ArcCos Notations Traditional name Inverse cosine Traditional notation cos Mathematica StandardForm notation ArcCos Primary definition 0..0.000.0 log The function cos can also be defined by the formula cos sin. The function cos can be defined also as the inverse function for cosw: w cos if and only if cosw. Specific values Values at fixed points 0..0.000.0 cos 0 0..0.000.0 5 0..0.000.0 7
http://functions.wolfram.com 0..0.000.0 5 5 0..0.0005.0 5 5 0..0.0006.0 cos 8 0..0.0007.0 cos 5 8 0..0.0008.0 cos 0..0.0009.0 cos 0..0.000.0 cos 5 5 0 0..0.00.0 cos 5 5 7 0 0..0.00.0 0..0.00.0 0..0.00.0 5 5 0..0.005.0 5 5 0..0.006.0 6
http://functions.wolfram.com 0..0.007.0 5 6 0..0.008.0 cos 8 0..0.009.0 cos 7 8 0..0.000.0 cos 5 5 0 0..0.00.0 cos 5 5 0..0.00.0 cos 0..0.00.0 0..0.00.0 cos 0 0..0.005.0 cos 9 0 0..0.006.0 cos log 0..0.007.0 cos log Values at infinities 0..0.008.0 cos 0..0.009.0 cos 0..0.000.0 cos 0..0.00.0 cos
http://functions.wolfram.com 0..0.00.0 cos General characteristics Domain and analyticity cos is an analytical function of, which is defined over the whole complex -plane. 0..0.000.0 cos Symmetries and periodicities Mirror symmetry 0..0.000.0 cos cos ;,, Periodicity No periodicity Poles and essential singularities The function cos does not have poles and essential singularities. 0..0.000.0 ing cos Branch points The function cos has three branch points: ±,. 0..0.000.0 cos,, 0..0.0005.0 cos, 0..0.0006.0 cos, 0..0.0007.0 cos, log Branch cuts The function cos is a single-valued function on the -plane cut along the intervals, and,. The function cos is continuous from above on the interval, and from below on the interval,. 0..0.0008.0 cos,,,,,
http://functions.wolfram.com 5 0..0.0009.0 lim Ε0 cos x Ε cos x ; x 0..0.000.0 lim Ε0 cos x Ε cos x ; x 0..0.00.0 lim Ε0 cos x Ε cos x ; x 0..0.00.0 lim Ε0 cos x Ε cos x ; x Analytic continuations The analytic continuation of cos has infinitely many sheets; the values of cos cos cos k ; k. are Series representations Generalied power series Expansions at 0 cos 0..06.000.0 6 5 ; 0 0..06.000.0 k k ; k k k 0 0..06.000.0 F, ; ; ;, 0..06.000.0 cos O ; 0 Expansions at 0..06.0005.0 cos 60 ; 0..06.0006.0 cos k 0 k ; k k k k 0..06.0007.0 cos F, ; ;
http://functions.wolfram.com 6 0..06.0008.0 cos O ; Expansions at 0..06.0009.0 cos 60 ; 0..06.000.0 cos k 0 k ; k k k k 0..06.00.0 cos F, ; ; 0..06.00.0 cos O ; Expansions at 0..06.00.0 log 6 ; 0..06.00.0 log k ; k k k k 0..06.005.0 log F,, ;, ; ;, 0..06.006.0 cos log O ; Residue representations 0..06.007.0 cos j s s res s s j ; s 0..06.008.0 s res s s s j 0 s j ;
http://functions.wolfram.com 7 Integral representations On the real axis Of the direct function 0..07.000.0 cos t t Contour integral representations 0..07.000.0 s s s s s ; Arg 0..07.000.0 s s s s s ; Arg 0..07.000.0 Γ s s s s s ; 0 Γ Γ 0..07.0005.0 Γ s Γ s s Arg s s ; 0 Γ Arg Continued fraction representations 0..0.000.0 5 7 5 6 9 ;,, 0..0.000.0 k k k ;,,, k
http://functions.wolfram.com 8 Differential equations Ordinary linear differential equations and wronskians For the direct function itself 0...000.0 w w 0 ; w cos w0 w 0 0...000.0 w w 0 ; w c c cos 0...000.0 W, cos 0...000.0 w ; w cos w0 Transformations Transformations and argument simplifications Argument involving basic arithmetic operations 0..6.000.0 cos cos 0..6.000.0 cos a b c m b c m b m m c cos a b m m c ; m 0..6.000.0 cos cos 0..6.000.0 cos cos 0..6.0005.0 cos cos 0..6.0006.0 cos cos 0..6.0007.0 cos cos
