Упр.28.6 ГДЗ Мордковича 10 класс профильный уровень (Алгебра)

1. а) $$\operatorname{tg} 25^\circ+\operatorname{tg} 35^\circ$$; б) $$\operatorname{tg}\frac{\pi}{5}-\operatorname{tg}\frac{\pi}{10}$$; в) $$\operatorname{tg} 20^\circ+\operatorname{tg} 40^\circ$$; г) $$\operatorname{tg}\frac{\pi}{3}-\operatorname{tg}\frac{\pi}{4}$$.

Используем формулу:

$$\tg \alpha \pm \tg \beta=\frac{\sin \alpha \cos \beta \pm \sin \beta \cos \alpha}{\cos \alpha \cos \beta}.$$

1. $$\tg 25^\circ+\tg 35^\circ=\frac{\sin 25^\circ}{\cos 25^\circ}+\frac{\sin 35^\circ}{\cos 35^\circ}$$

   $$=\frac{\sin 25^\circ \cos 35^\circ+\sin 35^\circ \cos 25^\circ}{\cos 25^\circ \cos 35^\circ}$$

   $$=\frac{\sin(25^\circ+35^\circ)}{\cos 25^\circ \cos 35^\circ}=\frac{\sin 60^\circ}{\cos 25^\circ \cos 35^\circ}.$$

2. $$\tg \frac{\pi}{5}-\tg \frac{\pi}{10}=\frac{\sin \frac{\pi}{5}}{\cos \frac{\pi}{5}}-\frac{\sin \frac{\pi}{10}}{\cos \frac{\pi}{10}}$$

   $$=\frac{\sin \frac{\pi}{5}\cos \frac{\pi}{10}-\sin \frac{\pi}{10}\cos \frac{\pi}{5}}{\cos \frac{\pi}{5}\cos \frac{\pi}{10}}$$

   $$=\frac{\sin\left(\frac{\pi}{5}-\frac{\pi}{10}\right)}{\cos \frac{\pi}{5}\cos \frac{\pi}{10}}=\frac{\sin \frac{\pi}{10}}{\cos \frac{\pi}{5}\cos \frac{\pi}{10}}.$$

3. $$\tg 20^\circ+\tg 40^\circ=\frac{\sin 20^\circ}{\cos 20^\circ}+\frac{\sin 40^\circ}{\cos 40^\circ}$$

   $$=\frac{\sin 20^\circ \cos 40^\circ+\sin 40^\circ \cos 20^\circ}{\cos 20^\circ \cos 40^\circ}$$

   $$=\frac{\sin(20^\circ+40^\circ)}{\cos 20^\circ \cos 40^\circ}=\frac{\sin 60^\circ}{\cos 20^\circ \cos 40^\circ}.$$

4. $$\tg \frac{\pi}{3}-\tg \frac{\pi}{4}=\frac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{3}}-\frac{\sin \frac{\pi}{4}}{\cos \frac{\pi}{4}}$$

   $$=\frac{\sin \frac{\pi}{3}\cos \frac{\pi}{4}-\sin \frac{\pi}{4}\cos \frac{\pi}{3}}{\cos \frac{\pi}{3}\cos \frac{\pi}{4}}$$

   $$=\frac{\sin\left(\frac{\pi}{3}-\frac{\pi}{4}\right)}{\cos \frac{\pi}{3}\cos \frac{\pi}{4}}=\frac{\sin \frac{\pi}{12}}{\cos \frac{\pi}{3}\cos \frac{\pi}{4}}.$$

Ответ: а) $$\frac{\sin 60^\circ}{\cos 25^\circ \cos 35^\circ}$$; б) $$\frac{\sin \frac{\pi}{10}}{\cos \frac{\pi}{5}\cos \frac{\pi}{10}}$$; в) $$\frac{\sin 60^\circ}{\cos 20^\circ \cos 40^\circ}$$; г) $$\frac{\sin \frac{\pi}{12}}{\cos \frac{\pi}{3}\cos \frac{\pi}{4}}$$.