Solve: Since $\arctan \frac{1}{5}$ and $\arctan \frac{1}{239}$ are the principal values of the arguments of the complex numbers $z_{1}=5+\mathrm{i}$ and $z_{2}=239+\mathrm{i}$, respectively.
And from $\frac{z_{1}^{4}}{z_{2}}=\frac{(5+\mathrm{i})^{4}}{239+\mathrm{i}}=\frac{476+480 \mathrm{i}}{239+\mathrm{i}}=\frac{114244}{239^{2}+1}+\frac{114244}{239^{2}+1} \mathrm{i}$,
we know that $\theta=\arg \left(\frac{z_{1}^{4}}{z_{2}}\right)=\arctan \left(\frac{114244}{239^{2}+1} \cdot \frac{239^{2}+1}{114244}\right)=\arctan 1=\frac{\pi}{4}$.
Example 4 Find the size of the angle $\theta=4 \arctan \frac{1}{5}-\arctan \frac{1}{239}$.