Найдите значения выражения: cos 107° cos 17° + sin 107° sin 17°; cos 36° cos 24° - sin 36° sin 24°...

Для решения этих примеров мы воспользуемся формулами сложения для тригонометрических функций: 1. $\cos(\alpha - \beta) = \cos\alpha \cos\beta + \sin\alpha \sin\beta$ 2. $\cos(\alpha + \beta) = \cos\alpha \cos\beta - \sin\alpha \sin\beta$ 3. $\sin(\alpha + \beta) = \sin\alpha \cos\beta + \cos\alpha \sin\beta$ 4. $\sin(\alpha - \beta) = \sin\alpha \cos\beta - \cos\alpha \sin\beta$ **Решение левого столбца:** - $\cos 107^{\circ} \cos 17^{\circ} + \sin 107^{\circ} \sin 17^{\circ} = \cos(107^{\circ} - 17^{\circ}) = \cos 90^{\circ} = 0$ - $\cos 36^{\circ} \cos 24^{\circ} - \sin 36^{\circ} \sin 24^{\circ} = \cos(36^{\circ} + 24^{\circ}) = \cos 60^{\circ} = \frac{1}{2}$ - $\cos \frac{\pi}{12} \cos \frac{\pi}{4} - \sin \frac{\pi}{12} \sin \frac{\pi}{4} = \cos(\frac{\pi}{12} + \frac{\pi}{4}) = \cos(\frac{\pi + 3\pi}{12}) = \cos \frac{4\pi}{12} = \cos \frac{\pi}{3} = \frac{1}{2}$ - $\sin \frac{\pi}{12} \cos \frac{\pi}{4} - \cos \frac{\pi}{12} \sin \frac{\pi}{4} = \sin(\frac{\pi}{12} - \frac{\pi}{4}) = \sin(\frac{\pi - 3\pi}{12}) = \sin(-\frac{2\pi}{12}) = -\sin \frac{\pi}{6} = -\frac{1}{2}$ **Решение правого столбца:** а) $\cos \frac{5\pi}{8} \cos \frac{3\pi}{8} + \sin \frac{5\pi}{8} \sin \frac{3\pi}{8} = \cos(\frac{5\pi}{8} - \frac{3\pi}{8}) = \cos \frac{2\pi}{8} = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ б) $\sin \frac{2\pi}{15} \cos \frac{\pi}{5} + \cos \frac{2\pi}{15} \sin \frac{\pi}{5} = \sin(\frac{2\pi}{15} + \frac{\pi}{5}) = \sin(\frac{2\pi + 3\pi}{15}) = \sin \frac{5\pi}{15} = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2}$ в) $\sin 63^{\circ} \cos 27^{\circ} + \cos 63^{\circ} \sin 27^{\circ} = \sin(63^{\circ} + 27^{\circ}) = \sin 90^{\circ} = 1$ г) $\sin 51^{\circ} \cos 21^{\circ} - \cos 51^{\circ} \sin 21^{\circ} = \sin(51^{\circ} - 21^{\circ}) = \sin 30^{\circ} = \frac{1}{2}$