6. Calculate: $\arctan \frac{1}{3}+\arctan \frac{1}{5}+\arcsin \frac{1}{\sqrt{50}}+\arcsin \frac{1}{\sqrt{65}}$ $=$ . $\qquad$
6. $\frac{\pi}{4}$.

Consider the complex numbers:
$$
z_{1}=3+\mathrm{i}, z_{2}=5+\mathrm{i}, z_{3}=7+\mathrm{i}, z_{4}=8+\mathrm{i} \text {. }
$$

It is easy to see that,
$$
\begin{array}{l}
\arg z_{1}=\arctan \frac{1}{3}, \arg z_{2}=\arctan \frac{1}{5}, \\
\arg z_{3}=\arcsin \frac{1}{\sqrt{50}}, \arg z_{4}=\arcsin \frac{1}{\sqrt{65}} .
\end{array}
$$

Therefore, $\arg z_{1}+\arg z_{2}+\arg z_{3}+\arg z_{4}$
$$
=\arg z_{1} z_{2} z_{3} z_{4}=\arg (650+650 \mathrm{i})=\frac{\pi}{4} \text {. }
$$