frac{5\pi}{6}\) 5) \(300^\circ = \frac{300\pi}{180} = \frac{5\pi}{3}\) 6) \(315^\circ = \frac{315\pi}{180} = \frac{7\pi}{4}\) 7) \(765^\circ = \frac{765\pi}{180} = \frac{17\pi}{4}\) 8) \(675^\circ = \frac{675\pi}{180} = \frac{15\pi}{4}\) Задание 2. Выразите в градусах. Для перевода подставляем \(\pi = 180^\circ\). 1) \(\frac{\pi}{15} = \frac{180^\circ}{15} = 12^\circ\) 2) \(\frac{2\pi}{3} = \frac{2 \cdot 180^\circ}{3} = 120^\circ\) 3) \(0,25\pi = \frac{180^\circ}{4} = 45^\circ\) 4) \(\frac{\pi}{12} = \frac{180^\circ}{12} = 15^\circ\) 5) \(\frac{11\pi}{6} = \frac{11 \cdot 180^\circ}{6} = 330^\circ\) 6) \(\frac{21}{4}\pi = \frac{21 \cdot 180^\circ}{4} = 945^\circ\) 7) \(\frac{\pi}{8} = \frac{180^\circ}{8} = 22,5^\circ\) 8) \(1,5\pi = 1,5 \cdot 180^\circ = 270^\circ\) 9) \(-\frac{31}{6}\pi = -\frac{31 \cdot 180^\circ}{6} = -930^\circ\) 10) \(\frac{7\pi}{9} = \frac{7 \cdot 180^\circ}{9} = 140^\circ\) 11) \(3\pi = 3 \cdot 180^\circ = 540^\circ\) 12) \(\frac{101}{12}\pi = \frac{101 \cdot 180^\circ}{12} = 1515^\circ\) Задание 4. Вычислите. 1) \(2\sin 30^\circ - \text{tg } 45^\circ + \text{ctg } 30^\circ = 2 \cdot \frac{1}{2} - 1 + \sqrt{3} = 1 - 1 + \sqrt{3} = \sqrt{3}\) 2) \(\text{tg } 60^\circ + 2\cos 45^\circ - \sqrt{3}\text{ctg } 45^\circ = \sqrt{3} + 2 \cdot \frac{\sqrt{2}}{2} - \sqrt{3} \cdot 1 = \sqrt{3} + \sqrt{2} - \sqrt{3} = \sqrt{2}\) 3) \(6\cos 30^\circ - 3\text{tg } 60^\circ + 2\sin 45^\circ = 6 \cdot \frac{\sqrt{3}}{2} - 3 \cdot \sqrt{3} + 2 \cdot \frac{\sqrt{2}}{2} = 3\sqrt{3} - 3\sqrt{3} + \sqrt{2} = \sqrt{2}\) 4) \(\sqrt{3}\text{tg } 30^\circ + 4\sin 30^\circ - \sqrt{3}\text{ctg } 30^\circ = \sqrt{3} \cdot \frac{\sqrt{3}}{3} + 4 \cdot \frac{1}{2} - \sqrt{3} \cdot \sqrt{3} = 1 + 2 - 3 = 0\) 5) \(\sqrt{3}\sin \frac{\pi}{3} - 2\cos \frac{\pi}{6} + \frac{\sqrt{3}}{2}\text{tg } \frac{\pi}{3} = \sqrt{3} \cdot \frac{\sqrt{3}}{2} - 2 \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \cdot \sqrt{3} = \frac{3}{2} - \sqrt{3} + \frac{3}{2} = 3 - \sqrt{3}\) 6) \(2\cos \frac{\pi}{3} + 2\sin \frac{\pi}{6} - 2\sin \frac{\pi}{4} = 2 \cdot \frac{1}{2} + 2 \cdot \frac{1}{2} - 2 \cdot \frac{\sqrt{2}}{2} = 1 + 1 - \sqrt{2} = 2 - \sqrt{2}\) 7) \(\sqrt{3}\cos \frac{\pi}{6} + 2\sin \frac{\pi}{3} - \frac{\sqrt{3}}{2}\text{ctg } \frac{\pi}{6} = \sqrt{3} \cdot \frac{\sqrt{3}}{2} + 2 \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} \cdot \sqrt{3} = \frac{3}{2} + \sqrt{3} - \frac{3}{2} = \sqrt{3}\) 8) \(\sqrt{2}\cos \frac{\pi}{4} - 2\sin \frac{\pi}{6} + \text{ctg } \frac{\pi}{6} = \sqrt{2} \cdot \frac{\sqrt{2}}{2} - 2 \cdot \frac{1}{2} + \sqrt{3} = 1 - 1 + \sqrt{3} = \sqrt{3}\) 9) \(2\sin \pi - \cos 0 + \text{tg } 0 + 3\cos \frac{\pi}{2} - \sin \frac{3\pi}{2} = 2 \cdot 0 - 1 + 0 + 3 \cdot 0 - (-1) = -1 + 1 = 0\) 10) \(5\sin 90^\circ + 2\cos 0^\circ - 2\sin 270^\circ + 10\cos 180^\circ = 5 \cdot 1 + 2 \cdot 1 - 2 \cdot (-1) + 10 \cdot (-1) = 5 + 2 + 2 - 10 = -1\)