Question

solve 0=(3^0.5)tan(x+(pi/6))+1 and find all general solutions

Answer 1

(3^0.5)tan(x+(pi/6))+1 = 0
I am going to let x=pi/6 = y for easier typing

(3^0.5)tan(y)+1 = 0

tany = -1/√3 ...... ah!, tan 30º = +1/√3

so y must be in the second or fourth quadrants.
y = 150º or y = 330º or (5pi/6, 11pi/6)


then x+pi/6 = 5pi/6
x = 2pi/3
or
x+pi/6 = 11pi/6
x = 5pi/3

the period of your function is pi

so the general solutions:
x = 2pi/3 + kpi or
x = 5pi/3 + kpi , where k is an integer.