3tg (pi*x) + 5ctg (pi*x) + 8=0

Решение:

3tg (pi*x) + 5ctg (pi*x) + 8=0

3tg (pi*x) + 5*1/tg (pi*x) + 8=0

tg (pi*x) = t

3t^2+8t+5=0

t = (-4+-sqrt (16-15)) / 3

t1 = (-4+1) / 3=-1

t2=-5/3

tg (pi*x) = - 1

pi*x=-pi/4+pi*k

x=-1/4+k

tg (pi*x) = - 5/3

x=-1/pi*arctg (5/3) + k.