arccot or arcctg — trigonometric arc cotangent function

1. Definition

Arc cotangent is inverse of the cotangent function.

arccotx ≡ cot^invx

Arc cotangent is monotone function defined everywhere on real axis. Its graph is depicted below — fig. 1.

Fig. 1. Graph of the arc cotangent function y = arccotx.

Function codomain is limited to the range (0, π).

Complementary angle:

arctanx + arccotx = π/2

and as consequence:

arccot tan φ = π/2 − φ

Negative argument:

arccot(−x) = π − arccotx

Reciprocal argument:

arccot(1/x) = arctanx for x > 0,
arccot(1/x) = arctanx + π for x < 0

Sum and difference:

arccotx + arccoty = arccot[(xy − 1) /(x + y)]
arccotx − arccoty = arccot[(xy + 1) /(y − x)]

Some argument values:

Table 1. Arc cotangent for some argument values.

Arc cotangent derivative:

arccot′x = −1 /(1 + x²)

Indefinite integral of the arc cotangent:

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

where C is an arbitrary constant.

To calculate arc cotangent of the number:

arccot(−1);

To get arc cotangent of the complex number:

arccot(−1+i);

To get arc cotangent of the current result:

arccot(rslt);

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

arccot(mem[z]);

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

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