arccos — trigonometric arc cosine function

1. Definition

Arc cosine is inverse of the cosine function.

arccosx ≡ cos^invx

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 = arccosx.

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

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:

Table 1. Arc cosine for some argument values.

Arc cosine derivative:

arccos′x = −1 /√(1 − x²)

Indefinite integral of the arc cosine:

∫ arccosx dx = x arccosx − √(1 − x²) + C

where C is an arbitrary constant.

To calculate arc cosine of the number:

arccos(−1);

To get arc cosine of the complex number:

arccos(−1+i);

To get arc cosine of the current result:

arccos(rslt);

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

arccos(mem[z]);

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

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