Question

Solve, finding all solutions in the interval [0, 2π). tan(3x) = -1

Answer 1

tan (3x) = -1
I know that 3x must be in quadrants II or IV
and I know that tan 45° = +1 or tan π/4 = 1
so 3x = π - π/4 OR 3x = 2π - π/4
3x = 3π/4 or 3x = 7π/4
x = π/4 or x = 7π/12

now the period of tan 3x is π/3
so adding multiples of π/3 to any answer will yield a new answer.

x = π/4
x = π/4 + π/3 = 7π/12 , we already have that one
x = 7π/12 + π/3 = 11π/12
x = 11π/12 + π/3 = 5π/4
x = 5π/4 + π/3 = 19π/12
x = 19π/12 + π/3 = 23π/12
The next one will be > 2π

So we have the above 6 solutions in the required domain.

btw, in degrees,
x = 45°, 105°, 165°, 225°, 285, and 345°

check two of them:
tan(3(105)) = -1
tan(3(345) = -1