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Article 11 — Appendix A.9

arctan or arctg — trigonometric arc tangent function

Category. Mathematics.

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

References. This article is a part of scientific calculator Li-L, scientific calculator Li-X, scientific calculator Li-Lc and scientific calculator Li-Xc products.

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 = arctan x 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[(x − y) /(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]);