8. Вычислить: 8.1. sin 40° - cos 315°; 8.2. cos 690° - sin 780°; 8.3. sin \(\frac{11\pi}{6}\) + cos \(\frac{5\pi}{3}\); 8.4. sin \(\frac{7\pi}{4}\) + cos \(\frac{7\pi}{4}\)

Решение:

1. 8.1. \( \sin 40^{\circ} - \cos 315^{\circ} \)
   \( \cos 315^{\circ} = \cos(360^{\circ} - 45^{\circ}) = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \)
   \( \sin 40^{\circ} - \frac{\sqrt{2}}{2} \)
2. 8.2. \( \cos 690^{\circ} - \sin 780^{\circ} \)
   \( \cos 690^{\circ} = \cos(690^{\circ} - 2 \cdot 360^{\circ}) = \cos(-30^{\circ}) = \cos 30^{\circ} = \frac{\sqrt{3}}{2} \)
   \( \sin 780^{\circ} = \sin(780^{\circ} - 2 \cdot 360^{\circ}) = \sin(60^{\circ}) = \frac{\sqrt{3}}{2} \)
   \( \frac{\sqrt{3}}{2} - \frac{\sqrt{3}}{2} = 0 \)
3. 8.3. \( \sin \frac{11\pi}{6} + \cos \frac{5\pi}{3} \)
   \( \sin \frac{11\pi}{6} = \sin(2\pi - \frac{\pi}{6}) = -\sin \frac{\pi}{6} = -\frac{1}{2} \)
   \( \cos \frac{5\pi}{3} = \cos(2\pi - \frac{\pi}{3}) = \cos \frac{\pi}{3} = \frac{1}{2} \)
   \( -\frac{1}{2} + \frac{1}{2} = 0 \)
4. 8.4. \( \sin \frac{7\pi}{4} + \cos \frac{7\pi}{4} \)
   \( \sin \frac{7\pi}{4} = \sin(2\pi - \frac{\pi}{4}) = -\sin \frac{\pi}{4} = -\frac{\sqrt{2}}{2} \)
   \( \cos \frac{7\pi}{4} = \cos(2\pi - \frac{\pi}{4}) = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \)
   \( -\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = 0 \)

Ответ: 8.1. \( \sin 40^{\circ} - \frac{\sqrt{2}}{2} \); 8.2. 0; 8.3. 0; 8.4. 0.