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Cos

Status: Stable

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

Description

Cos[z]

gives the cosine of z (argument in radians).

Notes Cos is Listable. Numeric inputs route to libm / MPFR; rational multiples of Pi reduce to exact values.

Examples (8)

Every input below was run against the current Mathilda build and its output recorded.

Applications (8)

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

In[2]:= Cos[{0, Pi/3, Pi}]
Out[2]= {1, 1/2, -1}

In[3]:= Cos[ArcCos[x]]
Out[3]= x

In[4]:= Cos[Pi/12]
Out[4]= 1/4 (Sqrt[2] + Sqrt[6])

In[5]:= Cos[Pi/5]
Out[5]= 1/4 (1 + Sqrt[5])

In[6]:= TrigExpand[Cos[a + b]]
Out[6]= Cos[a] Cos[b] - Sin[a] Sin[b]

In[7]:= Series[Cos[x], {x, 0, 8}]
Out[7]= 1 - 1/2 x^2 + 1/24 x^4 - 1/720 x^6 + 1/40320 x^8 + O[x]^9

In[8]:= N[Cos[1], 40]
Out[8]= 0.54030230586813971740093660744297660373228

Performance

Against other systems, from the benchmark suite (same input, results cross-checked for agreement):

case Mathilda Wolfram Python
NSolve all roots, degree 50 15.9 s 0.907 s 0.184 s
Simplify quartic-to-Cos[4x] 2.35 s 0.003 s 7.45 s
TrigReduce product of 4 sines 0.894 s 0.14 s 16.5 s
FullSimplify nested radical 0.185 s 0.003 s 1.28 s
Simplify log-exp collapse 0.059 s 0.004 s 1.35 s
Simplify trig Pythagorean 0.048 s 0.004 s 3.69 s

Implementation notes

Algorithm. builtin_cos (src/trig.c) applies a fixed cascade of reductions, returning the first that fires. (1) strip_inverse_call folds Cos[ArcCos[x]] -> x. (2) try_simp_forward_of_inverse handles Cos[ArcSin[x]] -> Sqrt[1-x^2] and Cos[ArcTan[x]] -> 1/Sqrt[1+x^2], building the unevaluated tree and letting the evaluator canonicalise. (3) even_fold uses evenness Cos[-x] -> Cos[x]. (4) trig_i_fold rewrites Cos[I y] -> Cosh[y]. (5) Cos[0] -> 1. (6) For a rational-multiple-of-Pi argument (recognised by extract_pi_multiplier matching Pi or Times[Rational[n,d], Pi]), exact_cos reduces n/d mod 2π using even symmetry, maps into [0, π/2] tracking a sign, and returns the closed surd form from a hardcoded table for denominators 1,2,3,4,5,6,10,12. (7) Numeric fallback: MPFR via mpfr_cos/mpfr_complex_cos for arbitrary-precision args, else get_approx + C99 ccos for already-inexact (real or complex) inputs. Anything else returns NULL (stays symbolic).

Data structures. Plain Expr* trees throughout; exact values are assembled with make_times/make_plus/make_sqrt/make_rational helpers. Pi multiples are carried as int64_t n, d.

Attributes: Listable, NumericFunction, Protected.

References

Notes & additional examples

Notes

The argument is in radians; rational multiples of Pi reduce to exact values while numeric inputs route to libm / MPFR. Cos is Listable.