Example 4 Find the value: $\arctan \frac{1}{3}+\arcsin \frac{1}{\sqrt{26}}+\arccos \frac{7}{\sqrt{50}}+\operatorname{arccot} 8$.
Solve: By the geometric meaning of complex number multiplication, the value of the above expression is equal to an argument of the complex number $z=(3+\mathrm{i})(7+\mathrm{i})(5+\mathrm{i})(8+\mathrm{i})$. Since each angle in the expression is acute, the value we are looking for is the principal value of the argument of $z$.
$z=650(1+\mathrm{i})=650 \sqrt{2}\left(\cos \frac{\pi}{4}+\mathrm{i} \sin \frac{\pi}{4}\right), \therefore \arg z=\frac{\pi}{4}$, i.e., the value we are looking for is $\frac{\pi}{4}$.