Преобразование градусов в радианы: Решение задачи

Самостоятельная работа 2. Преобразования простейших тригонометрических выражений. 1. Выразить в радианной мере. Для перевода градусов в радианы используется формула: \[ \alpha_{рад} = \frac{\alpha^\circ \cdot \pi}{180^\circ} \] Решение по пунктам: а) \( 10^\circ, 135^\circ, -60^\circ \) \[ 10^\circ = \frac{10 \cdot \pi}{180} = \frac{\pi}{18} \] \[ 135^\circ = \frac{135 \cdot \pi}{180} = \frac{3\pi}{4} \] \[ -60^\circ = -\frac{60 \cdot \pi}{180} = -\frac{\pi}{3} \] б) \( 45^\circ, 160^\circ, -75^\circ \) \[ 45^\circ = \frac{45 \cdot \pi}{180} = \frac{\pi}{4} \] \[ 160^\circ = \frac{160 \cdot \pi}{180} = \frac{8\pi}{9} \] \[ -75^\circ = -\frac{75 \cdot \pi}{180} = -\frac{5\pi}{12} \] в) \( 18^\circ, 150^\circ, -130^\circ \) \[ 18^\circ = \frac{18 \cdot \pi}{180} = \frac{\pi}{10} \] \[ 150^\circ = \frac{150 \cdot \pi}{180} = \frac{5\pi}{6} \] \[ -130^\circ = -\frac{130 \cdot \pi}{180} = -\frac{13\pi}{18} \] г) \( 54^\circ, 135^\circ, -36^\circ \) \[ 54^\circ = \frac{54 \cdot \pi}{180} = \frac{3\pi}{10} \] \[ 135^\circ = \frac{135 \cdot \pi}{180} = \frac{3\pi}{4} \] \[ -36^\circ = -\frac{36 \cdot \pi}{180} = -\frac{\pi}{5} \] д) \( 15^\circ, 120^\circ, -180^\circ \) \[ 15^\circ = \frac{15 \cdot \pi}{180} = \frac{\pi}{12} \] \[ 120^\circ = \frac{120 \cdot \pi}{180} = \frac{2\pi}{3} \] \[ -180^\circ = -\frac{180 \cdot \pi}{180} = -\pi \] е) \( 20^\circ, 125^\circ, -36^\circ \) \[ 20^\circ = \frac{20 \cdot \pi}{180} = \frac{\pi}{9} \] \[ 125^\circ = \frac{125 \cdot \pi}{180} = \frac{25\pi}{36} \] \[ -36^\circ = -\frac{36 \cdot \pi}{180} = -\frac{\pi}{5} \] ж) \( 40^\circ, 225^\circ, -30^\circ \) \[ 40^\circ = \frac{40 \cdot \pi}{180} = \frac{2\pi}{9} \] \[ 225^\circ = \frac{225 \cdot \pi}{180} = \frac{5\pi}{4} \] \[ -30^\circ = -\frac{30 \cdot \pi}{180} = -\frac{\pi}{6} \] з) \( 45^\circ, 240^\circ, -18^\circ \) \[ 45^\circ = \frac{45 \cdot \pi}{180} = \frac{\pi}{4} \] \[ 240^\circ = \frac{240 \cdot \pi}{180} = \frac{4\pi}{3} \] \[ -18^\circ = -\frac{18 \cdot \pi}{180} = -\frac{\pi}{10} \] и) \( 36^\circ, 150^\circ, -210^\circ \) \[ 36^\circ = \frac{36 \cdot \pi}{180} = \frac{\pi}{5} \] \[ 150^\circ = \frac{150 \cdot \pi}{180} = \frac{5\pi}{6} \] \[ -210^\circ = -\frac{210 \cdot \pi}{180} = -\frac{7\pi}{6} \] к) \( 60^\circ, 72^\circ, -252^\circ \) \[ 60^\circ = \frac{60 \cdot \pi}{180} = \frac{\pi}{3} \] \[ 72^\circ = \frac{72 \cdot \pi}{180} = \frac{2\pi}{5} \] \[ -252^\circ = -\frac{252 \cdot \pi}{180} = -\frac{7\pi}{5} \] л) \( 72^\circ, 108^\circ, -270^\circ \) \[ 72^\circ = \frac{72 \cdot \pi}{180} = \frac{2\pi}{5} \] \[ 108^\circ = \frac{108 \cdot \pi}{180} = \frac{3\pi}{5} \] \[ -270^\circ = -\frac{270 \cdot \pi}{180} = -\frac{3\pi}{2} \] м) \( 120^\circ, 135^\circ, -144^\circ \) \[ 120^\circ = \frac{120 \cdot \pi}{180} = \frac{2\pi}{3} \] \[ 135^\circ = \frac{135 \cdot \pi}{180} = \frac{3\pi}{4} \] \[ -144^\circ = -\frac{144 \cdot \pi}{180} = -\frac{4\pi}{5} \] н) \( 75^\circ, 210^\circ, -36^\circ \) \[ 75^\circ = \frac{75 \cdot \pi}{180} = \frac{5\pi}{12} \] \[ 210^\circ = \frac{210 \cdot \pi}{180} = \frac{7\pi}{6} \] \[ -36^\circ = -\frac{36 \cdot \pi}{180} = -\frac{\pi}{5} \] о) \( 100^\circ, 54^\circ, -90^\circ \) \[ 100^\circ = \frac{100 \cdot \pi}{180} = \frac{5\pi}{9} \] \[ 54^\circ = \frac{54 \cdot \pi}{180} = \frac{3\pi}{10} \] \[ -90^\circ = -\frac{90 \cdot \pi}{180} = -\frac{\pi}{2} \] п) \( 45^\circ, 160^\circ, -75^\circ \) (дублирует пункт б) \[ 45^\circ = \frac{\pi}{4}, \quad 160^\circ = \frac{8\pi}{9}, \quad -75^\circ = -\frac{5\pi}{12} \] р) \( 216^\circ, 15^\circ, -60^\circ \) \[ 216^\circ = \frac{216 \cdot \pi}{180} = \frac{6\pi}{5} \] \[ 15^\circ = \frac{15 \cdot \pi}{180} = \frac{\pi}{12} \] \[ -60^\circ = -\frac{60 \cdot \pi}{180} = -\frac{\pi}{3} \] с) \( 130^\circ, 72^\circ, -180^\circ \) \[ 130^\circ = \frac{130 \cdot \pi}{180} = \frac{13\pi}{18} \] \[ 72^\circ = \frac{72 \cdot \pi}{180} = \frac{2\pi}{5} \] \[ -180^\circ = -\pi \] т) \( 54^\circ, 120^\circ, -150^\circ \) \[ 54^\circ = \frac{3\pi}{10} \] \[ 120^\circ = \frac{2\pi}{3} \] \[ -150^\circ = -\frac{150 \cdot \pi}{180} = -\frac{5\pi}{6} \]