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ArcSin

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

ArcSin[z]
    gives the principal inverse sine of z, in [-Pi/2, Pi/2] for real z
    in [-1, 1].
ArcSin is Listable. Branch cuts run along the real axis with |z| > 1.

Examples

No verified examples yet for this function.

Implementation notes

Algorithm. builtin_arcsin (src/trig.c): (1) odd_fold uses oddness ArcSin[-x] -> -ArcSin[x]. (2) trig_i_fold applies the principal-branch identity ArcSin[I y] -> I ArcSinh[y]. (3) Exact inversion via exact_arcsin, which brute-forces the forward table: for each denominator d in {1,2,3,4,5,6,10,12} and each n it computes exact_sin(n,d) and, if it expr_eq-matches the argument, returns n/d * Pi. (4) Numeric fallback: MPFR via mpfr_asin/mpfr_complex_asin, else get_approx + C99 casin for inexact inputs. On the real branch cut x>1 the imaginary part is negated to match Mathematica's lower-side convention (C99 lands on the upper side). Otherwise NULL.

Data structures. Expr* trees; the exact-inversion search reuses the forward exact_sin table rather than a separate inverse table.

Attributes: Listable, NumericFunction, Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= ArcSin[1/2]
Out[1]= 1/6 Pi

In[2]:= ArcSin[1]
Out[2]= 1/2 Pi

In[3]:= N[ArcSin[0.5]]
Out[3]= 0.523599

In[4]:= ArcSin[Sin[x]]
Out[4]= ArcSin[Sin[x]]

In[1]:= ArcSin[Sqrt[3]/2]
Out[1]= 1/3 Pi

In[2]:= ArcSin[I]
Out[2]= I ArcSinh[1]

In[1]:= D[ArcSin[x], x]
Out[1]= 1/Sqrt[1 - x^2]

In[2]:= Series[ArcSin[x], {x, 0, 7}]
Out[2]= x + 1/6 x^3 + 3/40 x^5 + 5/112 x^7 + O[x]^8

Notes

ArcSin[z] gives the principal inverse sine, in [-Pi/2, Pi/2] for real z in [-1, 1]. The inverse-of-forward composition ArcSin[Sin[x]] is deliberately not folded to x, since that holds only on the principal branch. Exact angles such as ArcSin[Sqrt[3]/2] == 1/3 Pi are recognised, and imaginary arguments map to the hyperbolic inverse, ArcSin[I] == I ArcSinh[1]. Differentiation gives the familiar 1/Sqrt[1 - x^2], whose Maclaurin expansion reproduces the classical ArcSin series x + x^3/6 + 3 x^5/40 + 5 x^7/112 + .... ArcSin is Listable.