Question

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

Answer 1

`arctan(1/2)=0.46364760900081" " "`radian
`arctan(1/2)=26^@ 33' 54.1842''`

Explanation:

these are calculator values

Answer 2

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. .

Answer 3

`arctan(1/2)`

is the best expression for the exact value of `arctan(1/2)`.

Explanation:

There's essentially no way to find an "exact" value of `arctan(1/2)` 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(1/2)`

is the exact value of `arctan(1/2)`.

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(pm 1)` are rational fractions of a circle.