The Art of Interface

# arctan or arctg — trigonometric arc tangent function

Category. Mathematics.

Abstract. Trigonometric arc tangent: definition, graph, properties, identities and table of values for some arguments.

## 1. Definition

Arc tangent is inverse of the tangent function.

## 2. Graph

Arc tangent is monotone antisymmetric function defined everywhere on real axis. Its graph is depicted below in fig. 1. Fig. 1. Graph of the arc tangent function y = arctanx.

Function codomain is limited to the range (−π/2, π/2).

## 3. Identities

Complementary angle:

arctanx + arccotx = π/2

and as consequence:

arctan cot φ = π/2 − φ

Negative argument:

arctan(−x) = −arctanx

Reciprocal argument:

arctan(1/x) = arccotx for x > 0,
arctan(1/x) = arccotx − π for x < 0

Sum and difference:

arctanx + arctany = arctan[(x + y) /(1 − xy)]
arctanx − arctany = arctan[(xy) /(1 + xy)]

Some argument values:

Argument xValue arctanx
00
2 − √3π/12
√(1 − 2 /√5)π/10
√2 − 1π/8
√3 /3π/6
√(5 − 2√5)π/5
1π/4
√(1 + 2 /√5)3π/10
√3π/3
√2 + 13π/8
√(5 + 2√5)2π/5
2 + √35π/12
Table 1. Arc tangent for some argument values.

## 4. Support

Trigonometric arc tangent function arctan or arctg is supported in:

Trigonometric arc tangent function of the complex argument arctan or arctg is supported in:

## 5. How to use

To calculate arc tangent of the number:

``arctan(-1);``

To calculate arc tangent of the current result:

``arctan(Rslt);``

To calculate arc tangent of the number x in memory:

``arctan(Mem[x]);``