http://functions.wolfram.com 9 0..6.0008.0 cos cos 0..6.0009.0 cos ;,, Products, sums, and powers of the direct function Sums of the direct function 0..6.000.0 cos x cos y sgnx y cos x y x y sgnx y ; x y x y 0 0..6.00.0 cos x cos y sgnx y sgnx y cos x y x y ; x y x y 0..6.00.0 cos x cos y sgnx y cos x y x y ; x y x y 0 0..6.00.0 cos x cos y sgnx y cos x y x y ; x y x y Related transformations Sums involving the direct function 0..6.00.0 cos x sin y sgnx y cos x y x y ; x y x y 0 0..6.005.0 cos x sin y sgnx y cos x y x y ; x y x y 0 0..6.006.0 cos x sin y sgnx y cos x y x y sgnx y ; x y x y 0 0..6.007.0 cos x sin y sgnx y sgnx y cos x y x y ; x y x y Identities Functional identities 0..7.000.0 cos w w cosw w ; w cos
http://functions.wolfram.com 0 Complex characteristics Real part 0..9.000.0 Recos x y cos x y 0..9.000.0 Recos x y tan x y x y cos sin tan x y, x y x y x y Imaginary part 0..9.000.0 Imcos x y sgny logx X ; X x y 0..9.000.0 x y ; x y, x y, tan x y, x y y, x x y x y, x y, Imcos x y log y x y x y cos tan x y, x y sin tan x y, x y x y x y Absolute value 0..9.0005.0 cos x y tan x y x y cos tan x y, x y y, sin tan x y, x y x y x y x ^ x log y x y x y cos tan x y, x y sin tan x y, x y x y x y x
http://functions.wolfram.com Argument 0..9.0006.0 Argcos x y tan tan x y x y cos tan x y, x y y, sin tan x y, x y x y x y x, log y x y x y cos tan x y, x y Conjugate value sin tan x y, x y x y x y 0..9.0007.0 cos x y tan x y x y cos tan x y, x y y, sin tan x y, x y x y x y x log y x y x y cos tan x y, x y sin tan x y, x y x y x y x ^ x Differentiation Low-order differentiation 0..0.000.0 cos 0..0.000.0 cos Symbolic differentiation 0..0.000.0 n cos n F n n, n, ; Fractional integro-differentiation, n ; ; n 0..0.000.0 Α cos Α Α Α Α Α F, Α, ;, Α ;
http://functions.wolfram.com Integration Indefinite integration For the direct function itself 0...000.0 cos 0...000.0 cos cos log cos Li cos 0...000.0 cos cos E sin Fsin 0...000.0 Α cos cos Α Α Α Α Α F Α, ; Α ; 0...0005.0 cos b a cos b a b sin b a b a a a 0...0006.0 cos b a a cos b a b sin b a b a b a a 0...0007.0 cos a b cos b a sin b tan b b tan cos b a cos b a sin b log cos ba b b cos b a sin b Li b b cos ba Li b b cos ba log b b cos ba 0...0008.0 cos Sicos 0...0009.0 cos n cos n cos n n, cos cos n cos n n, cos 0...000.0 cos n n cos n cos n n, cos n cos n cos n n, cos
http://functions.wolfram.com Definite integration For the direct function itself 0 0...00.0 t cos tt 8 0...00.0 cos t t 0 t 5 0...00.0 a t a cos tt ; Rea 0 a a Involving the direct function 0...00.0 logt cos tt log 0 Representations through more general functions Through hypergeometric functions Involving F 0..6.000.0 F, ; ; 0..6.000.0 cos F, ; ; 0..6.000.0 cos F, ; ; Involving p F q 0..6.000.0 log F,, ;, ; ;, Through Meijer G Classical cases for the direct function itself 0..6.0005.0 cos G,,, 0,
http://functions.wolfram.com 0..6.0006.0 cos G,,,, 0..6.0007.0 cos G,,,, 0 0..6.0008.0 cos G,, 0..6.0009.0 cos G,,,, 0 ; Im 0,, 0 ; Im 0 0..6.000.0 cos G,,, 0, Classical cases involving algebraic functions in the arguments 0..6.00.0 cos G,, 0..6.00.0 cos G,, 0..6.00.0 cos G,, 0..6.00.0 cos G,,,,,,, 0, 0,, 0 ;, 0 Classical cases involving unit step Θ,,,,, 0 ;, 0, 0,, 0 ;, 0 0..6.005.0 Θ cos,0 G,, 0, ;, 0 0..6.006.0 Θ cos 0, G,, 0, 0
