Найти значение выражения: 1) cos 630° − sin 1470° − ctg 1125°

Для решения этих задач воспользуемся периодичностью тригонометрических функций: $360^\circ$ ($2\pi$) для синуса и косинуса, $180^\circ$ ($\pi$) для тангенса и котангенса. ### Задание 530 1) $\cos 630^\circ - \sin 1470^\circ - \text{ctg } 1125^\circ$ - $\cos(630^\circ) = \cos(630^\circ - 720^\circ) = \cos(-90^\circ) = 0$ - $\sin(1470^\circ) = \sin(1470^\circ - 1440^\circ) = \sin(30^\circ) = 0{,}5$ - $\text{ctg}(1125^\circ) = \text{ctg}(1125^\circ - 1080^\circ) = \text{ctg}(45^\circ) = 1$ - $0 - 0{,}5 - 1 = -1{,}5$ 2) $\text{tg } 1800^\circ - \sin 495^\circ + \cos 945^\circ$ - $\text{tg}(1800^\circ) = \text{tg}(0^\circ) = 0$ - $\sin(495^\circ) = \sin(495^\circ - 360^\circ) = \sin(135^\circ) = \frac{\sqrt{2}}{2}$ - $\cos(945^\circ) = \cos(945^\circ - 720^\circ) = \cos(225^\circ) = -\frac{\sqrt{2}}{2}$ - $0 - \frac{\sqrt{2}}{2} + (-\frac{\sqrt{2}}{2}) = -\sqrt{2}$ 3) $3 \cos 3660^\circ + \sin(-1560^\circ) + \cos(-450^\circ)$ - $3 \cos(60^\circ) + \sin(-120^\circ) + \cos(-90^\circ) = 3 \cdot 0{,}5 - \frac{\sqrt{3}}{2} + 0 = 1{,}5 - \frac{\sqrt{3}}{2} = \frac{3-\sqrt{3}}{2}$ 4) $\cos 4455^\circ - \cos(-945^\circ) + \text{tg } 1035^\circ - \text{ctg }(-1500^\circ)$ - $\cos(135^\circ) - \cos(-225^\circ) + \text{tg}(315^\circ) - \text{ctg}(-60^\circ) = -\frac{\sqrt{2}}{2} - (-\frac{\sqrt{2}}{2}) - 1 + \frac{\sqrt{3}}{3} = \frac{\sqrt{3}}{3} - 1$ ### Задание 531 1) $\cos \frac{23\pi}{4} - \sin \frac{15\pi}{4} - \text{ctg } (-\frac{11\pi}{2})$ - $\cos(\frac{23\pi}{4}) = \cos(-\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ - $\sin(\frac{15\pi}{4}) = \sin(-\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}$ - $\text{ctg}(-\frac{11\pi}{2}) = \text{ctg}(\frac{\pi}{2}) = 0$ - $\frac{\sqrt{2}}{2} - (-\frac{\sqrt{2}}{2}) - 0 = \sqrt{2}$ 2) $\sin \frac{25\pi}{3} - \cos (-\frac{17\pi}{2}) - \text{tg } \frac{10\pi}{3}$ - $\sin(\frac{\pi}{3}) - \cos(-\frac{\pi}{2}) - \text{tg}(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} - 0 - \sqrt{3} = -\frac{\sqrt{3}}{2}$ 3) $\sin(-7\pi) - 2 \cos \frac{31\pi}{3} - \text{tg } \frac{7\pi}{4}$ - $0 - 2 \cos(\frac{\pi}{3}) - \text{tg}(-\frac{\pi}{4}) = 0 - 2 \cdot 0{,}5 - (-1) = 0$ 4) $\cos (-9\pi) + 2 \sin (-\frac{49\pi}{6}) - \text{ctg } (-\frac{21\pi}{4})$ - $\cos(\pi) + 2 \sin(-\frac{\pi}{6}) - \text{ctg}(\frac{3\pi}{4}) = -1 + 2 \cdot (-0{,}5) - (-1) = -1 - 1 + 1 = -1$