№2. а) Найдите значение выражения 1) sin 240° 2) cos 11π/4 3) tg 7π/6 4) 2 sin 9π/4 cos 9π/4. b) Найдите значение выражения 1) cos 240° 2) sin 11π/3 3) tg 7π/6 4) 2 sin 3π/4 cos 3π/4.

Решение:

а) Находим значения выражений:

1. \( \sin 240^{\circ} = \sin (180^{\circ} + 60^{\circ}) = -\sin 60^{\circ} = -\frac{\sqrt{3}}{2} \)
2. \( \cos \frac{11\pi}{4} = \cos \left( 2\pi + \frac{3\pi}{4} \right) = \cos \frac{3\pi}{4} = \cos \left( \pi - \frac{\pi}{4} \right) = -\cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2} \)
3. \( \text{tg } \frac{7\pi}{6} = \text{tg } \left( \pi + \frac{\pi}{6} \right) = \text{tg } \frac{\pi}{6} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \)
4. \( 2 \sin \frac{9\pi}{4} \cos \frac{9\pi}{4} = \sin \left( 2 \cdot \frac{9\pi}{4} \right) = \sin \frac{9\pi}{2} = \sin \left( 4\pi + \frac{\pi}{2} \right) = \sin \frac{\pi}{2} = 1 \)

b) Находим значения выражений:

1. \( \cos 240^{\circ} = \cos (180^{\circ} + 60^{\circ}) = -\cos 60^{\circ} = -\frac{1}{2} \)
2. \( \sin \frac{11\pi}{3} = \sin \left( 4\pi - \frac{\pi}{3} \right) = \sin \left(-\frac{\pi}{3}\right) = -\sin \frac{\pi}{3} = -\frac{\sqrt{3}}{2} \)
3. \( \text{tg } \frac{7\pi}{6} = \text{tg } \left( \pi + \frac{\pi}{6} \right) = \text{tg } \frac{\pi}{6} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \)
4. \( 2 \sin \frac{3\pi}{4} \cos \frac{3\pi}{4} = \sin \left( 2 \cdot \frac{3\pi}{4} \right) = \sin \frac{3\pi}{2} = -1 \)

Ответ: а) 1) -√3/2; 2) -√2/2; 3) √3/3; 4) 1. б) 1) -1/2; 2) -√3/2; 3) √3/3; 4) -1.