Для начала вычислим значения ctg(5π/3) = 1/tg(5π/3) и sin(8π/3):

tg(5π/3) = sin(5π/3) / cos(5π/3) = √3 / (-1/2) = -2√3

ctg(5π/3) = 1 / tg(5π/3) = -1 / 2√3 = -√3 / 6

sin(8π/3) = sin(2π + 2π/3) = sin(2π/3) = √3 / 2

Теперь найдем значение выражения:

6 ctg(5π/3) sin(8π/3) = 6 (-√3 / 6) (√3 / 2) = -3

Итак, значение выражения равно -3.