The minimum positive period of y = a tan (ω x+φ)+b is

T=π/|ω|

According to the nature of periodic function, if it is a periodic function, it must be: f(x)=f(x+T).

Assuming that this function is periodic and the period is t, there is f (x+t) = atan [ω (x+t)+ψ] = atan [ω x+ψ+ω t] = f (x) = atan (ω x+ψ).

tan[ωx+ψ+ωT]=tan(ωx+ψ)

According to the inductive formula, tan(x+π)=tan(x).

So: ωT=π.

T=π/ω

So there is a non-zero constant t, which makes f(x)=f(x+T) hold, so it is a periodic function, and the small positive period is π/ω.

The extended data is tanθ=y/x in rectangular coordinate system, and trigonometric function is a kind of transcendental function in elementary functions in mathematics. Their essence is the mapping between the set of arbitrary angles and a set of ratio variables. The usual trigonometric function is defined in the plane rectangular coordinate system, and its domain is the whole real number domain.

The other is defined in a right triangle, but it is incomplete. Modern mathematics describes them as the limit of infinite sequence and the solution of differential equation, and extends their definitions to complex system. Because of the periodicity of trigonometric function, it does not have the inverse function in the sense of single-valued function.