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Article 11 — Appendix A.8

arcsin — trigonometric arc sine function

Category. Mathematics.

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

2. Graph

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

Fig. 1. Graph of the arc sine function y = arcsin x Fig. 1. Graph of the arc sine function y = arcsinx.

Function codomain is limited to the range [−π/2, π/2].

3. Identities

Complementary angle:

arcsinx + arccosx = π/2

and as consequence:

arcsin cos φ = π/2 − φ

Negative argument:

arcsin(−x) = −arcsinx

Reciprocal argument:

arcsin(1/x) = arccscx

Sum and difference:

arcsinx + arcsiny = arcsin[x√(1 − y2) + y√(1 − x2)]
arcsinx − arcsiny = arcsin[x√(1 − y2) − y√(1 − x2)]

Some argument values:

Argument xValue arcsinx
00
(√6 − √2) /4π/12
(√5 − 1) /4π/10
√(2 − √2) /2π/8
1 /2π/6
√(10 - 2√5) /4π/5
1 /√2π/4
(√5 + 1) /43π/10
√3 /2π/3
√(2 + √2) /23π/8
√(10 + 2√5) /42π/5
(√6 + √2) /45π/12
1π/2
Table 1. Arc sine for some argument values.

4. Support

Trigonometric arc sine function arsin is supported in:

Trigonometric arc sine function of the complex argument arsin is supported in:

5. How to use

To calculate arc sine of the number:

arcsin(-1);

To calculate arc sine of the current result:

arcsin(Rslt);

To calculate arc sine of the number x in memory:

arcsin(Mem[x]);