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

Решение:

№2. а)

1. \( \sin 240^{\circ} \)

\( 240^{\circ} = 180^{\circ} + 60^{\circ} \). \( \sin(180^{\circ} + 60^{\circ}) = -\sin 60^{\circ} = -\frac{\sqrt{3}}{2} \).

2. \( \cos \frac{7\pi}{4} \)

\( \frac{7\pi}{4} = 2\pi - \frac{\pi}{4} \). \( \cos(2\pi - \frac{\pi}{4}) = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

3. \( \operatorname{tg} \frac{11\pi}{6} \)

\( \frac{11\pi}{6} = 2\pi - \frac{\pi}{6} \). \( \operatorname{tg}(2\pi - \frac{\pi}{6}) = -\operatorname{tg} \frac{\pi}{6} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \).

4. \( 2 \sin \frac{3\pi}{4} - \cos \frac{9\pi}{4} \)

\( \sin \frac{3\pi}{4} = \sin(\pi - \frac{\pi}{4}) = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

\( \cos \frac{9\pi}{4} = \cos(2\pi + \frac{\pi}{4}) = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

\( 2 \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = \sqrt{2} - \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} \).

№2. b)

1. \( \cos 240^{\circ} \)

\( 240^{\circ} = 180^{\circ} + 60^{\circ} \). \( \cos(180^{\circ} + 60^{\circ}) = -\cos 60^{\circ} = -\frac{1}{2} \).

2. \( \sin \frac{7\pi}{3} \)

\( \frac{7\pi}{3} = 2\pi + \frac{\pi}{3} \). \( \sin(2\pi + \frac{\pi}{3}) = \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \).

3. \( \operatorname{tg} \frac{11\pi}{6} \)

\( \frac{11\pi}{6} = 2\pi - \frac{\pi}{6} \). \( \operatorname{tg}(2\pi - \frac{\pi}{6}) = -\operatorname{tg} \frac{\pi}{6} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3} \).

4. \( 2 \sin \frac{3\pi}{4} - \cos \frac{9\pi}{4} \)

\( \sin \frac{3\pi}{4} = \sin(\pi - \frac{\pi}{4}) = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

\( \cos \frac{9\pi}{4} = \cos(2\pi + \frac{\pi}{4}) = \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \).

\( 2 \cdot \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = \sqrt{2} - \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2} \).

Ответ: а) 1) \( -\frac{\sqrt{3}}{2} \); 2) \( \frac{\sqrt{2}}{2} \); 3) \( -\frac{\sqrt{3}}{3} \); 4) \( \frac{\sqrt{2}}{2} \). b) 1) \( -\frac{1}{2} \); 2) \( \frac{\sqrt{3}}{2} \); 3) \( -\frac{\sqrt{3}}{3} \); 4) \( \frac{\sqrt{2}}{2} \).