# How do you evaluate arctan 133.3?

Oct 23, 2017

$\arctan \left(133.3\right) \approx 1.56329459205$

#### Explanation:

Note that:

$\arctan x = x - {x}^{3} / 3 + {x}^{5} / 5 - {x}^{7} / 7 + \ldots$

This is practical to use for small values of $x$, but what about $133.3$?

The trick is that:

$\arctan \left(x\right) = \frac{\pi}{2} - \arctan \left(\frac{1}{x}\right)$

So we can evaluate $\arctan \left(\frac{1}{133.3}\right)$ and subtract it from $\frac{\pi}{2}$.

$\arctan \left(\frac{1}{133.3}\right) = \arctan \left(\frac{10}{1333}\right)$

$\textcolor{w h i t e}{\arctan \left(\frac{1}{133.3}\right)} \approx \frac{10}{1333} - {10}^{3} / \left({1333}^{3} \cdot 3\right)$

$\textcolor{w h i t e}{\arctan \left(\frac{1}{133.3}\right)} \approx \frac{10}{1333} - \frac{1000}{7105779111}$

$\textcolor{w h i t e}{\arctan \left(\frac{1}{133.3}\right)} \approx 0.00750173474$

So

$\arctan \left(133.3\right) \approx \frac{\pi}{2} - 0.00750173474 \approx 1.56329459205$