Упр.1274 ГДЗ Алимов 10-11 класс (Алгебра)

1. $$\operatorname{ctg}(\operatorname{arctg}\sqrt{3})$$
2. $$\operatorname{ctg}(\operatorname{arctg}1)$$
3. $$\sin(\operatorname{arctg}(-\sqrt{3}))$$
4. $$\sin\left(\operatorname{arctg}\frac{1}{\sqrt{3}}\right)$$
5. $$\cos(\operatorname{arctg}1)$$
6. $$\cos(\operatorname{arctg}(-\sqrt{3}))$$

1) $$\ctg(\arctg \sqrt{3})=\ctg \frac{\pi}{3}=\frac{1}{\tg \frac{\pi}{3}}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}.$$

2) $$\ctg(\arctg 1)=\ctg \frac{\pi}{4}=1.$$

3) $$\sin(\arctg(-\sqrt{3}))=\sin\left(-\arctg \sqrt{3}\right)=\sin\left(-\frac{\pi}{3}\right)=-\frac{\sqrt{3}}{2}.$$

4) $$\sin\left(\arctg \frac{1}{\sqrt{3}}\right)=\sin \frac{\pi}{6}=\frac{1}{2}.$$

5) $$\cos(\arctg 1)=\cos \frac{\pi}{4}=\frac{\sqrt{2}}{2}.$$

6) $$\cos(\arctg(-\sqrt{3}))=\cos\left(-\arctg \sqrt{3}\right)=\cos\left(-\frac{\pi}{3}\right)=\frac{1}{2}.$$

Ответ: $$\frac{\sqrt{3}}{3};\ 1;\ -\frac{\sqrt{3}}{2};\ \frac{1}{2};\ \frac{\sqrt{2}}{2};\ \frac{1}{2}.$$