Задание 11. Найдите значение выражения 1) 30tg3°tg87°-43 2) -42tg34°tg56°+6 3) 46tg7°tg83°-57 4) -27tg33°tg57°+48 5) 12tg20°tg70°-7 6) -50tg9°tg81°+31

Краткое пояснение:

Для решения этих примеров будем использовать свойство тангенса для смежных углов: \( \text{tg}(90^{\circ} - \alpha) = \text{ctg}(\alpha) \) и \( \text{tg}(\alpha) \cdot \text{ctg}(\alpha) = 1 \).

Пошаговое решение:

1. 1. \[ 30\text{tg}3^{\circ}\text{tg}87^{\circ} - 43 \]Так как \(\text{tg}87^{\circ} = \text{tg}(90^{\circ}-3^{\circ}) = \text{ctg}3^{\circ}\), то:\[ 30\text{tg}3^{\circ}\text{ctg}3^{\circ} - 43 = 30
   1 - 43 = 30 - 43 = -13 \]
2. 2. \[ -42\text{tg}34^{\circ}\text{tg}56^{\circ} + 6 \]Так как \(\text{tg}56^{\circ} = \text{tg}(90^{\circ}-34^{\circ}) = \text{ctg}34^{\circ}\), то:\[ -42\text{tg}34^{\circ}\text{ctg}34^{\circ} + 6 = -42
   1 + 6 = -42 + 6 = -36 \]
3. 3. \[ 46\text{tg}7^{\circ}\text{tg}83^{\circ} - 57 \]Так как \(\text{tg}83^{\circ} = \text{tg}(90^{\circ}-7^{\circ}) = \text{ctg}7^{\circ}\), то:\[ 46\text{tg}7^{\circ}\text{ctg}7^{\circ} - 57 = 46
   1 - 57 = 46 - 57 = -11 \]
4. 4. \[ -27\text{tg}33^{\circ}\text{tg}57^{\circ} + 48 \]Так как \(\text{tg}57^{\circ} = \text{tg}(90^{\circ}-33^{\circ}) = \text{ctg}33^{\circ}\), то:\[ -27\text{tg}33^{\circ}\text{ctg}33^{\circ} + 48 = -27
   1 + 48 = -27 + 48 = 21 \]
5. 5. \[ 12\text{tg}20^{\circ}\text{tg}70^{\circ} - 7 \]Так как \(\text{tg}70^{\circ} = \text{tg}(90^{\circ}-20^{\circ}) = \text{ctg}20^{\circ}\), то:\[ 12\text{tg}20^{\circ}\text{ctg}20^{\circ} - 7 = 12
   1 - 7 = 12 - 7 = 5 \]
6. 6. \[ -50\text{tg}9^{\circ}\text{tg}81^{\circ} + 31 \]Так как \(\text{tg}81^{\circ} = \text{tg}(90^{\circ}-9^{\circ}) = \text{ctg}9^{\circ}\), то:\[ -50\text{tg}9^{\circ}\text{ctg}9^{\circ} + 31 = -50
   1 + 31 = -50 + 31 = -19 \]

Ответ: 1) -13; 2) -36; 3) -11; 4) 21; 5) 5; 6) -19