Prove that $\tan^{-1}{\frac{1}{5}} \approx \frac{\pi}{16}$ using complex number method 
Prove that $\tan^{-1}{\frac{1}{5}} \approx \frac{\pi}{16}$ using
  complex number method.
Hint: Take $z=5+i$

What is meant by complex number method in this problem? I couldn't think of do the above proof using complex numbers. Any suggestions ?

We may notice that 
$$ (5+i)^8 = -3824 + 456960 i \tag{1} $$
is a complex number relatively close to the imaginary axis. By applying $\text{arg}$ to both sides of $(1)$,
$$ 8\arctan\frac{1}{5}\approx \frac{\pi}{2} \tag{2} $$
follows.
