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Article 11 — Appendix A.2

arccos — trigonometric arc cosine function

Category. Mathematics.

Abstract. Trigonometric arc cosine: 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 cosine is inverse of the cosine function.

2. Graph

Arc cosine is monotone function defined in the range [-1, 1]. Its graph is depicted below in fig. 1.

Fig. 1. Graph of the arc cosine function y = arccos x Fig. 1. Graph of the arc cosine function y = arccosx.

Function codomain is limited to the range [0, π].

3. Identities

Complementary angle:

arcsinx + arccosx = π/2

and as consequence:

arccos sin φ = π/2 − φ

Negative argument:

arccos(−x) = π − arccosx

Reciprocal argument:

arcos(1/x) = arcsecx

Sum and difference:

arccosx + arccosy = arccos{xy − √[(1 − x2)(1 − y2)]}
arccosx − arccosy = arccos{xy + √[(1 − x2)(1 − y2)]}

Some argument values:

Argument xValue arccosx
0π/2
(√6 − √2) /45π/12
(√5 − 1) /42π/5
√(2 − √2) /23π/8
1 /2π/3
√(10 - 2√5) /43π/10
1 /√2π/4
(√5 + 1) /4π/5
√3 /2π/6
√(2 + √2) /2π/8
√(10 + 2√5) /4π/10
(√6 + √2) /4π/12
10
Table 1. Arc cosine for some argument values.

4. Support

Trigonometric arc cosine function arccos is supported in:

Trigonometric arc cosine function of the complex argument arccos is supported in:

5. How to use

To calculate arc cosine of the number:

arccos(-1);

To calculate arc cosine of the current result:

arccos(Rslt);

To calculate arc cosine of the number x in memory:

arccos(Mem[x]);