http://functions.wolfram.com 5 Generalied cases for the direct function itself 0..6.007.0 cos G,,, 0..6.008.0 cos G,,,,,,, 0 Generalied cases involving algebraic functions in the arguments 0..6.009.0 cos G,,,,,, 0..6.000.0 cos G,,,, 0, 0..6.00.0 cos G,,, 0..6.00.0, 0 ;, 0,, 0,,, cos G,,, 0, Generalied cases involving unit step Θ,, 0 ;, 0,, 0 0..6.00.0 Θ cos,0 G,,, 0, 0..6.00.0 Θ cos 0, G,, Through other functions Involving inverse Jacobi functions 0..6.005.0 cos cd 0 0..6.006.0 cos cn 0 0..6.007.0 cos dc 0 0..6.008.0 ds 0, 0, 0
http://functions.wolfram.com 6 0..6.009.0 cos nc 0 0..6.000.0 ns 0 0..6.00.0 sd 0 0..6.00.0 sn 0 Involving some hypergeometric-type functions 0..6.00.0, 0..6.00.0, Representations through equivalent functions With inverse function 0..7.000.0 cos cos ; 0 Re Re 0 Im 0 Re Im 0 0..7.000.0 cos cos ; Re 0 Re 0 Im 0 Re Im 0 0..7.000.0 cos cos k k k ; k Re k Re k Im 0 Re k Im 0 k 0..7.000.0 cos cos Re Re Re Re ΘIm Re 0..7.0005.0 coscos With related functions Involving log 0..7.0006.0 log 0..7.0007.0 cos log ; Re Im 0 Re 0
http://functions.wolfram.com 7 0..7.0008.0 cos log 0..7.0009.0 cos log log 0..7.000.0 cos log Involving sin 0..7.00.0 sin 0..7.00.0 cos sin 0..7.00.0 cos sin 0..7.00.0 cos sin 0..7.005.0 cos sin 0..7.006.0 cos sin 0..7.007.0 cos sin Involving tan 0..7.008.0 cos tan ;, 0..7.009.0 cos tan ; 0..7.000.0 cos tan ;,, 0..7.00.0 cos tan
http://functions.wolfram.com 8 0..7.00.0 tan 0..7.00.0 cos tan ;,, 0..7.00.0 cos r c r c r r c c r r c c tan r c r c r c 0..7.005.0 a a a a a a tan a 0..7.006.0 0..7.007.0 cos tan tan 0..7.008.0 cot Involving cot 0..7.009.0 cos cot ; 0..7.000.0 cos cot 0..7.00.0 a c a c cot a c a c a a c a c c a c Involving csc 0..7.00.0 cos csc 0..7.00.0 csc
http://functions.wolfram.com 9 0..7.00.0 cos csc 0..7.005.0 csc 0..7.006.0 cos csc ; Re 0 0..7.007.0 cos csc 0..7.008.0 cos r c c r r c c c r r c r csc c r 0..7.009.0 a a a a a csc a 0..7.000.0 cos csc Involving sec 0..7.00.0 cos sec 0..7.00.0 cos sec 0..7.00.0 sec ;, 0 Involving sinh 0..7.00.0 cos sinh 0..7.005.0 cos sinh 0..7.006.0 cos sinh
http://functions.wolfram.com 0 0..7.007.0 cos sinh 0..7.008.0 cos a c a c sinh a c a c Involving cosh 0..7.009.0 cos cosh 0..7.0050.0 cos cosh 0..7.005.0 cos cosh ; Im 0 0..7.005.0 cos cosh ; Im 0 Involving tanh 0..7.005.0 cos tanh ;,, 0..7.005.0 tanh 0..7.0055.0 cos tanh ;,, 0..7.0056.0 cos tanh 0..7.0057.0 cos r c r c r r c c r r c c r c tanh r c r c 0..7.0058.0 a a a a a a a tanh a a 0..7.0059.0 cos tanh
http://functions.wolfram.com Involving coth 0..7.0060.0 cos coth 0..7.006.0 cos coth 0..7.006.0 coth Involving csch 0..7.006.0 cos csch 0..7.006.0 0..7.0065.0 cos csch ; Re 0 0..7.0066.0 cos csch ; Re Im 0 0..7.0067.0 cos csch 0..7.0068.0 cos r c c r r c r c c r c r r c 0..7.0069.0 a a a a a csch a 0..7.0070.0 csch 0..7.007.0 csch csch c r csch c r
http://functions.wolfram.com Involving sech 0..7.007.0 sech Inequalities 0..9.000.0 cos x 0 ; x x 0..9.000.0 cos x ; x x Zeros 0..0.000.0 cos 0 ; History J. Herschel (8) introduced the notation cos The function cos is often encountered in mathematics and the natural sciences.
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