For the sine and cosine functions their period is $2\pi$, so their first period occurs between
$0<\theta<2\pi$
and their outputs repeat every $2\pi$.
This makes sense visually if thinking about a unit circle. As an angle measure $\theta$ goes from $0$ to $2\pi$ it will have gone through a complete rotation, giving every possible output of functions sine and cosine. $\theta$ rotating further will simply cause the outputs of sine and cosine to repeat, and of course their outputs will consistently repeat every $2\pi$.
The period of tangent is $\pi$, so why isn't its first period over the interval
$0<x<\pi$ instead like it is for sine and cosine?
Why have angle measure $\theta$ start at $-\frac{\pi}{2}$ of the unit cricle instead of $0$ for tangent?
You could certainly choose the interval from $0$ to $\pi$ as the "first period" of the tangent function - the choice of where to start is mostly arbitrary. But as @Thoth points out, the tangent function has asymptotes and is undefined at the points $\{(n+\frac 1 2)\pi:n\in \mathbb Z\}$, so it makes sense to view it as defined on the open interval $(-\frac \pi 2, \frac \pi 2)$ and repeated from there. Looking at a graph of the tangent function should make this clearer.