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Упражнения Вычислите (413-423). 413. a) \( \operatorname{tg} 75^{\circ} \); в) \( \operatorname{ctg} 105^{\circ} \); д) \( \operatorname{tg} \frac{7 \pi}{12} \); б) \( \operatorname{tg} 225^{\circ} \); г) \( \operatorname{ctg} \frac{\pi}{12} \); e) \( \operatorname{ctg} \frac{5 \pi}{12} \). 414. \( \operatorname{tg}\left(45^{\circ}+\alpha\right) \), если \( \operatorname{tg} \alpha=2 \). 415. \( \operatorname{ctg}\left(\frac{\pi}{6}-\alpha\right) \), если \( \operatorname{ctg} \alpha=3 \). 416. \( \operatorname{tg}\left(\frac{\pi}{3}-\alpha\right) \), если \( \sin \alpha=\frac{3}{4}, \quad \frac{\pi}{2}<\alpha<\pi \). 417. \( \operatorname{ctg}\left(\frac{\pi}{4}+\alpha\right) \), если \( \cos \alpha=\frac{12}{13}, \quad \pi<\alpha<\frac{3}{2} \pi \). 418. \( \operatorname{tg} \alpha \), если \( \operatorname{tg}\left(\frac{\pi}{4}-\alpha\right)=-2 \). 419. \( \operatorname{tg} \beta \), если \( \operatorname{tg}(\alpha+\beta)=-1, \quad \operatorname{tg} \alpha=3 \). 420. \( \operatorname{tg}(\alpha+\beta) \), если \( \sin \alpha=\frac{3}{4}, \quad \frac{\pi}{2}<\alpha<\pi \). 421. \( \operatorname{tg}(\alpha-\beta) \), если \( \cos \alpha=-0,8, \quad \pi<\alpha<\frac{3 \pi}{2} \). 422. а) \( \operatorname{tg}(\alpha+\beta) \), если \( \operatorname{tg} \alpha=-\frac{3}{4} \) и \( \operatorname{tg} \beta=2,4 \); б) \( \operatorname{ctg}(\alpha-\beta) \), если \( \operatorname{ctg} \alpha=\frac{4}{3} \) и \( \operatorname{ctg} \beta=-1 \). 423. a) \( \frac{\operatorname{tg} 27^{\circ}+\operatorname{tg} 18^{\circ}}{1-\operatorname{tg} 27^{\circ} \operatorname{tg} 18^{\circ}} \); г) \( \frac{\operatorname{ctg} \frac{3 \pi}{5} \operatorname{ctg} \frac{\pi}{10}+1}{\operatorname{ctg} \frac{3 \pi}{5}-\operatorname{ctg} \frac{\pi}{10}} \); 121
Trigonometry Jan 18, 2025
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