The Art of Mathematics

# arccos — trigonometric arc cosine function

## 1. Definition

Arc cosine is inverse of the cosine function.

arccosxcosinvx

## 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 = 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. Derivative and indefinite integral

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.

## 5. How to use

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]);``

## 6. Support

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.