# How do you find the exact value of Arctan(1/2)?

$\arctan \left(\frac{1}{2}\right) = 0.46364760900081 \text{ " }$radian
$\arctan \left(\frac{1}{2}\right) = {26}^{\circ} 33 ' 54.1842 ' '$

#### Explanation:

these are calculator values

Feb 21, 2016

In [0, 2$\pi$], there are two angles 26.56505118 deg and 206.56595118 deg, nearly.

#### Explanation:

tan x can be any number on the real line, including rational numbers i.e. integer/integer.
Inversely, the angle(s) are transcendental numbers (sans 0 for 0), in radian measure, that might approximate to rational numbers, in degree measure. For example, arctan 1 = $\pi$.4 = 45 deg.
This is a matter of our convenience, by dividing $\pi$ radian into 180 equal parts, in deg measure. .

May 14, 2018

$\arctan \left(\frac{1}{2}\right)$

is the best expression for the exact value of $\arctan \left(\frac{1}{2}\right)$.

#### Explanation:

There's essentially no way to find an "exact" value of $\arctan \left(\frac{1}{2}\right)$ expressed as a finite expression of integers combined via addition, subtraction, multiplication, division, and root taking.

By the typically vacuous arithmetic of the real numbers

$\arctan \left(\frac{1}{2}\right)$

is the exact value of $\arctan \left(\frac{1}{2}\right)$.

In general the relationship between a slope (which is what a tangent is) and an angle is transcendental. Among rational tangents, only $\arctan 0$ and $\arctan \left(\pm 1\right)$ are rational fractions of a circle.