4. Найдите значения выражения: I) 2 sin π/3 ⋅ cos π/3 + 3 tg π/4 - ctg π/2 II) sin (-π/3) + cos π/6 - tg π/3 + ctg π/3 III) 2 sin π/6 - 3 tg π/4 + ctg π/6 IV) 3 tg π/4 + 2 sin π/2 - 3 tg 0 - 2 ctg π/4 V) 5 sin π/4 + 4 cos 0 - 3 sin π/2 + cos π

4. Вычисление значений выражений:

• I) \( 2 \sin \frac{\pi}{3} \cdot \cos \frac{\pi}{3} + 3 \operatorname{tg} \frac{\pi}{4} - \operatorname{ctg} \frac{\pi}{2} = 2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2} + 3 \cdot 1 - 0 = \frac{\sqrt{3}}{2} + 3 \)
• II) \( \sin \left(-\frac{\pi}{3}\right) + \cos \frac{\pi}{6} - \operatorname{tg} \frac{\pi}{3} + \operatorname{ctg} \frac{\pi}{3} = -\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} - \sqrt{3} + \frac{1}{\sqrt{3}} = -\sqrt{3} + \frac{\sqrt{3}}{3} = -\frac{3\sqrt{3}}{3} + \frac{\sqrt{3}}{3} = -\frac{2\sqrt{3}}{3} \)
• III) \( 2 \sin \frac{\pi}{6} - 3 \operatorname{tg} \frac{\pi}{4} + \operatorname{ctg} \frac{\pi}{6} = 2 \cdot \frac{1}{2} - 3 \cdot 1 + \sqrt{3} = 1 - 3 + \sqrt{3} = -2 + \sqrt{3} \)
• IV) \( 3 \operatorname{tg} \frac{\pi}{4} + 2 \sin \frac{\pi}{2} - 3 \operatorname{tg} 0 - 2 \operatorname{ctg} \frac{\pi}{4} = 3 \cdot 1 + 2 \cdot 1 - 3 \cdot 0 - 2 \cdot 1 = 3 + 2 - 0 - 2 = 3 \)
• V) \( 5 \sin \frac{\pi}{4} + 4 \cos 0 - 3 \sin \frac{\pi}{2} + \cos \pi = 5 \cdot \frac{\sqrt{2}}{2} + 4 \cdot 1 - 3 \cdot 1 + (-1) = \frac{5\sqrt{2}}{2} + 4 - 3 - 1 = \frac{5\sqrt{2}}{2} \)

Ответ: I) \( 3 + \frac{\sqrt{3}}{2} \), II) \( -\frac{2\sqrt{3}}{3} \), III) \( -2 + \sqrt{3} \), IV) 3, V) \( \frac{5\sqrt{2}}{2} \).