23. Вычислить значения выражений: 23.1. cos105° + cos75°; 23.2. sin105° - sin75°; 23.3. cos \(\frac{11\pi}{12}\) + cos \(\frac{5\pi}{12}\); 23.4. cos \(\frac{11\pi}{12}\) - cos \(\frac{5\pi}{12}\); 23.5. sin \(\frac{7\pi}{12}\) - sin \(\frac{\pi}{12}\); 23.6. sin105° + sin165°

Решение:

1. 23.1. \( \cos 105^{\circ} + \cos 75^{\circ} \)
   Используем формулу суммы косинусов: \( \cos \alpha + \cos \beta = 2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} \)
   \( 2 \cos \frac{105^{\circ}+75^{\circ}}{2} \cos \frac{105^{\circ}-75^{\circ}}{2} = 2 \cos \frac{180^{\circ}}{2} \cos \frac{30^{\circ}}{2} = 2 \cos 90^{\circ} \cos 15^{\circ} = 2 \cdot 0 \cdot \cos 15^{\circ} = 0 \)
2. 23.2. \( \sin 105^{\circ} - \sin 75^{\circ} \)
   Используем формулу разности синусов: \( \sin \alpha - \sin \beta = 2 \cos \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2} \)
   \( 2 \cos \frac{105^{\circ}+75^{\circ}}{2} \sin \frac{105^{\circ}-75^{\circ}}{2} = 2 \cos 90^{\circ} \sin 15^{\circ} = 2 \cdot 0 \cdot \sin 15^{\circ} = 0 \)
3. 23.3. \( \cos \frac{11\pi}{12} + \cos \frac{5\pi}{12} \)
   \( 2 \cos \frac{\frac{11\pi}{12}+\frac{5\pi}{12}}{2} \cos \frac{\frac{11\pi}{12}-\frac{5\pi}{12}}{2} = 2 \cos \frac{\frac{16\pi}{12}}{2} \cos \frac{\frac{6\pi}{12}}{2} = 2 \cos \frac{4\pi}{6} \cos \frac{\pi}{4} = 2 \cos \frac{2\pi}{3} \cos \frac{\pi}{4} \)
   \( 2 \cdot (-\frac{1}{2}) \cdot \frac{\sqrt{2}}{2} = -\frac{\sqrt{2}}{2} \)
4. 23.4. \( \cos \frac{11\pi}{12} - \cos \frac{5\pi}{12} \)
   Используем формулу разности косинусов: \( \cos \alpha - \cos \beta = -2 \sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2} \)
   \( -2 \sin \frac{\frac{11\pi}{12}+\frac{5\pi}{12}}{2} \sin \frac{\frac{11\pi}{12}-\frac{5\pi}{12}}{2} = -2 \sin \frac{16\pi}{24} \sin \frac{6\pi}{24} = -2 \sin \frac{2\pi}{3} \sin \frac{\pi}{4} \)
   \( -2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} = -\frac{\sqrt{6}}{2} \)
5. 23.5. \( \sin \frac{7\pi}{12} - \sin \frac{\pi}{12} \)
   \( 2 \cos \frac{\frac{7\pi}{12}+\frac{\pi}{12}}{2} \sin \frac{\frac{7\pi}{12}-\frac{\pi}{12}}{2} = 2 \cos \frac{8\pi}{24} \sin \frac{6\pi}{24} = 2 \cos \frac{\pi}{3} \sin \frac{\pi}{4} \)
   \( 2 \cdot \frac{1}{2} \cdot \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} \)
6. 23.6. \( \sin 105^{\circ} + \sin 165^{\circ} \)
   Используем формулу суммы синусов: \( \sin \alpha + \sin \beta = 2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2} \)
   \( 2 \sin \frac{105^{\circ}+165^{\circ}}{2} \cos \frac{105^{\circ}-165^{\circ}}{2} = 2 \sin \frac{270^{\circ}}{2} \cos \frac{-60^{\circ}}{2} = 2 \sin 135^{\circ} \cos (-30^{\circ}) \)
   \( \sin 135^{\circ} = \sin(180^{\circ} - 45^{\circ}) = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \)
   \( \cos (-30^{\circ}) = \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
   \( 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2} \)

Ответ: 23.1. 0; 23.2. 0; 23.3. \( -\frac{\sqrt{2}}{2} \); 23.4. \( -\frac{\sqrt{6}}{2} \); 23.5. \( \frac{\sqrt{2}}{2} \); 23.6. \( \frac{\sqrt{6}}{2} \).