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ArcCos

Status: Stable

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

Description

ArcCos[z]
    gives the principal inverse cosine of z, in [0, Pi] for real z
    in [-1, 1].
ArcCos is Listable. Branch cuts run along the real axis with |z| > 1.

Examples

No verified examples yet for this function.

Implementation notes

Algorithm. builtin_arccos (src/trig.c): (1) arc_pi_minus_fold applies the reflection ArcCos[-x] -> Pi - ArcCos[x] for superficially-negative arguments. (2) arccos_i_fold applies the imaginary-axis identity ArcCos[I y] -> Pi/2 - I ArcSinh[y]. (3) Exact inversion via exact_arccos, which scans n in [0,d] for d in {1,2,3,4,5,6,10,12}, computes exact_cos(n,d), and returns n/d * Pi on an expr_eq match. (4) Numeric fallback: MPFR via mpfr_acos/mpfr_complex_acos, else get_approx + C99 cacos; for real x>1 the imaginary part is negated to match Mathematica's +i*acosh(x) convention. Otherwise NULL.

Data structures. Expr* trees; exact inversion reuses the forward exact_cos 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]:= ArcCos[0]
Out[1]= 1/2 Pi

In[2]:= ArcCos[1/2]
Out[2]= 1/3 Pi

In[3]:= ArcCos[1]
Out[3]= 0

In[4]:= N[ArcCos[0.5]]
Out[4]= 1.0472

In[1]:= ArcCos[-1/2]
Out[1]= 2/3 Pi

In[2]:= ArcCos[Sqrt[2]/2]
Out[2]= 1/4 Pi

In[1]:= N[ArcCos[2], 20]
Out[1]= 0.0 + 1.31695789692481670862*I

Notes

ArcCos[z] gives the principal inverse cosine, in [0, Pi] for real z in [-1, 1]. Special exact angles are recognised: ArcCos[-1/2] is 2/3 Pi and ArcCos[Sqrt[2]/2] is 1/4 Pi. For |z| > 1 the result moves off the real axis onto the branch cut, so ArcCos[2] is purely imaginary, 1.3169... I. ArcCos is Listable.