arctan or arctg — trigonometric arc tangent function

1. Definition

Arc tangent is inverse of the tangent function.

arctanx ≡ tan^invx

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).

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[(x − y) /(1 + xy)]

Some argument values:

Table 1. Arc tangent for some argument values.

Arc tangent derivative:

arctan′x = 1 /(1 + x²)

Indefinite integral of the arc tangent:

∫ arctanx dx = x arctanx − ln(1 + x²) /2 + C

where C is an arbitrary constant.

To calculate arc tangent of the number:

arctan(−1);

To get arc tangent of the complex number:

arctan(−1−i);

To get arc tangent of the current result:

arctan(rslt);

To get arc tangent of the number z in calculator memory:

arctan(mem[z]);

Trigonometric arc tangent of the real argument is supported in free version of the Librow calculator.

Trigonometric arc tangent of the complex argument is supported in professional version of the Librow calculator.