(I) Definition Method
If for any independent variable $x$ within the domain, there exists a non-zero constant $T$ such that $f(x+T)=f(x)$ always holds, then $T$ is a period of the function $f(x)$. Usually, when we find the period, we are looking for its smallest positive period.
Example 1 Find the smallest positive period of the following functions according to the definition of periodic functions:
(1) $f(x)=|\sin x|+|\cos x|$;
(2) $f(x)=\tan \frac{2 x}{3}$.
(1) According to the definition of a periodic function, the period $T$ is the increment of the independent variable $x$ that causes the function value to repeat. The problem is to find the smallest positive number $T$ such that for every $x$ in the domain of the function, $f(x+T)=f(x)$ holds.

Let the smallest positive period be $T$. Then for all $x \in \mathbf{R}$, we have $|\sin (x+T)|+|\cos (x+T)|=|\sin x|+|\cos x|$. Let $x=0$, then $|\sin T|+|\cos T|=1$. Squaring both sides gives $|\sin 2 T|=0$, so $T=\frac{k \pi}{2}(k \in \mathbf{Z})$. When $T=\frac{\pi}{2}$, for all $x \in \mathbf{R}$, we have $\left|\sin \left(x+\frac{\pi}{2}\right)\right|+\left|\cos \left(x+\frac{\pi}{2}\right)\right|=|\cos x|+|\sin x|$. Further, we can use proof by contradiction to show that the smallest positive period of $f(x)$ is $\frac{\pi}{2}$.
(2) According to the definition of a periodic function, it is to find the smallest positive number $T$ such that for every $x$ in the domain of the function, $f(x+T)=f(x)$ always holds, while considering that the period of the tangent function $y=\tan x$ is $\pi$.
Since
$$
\tan \frac{2 x}{3}=\tan \left(\frac{2 x}{3}+\pi\right)=\tan \frac{2}{3}\left(x+\frac{3 \pi}{2}\right),
$$

i.e., $f\left(x+\frac{3 \pi}{2}\right)=\tan \frac{2}{3}\left(x+\frac{3 \pi}{2}\right)=\tan \frac{2 x}{3}=f(x)$ always holds. It can also be verified that $\frac{3 \pi}{2}$ is the smallest positive number that meets the condition. Therefore, the smallest positive period of the function $y=\tan \frac{2 x}{3}$ is $\frac{3 \pi}{2}$.
Note: Using the definition is one of the important methods for finding the period of a function.
(II) Formula Method
We already know: The smallest positive period of the function $y=A \sin (\omega x+\varphi)(\omega \neq 0)$ is $\frac{2 \pi}{|\omega|}$, and the smallest positive period of $y=A \cos (\omega x+\varphi)$ is also $\frac{2 \pi}{|\omega|}$. The smallest positive period of the function $y=A \tan (\omega x+\varphi)$ is $\frac{\pi}{|\omega|}$. Therefore, for functions that can be transformed into similar trigonometric forms, we can consider using the formula to find their smallest positive periods. Please see the following examples.