Category. Mathematics.
Abstract. Trigonometric arc sine: definition, plot, properties, identities and table of values for some arguments.
Reference. This article is a part of Librow scientific formula calculator project.
Arc sine is inverse of the sine function.
Arc sine is monotone antisymmetric function defined in the range [−1, 1]. Its plot is depicted below in fig. 1.
[画像:Fig. 1. Plot of the arc sine function y = arcsin x.] Fig. 1. Plot of the arc sine function y = arcsinx.Function codomain is limited to the range [−π/2, π/2].
Complementary angle:
arcsinx + arccosx = π/2and as consequence:
arcsin cos φ = π/2 − φNegative argument:
arcsin(−x) = −arcsinxReciprocal argument:
arcsin(1/x) = arccscxSum and difference:
arcsinx + arcsiny = arcsin[x√(1 − y2) + y√(1 − x2)]Some argument values:
| Argument x | Value arcsinx |
|---|---|
| 0 | 0 |
| (√6 − √2) /4 | π/12 |
| (√5 − 1) /4 | π/10 |
| √(2 − √2) /2 | π/8 |
| 1 /2 | π/6 |
| √(10 - 2√5) /4 | π/5 |
| 1 /√2 | π/4 |
| (√5 + 1) /4 | 3π/10 |
| √3 /2 | π/3 |
| √(2 + √2) /2 | 3π/8 |
| √(10 + 2√5) /4 | 2π/5 |
| (√6 + √2) /4 | 5π/12 |
| 1 | π/2 |
Trigonometric arc sine function arsin of the real argument is supported by free version of the Librow calculator.
Trigonometric arc sine function arsin of the complex argument is supported by professional version of the Librow calculator.
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]);