Практическая работа: Преобразование суммы тригонометрических функций в произведение

Практическая работа: Преобразование суммы тригонометрических функций в произведение. Задание 1. Представьте в виде произведения: 1) \( \sin 40^\circ + \sin 16^\circ = 2 \sin \frac{40^\circ + 16^\circ}{2} \cos \frac{40^\circ - 16^\circ}{2} = 2 \sin 28^\circ \cos 12^\circ \) 2) \( \sin 20^\circ - \sin 40^\circ = 2 \sin \frac{20^\circ - 40^\circ}{2} \cos \frac{20^\circ + 40^\circ}{2} = 2 \sin(-10^\circ) \cos 30^\circ = -2 \sin 10^\circ \cos 30^\circ \) 3) \( \cos 15^\circ + \cos 45^\circ = 2 \cos \frac{15^\circ + 45^\circ}{2} \cos \frac{15^\circ - 45^\circ}{2} = 2 \cos 30^\circ \cos(-15^\circ) = 2 \cos 30^\circ \cos 15^\circ \) 4) \( \cos 46^\circ - \cos 74^\circ = -2 \sin \frac{46^\circ + 74^\circ}{2} \sin \frac{46^\circ - 74^\circ}{2} = -2 \sin 60^\circ \sin(-14^\circ) = 2 \sin 60^\circ \sin 14^\circ \) 5) \( \sin \frac{\pi}{5} - \sin \frac{\pi}{10} = 2 \sin \frac{\frac{\pi}{5} - \frac{\pi}{10}}{2} \cos \frac{\frac{\pi}{5} + \frac{\pi}{10}}{2} = 2 \sin \frac{\pi}{20} \cos \frac{3\pi}{20} \) 6) \( \sin \frac{\pi}{3} + \sin \frac{\pi}{4} = 2 \sin \frac{\frac{\pi}{3} + \frac{\pi}{4}}{2} \cos \frac{\frac{\pi}{3} - \frac{\pi}{4}}{2} = 2 \sin \frac{7\pi}{24} \cos \frac{\pi}{24} \) 7) \( \cos \frac{\pi}{10} - \cos \frac{\pi}{20} = -2 \sin \frac{\frac{\pi}{10} + \frac{\pi}{20}}{2} \sin \frac{\frac{\pi}{10} - \frac{\pi}{20}}{2} = -2 \sin \frac{3\pi}{40} \sin \frac{\pi}{40} \) 8) \( \cos \frac{11\pi}{12} + \cos \frac{3\pi}{4} = 2 \cos \frac{\frac{11\pi}{12} + \frac{9\pi}{12}}{2} \cos \frac{\frac{11\pi}{12} - \frac{9\pi}{12}}{2} = 2 \cos \frac{20\pi}{24} \cos \frac{2\pi}{24} = 2 \cos \frac{5\pi}{6} \cos \frac{\pi}{12} \) 9) \( \sin 3t - \sin t = 2 \sin \frac{3t - t}{2} \cos \frac{3t + t}{2} = 2 \sin t \cos 2t \) 10) \( \cos(\alpha - 2\beta) - \cos(\alpha + 2\beta) = -2 \sin \frac{\alpha - 2\beta + \alpha + 2\beta}{2} \sin \frac{\alpha - 2\beta - \alpha - 2\beta}{2} = -2 \sin \alpha \sin(-2\beta) = 2 \sin \alpha \sin 2\beta \) 11) \( \cos 6t + \cos 4t = 2 \cos \frac{6t + 4t}{2} \cos \frac{6t - 4t}{2} = 2 \cos 5t \cos t \) 12) \( \sin(\alpha - 2\beta) - \sin(\alpha + 2\beta) = 2 \sin \frac{\alpha - 2\beta - \alpha - 2\beta}{2} \cos \frac{\alpha - 2\beta + \alpha + 2\beta}{2} = 2 \sin(-2\beta) \cos \alpha = -2 \sin 2\beta \cos \alpha \) 13) \( \text{tg } 25^\circ + \text{tg } 35^\circ = \frac{\sin(25^\circ + 35^\circ)}{\cos 25^\circ \cos 35^\circ} = \frac{\sin 60^\circ}{\cos 25^\circ \cos 35^\circ} \) 14) \( \text{tg } \frac{\pi}{5} - \text{tg } \frac{\pi}{10} = \frac{\sin(\frac{\pi}{5} - \frac{\pi}{10})}{\cos \frac{\pi}{5} \cos \frac{\pi}{10}} = \frac{\sin \frac{\pi}{10}}{\cos \frac{\pi}{5} \cos \frac{\pi}{10}} \) 15) \( \text{tg } 20^\circ + \text{tg } 40^\circ = \frac{\sin(20^\circ + 40^\circ)}{\cos 20^\circ \cos 40^\circ} = \frac{\sin 60^\circ}{\cos 20^\circ \cos 40^\circ} \) 16) \( \text{tg } \frac{\pi}{3} - \text{tg } \frac{\pi}{4} = \frac{\sin(\frac{\pi}{3} - \frac{\pi}{4})}{\cos \frac{\pi}{3} \cos \frac{\pi}{4}} = \frac{\sin \frac{\pi}{12}}{\cos \frac{\pi}{3} \cos \frac{\pi}{4}} \) Задание 2. Проверьте равенство: 1) \( \sin 35^\circ + \sin 25^\circ = 2 \sin \frac{35^\circ + 25^\circ}{2} \cos \frac{35^\circ - 25^\circ}{2} = 2 \sin 30^\circ \cos 5^\circ = 2 \cdot \frac{1}{2} \cdot \cos 5^\circ = \cos 5^\circ \). Равенство верно. 2) \( \sin 40^\circ + \cos 70^\circ = \sin 40^\circ + \sin(90^\circ - 70^\circ) = \sin 40^\circ + \sin 20^\circ = 2 \sin 30^\circ \cos 10^\circ = 2 \cdot \frac{1}{2} \cdot \cos 10^\circ = \cos 10^\circ \). Равенство верно. 3) \( \cos 12^\circ - \cos 48^\circ = -2 \sin \frac{12^\circ + 48^\circ}{2} \sin \frac{12^\circ - 48^\circ}{2} = -2 \sin 30^\circ \sin(-18^\circ) = 2 \cdot \frac{1}{2} \cdot \sin 18^\circ = \sin 18^\circ \). Равенство верно. 4) \( \cos 20^\circ - \sin 50^\circ = \cos 20^\circ - \cos(90^\circ - 50^\circ) = \cos 20^\circ - \cos 40^\circ = -2 \sin 30^\circ \sin(-10^\circ) = 2 \cdot \frac{1}{2} \cdot \sin 10^\circ = \sin 10^\circ \). Равенство верно.