Вычислите: 28.14. а) cos 68° - cos 22° / sin 68° - sin 22°

Для решения этих задач воспользуемся формулами суммы и разности тригонометрических функций: $\sin \alpha + \sin \beta = 2 \sin \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2}$ $\sin \alpha - \sin \beta = 2 \sin \frac{\alpha - \beta}{2} \cos \frac{\alpha + \beta}{2}$ $\cos \alpha - \cos \beta = -2 \sin \frac{\alpha + \beta}{2} \sin \frac{\alpha - \beta}{2}$ $\cos \alpha + \cos \beta = 2 \cos \frac{\alpha + \beta}{2} \cos \frac{\alpha - \beta}{2}$ а) $\frac{\cos 68^\circ - \cos 22^\circ}{\sin 68^\circ - \sin 22^\circ} = \frac{-2 \sin \frac{68^\circ+22^\circ}{2} \sin \frac{68^\circ-22^\circ}{2}}{2 \sin \frac{68^\circ-22^\circ}{2} \cos \frac{68^\circ+22^\circ}{2}} = \frac{-2 \sin 45^\circ \sin 23^\circ}{2 \sin 23^\circ \cos 45^\circ} = -\tan 45^\circ = -1$ б) $\frac{\sin \frac{7\pi}{18} - \sin \frac{\pi}{9}}{\cos \frac{7\pi}{18} - \cos \frac{\pi}{9}} = \frac{\sin \frac{7\pi}{18} - \sin \frac{2\pi}{18}}{\cos \frac{7\pi}{18} - \cos \frac{2\pi}{18}} = \frac{2 \sin \frac{\frac{7\pi}{18}-\frac{2\pi}{18}}{2} \cos \frac{\frac{7\pi}{18}+\frac{2\pi}{18}}{2}}{-2 \sin \frac{\frac{7\pi}{18}+\frac{2\pi}{18}}{2} \sin \frac{\frac{7\pi}{18}-\frac{2\pi}{18}}{2}} = \frac{\cos \frac{9\pi}{36}}{-\sin \frac{9\pi}{36}} = -\cot \frac{\pi}{4} = -1$ в) $\frac{\sin 130^\circ + \sin 110^\circ}{\cos 130^\circ + \cos 110^\circ} = \frac{2 \sin \frac{130^\circ+110^\circ}{2} \cos \frac{130^\circ-110^\circ}{2}}{2 \cos \frac{130^\circ+110^\circ}{2} \cos \frac{130^\circ-110^\circ}{2}} = \frac{\sin 120^\circ}{\cos 120^\circ} = \tan 120^\circ = -\sqrt{3}$ г) $\frac{\sin \frac{5\pi}{18} + \sin \frac{11\pi}{9}}{\cos \frac{5\pi}{18} + \cos \frac{11\pi}{9}} = \frac{\sin \frac{5\pi}{18} + \sin \frac{22\pi}{18}}{\cos \frac{5\pi}{18} + \cos \frac{22\pi}{18}} = \frac{2 \sin \frac{\frac{27\pi}{18}}{2} \cos \frac{\frac{-17\pi}{18}}{2}}{2 \cos \frac{\frac{27\pi}{18}}{2} \cos \frac{\frac{-17\pi}{18}}{2}} = \tan \frac{27\pi}{36} = \tan \frac{3\pi}{4} = -1$