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

3 Answers

#arctan(1/2)=0.46364760900081" " " #radian
#arctan(1/2)=26^@ 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(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.