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This assumes ARG to be an Add. The multiple of $\pi$ returned in the second position is always a Rational. Examples ======== >>> from sympy.functions.elementary.trigonometric import _peeloff_pi >>> from sympy import pi >>> from sympy.abc import x, y >>> _peeloff_pi(x + pi/2) (x, 1/2) >>> _peeloff_pi(x + 2*pi/3 + pi*y) (x + pi*y + pi/6, 1/2) F) rrJr make_argscoeffrr6appendHalf is_integeris_even)rr? rest_termsrZKm1m2r/r/r0 _peeloff_pios        rpc Cs|turtjS|s tjS|jr||t}|rz|\}}|jrYt|d}|dkrOt t t |d  }d|}||}t |} | |krNt | |}||}n t t |}||}|jrx|d} | dkrf|S| st|jdurptjStdS| |S|SdS|jrtjSdS)a6 When arg is a Number times $\pi$ (e.g. $3\pi/2$) then return the Number normalized to be in the range $[0, 2]$, else `None`. When an even multiple of $\pi$ is encountered, if it is multiplying something with known parity then the multiple is returned as 0 otherwise as 2. Examples ======== >>> from sympy.functions.elementary.trigonometric import _pi_coeff >>> from sympy import pi, Dummy >>> from sympy.abc import x >>> _pi_coeff(3*x*pi) 3*x >>> _pi_coeff(11*pi/7) 11/7 >>> _pi_coeff(-11*pi/7) 3/7 >>> _pi_coeff(4*pi) 0 >>> _pi_coeff(5*pi) 1 >>> _pi_coeff(5.0*pi) 1 >>> _pi_coeff(5.5*pi) 3/2 >>> _pi_coeff(2 + pi) >>> _pi_coeff(2*Dummy(integer=True)*pi) 2 >>> _pi_coeff(2*Dummy(even=True)*pi) 0 rqrreN)rrOnerJrPrg as_coeff_Mulis_FloatrRintroundr!evalfrrjrkrr7) rcyclescxcxrWpmcmic2r/r/r0r>sF%      r>c@seZdZdZd6ddZd7ddZedd Zee d d Z d8d dZ ddZ ddZ ddZddZddZddZddZddZdd Zd!d"Zd#d$Zd%d&Zd9d(d)Zd*d+Zd:d,d-Zd.d/Zd0d1Zd2d3Zd4d5ZdS);sina The sine function. Returns the sine of x (measured in radians). Explanation =========== This function will evaluate automatically in the case $x/\pi$ is some rational number [4]_. For example, if $x$ is a multiple of $\pi$, $\pi/2$, $\pi/3$, $\pi/4$, and $\pi/6$. Examples ======== >>> from sympy import sin, pi >>> from sympy.abc import x >>> sin(x**2).diff(x) 2*x*cos(x**2) >>> sin(1).diff(x) 0 >>> sin(pi) 0 >>> sin(pi/2) 1 >>> sin(pi/6) 1/2 >>> sin(pi/12) -sqrt(2)/4 + sqrt(6)/4 See Also ======== csc, cos, sec, tan, cot asin, acsc, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions .. [2] http://dlmf.nist.gov/4.14 .. [3] http://functions.wolfram.com/ElementaryFunctions/Sin .. 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[1] https://en.wikipedia.org/wiki/Trigonometric_functions .. [2] http://dlmf.nist.gov/4.14 .. 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Returns the cotangent of x (measured in radians). Explanation =========== See :class:`sin` for notes about automatic evaluation. Examples ======== >>> from sympy import cot, pi >>> from sympy.abc import x >>> cot(x**2).diff(x) 2*x*(-cot(x**2)**2 - 1) >>> cot(1).diff(x) 0 >>> cot(pi/12) sqrt(3) + 2 See Also ======== sin, csc, cos, sec, tan asin, acsc, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions .. [2] http://dlmf.nist.gov/4.14 .. [3] http://functions.wolfram.com/ElementaryFunctions/Cot NcCrr]rrr/r/r0r`rz cot.periodrqcCs |dkr tj|dSt||r)rrrrr/r/r0rcrz cot.fdiffcCrrrrr/r/r0rirz cot.inversec Csddlm}|jr&|tjurtjS|jrtjS|tjtjfvr&|tjtjS|tjur.tjSt ||rr?r@rCrF)$rrrrrr7rcrrr,rrrr1rrr.r>rjrrKrr|rirSrprrr5rrrr$rrr)rrrrrr?rrrrMrKr|rOrPr{r}cotmrr/r/r0ros            2                      zcot.evalcGs||dkr dt|S|dks|ddkrtjSt|}t|d}t|d}tj|ddd|d||||SNrrqre)rrrJrrr)rr{rrrr/r/r0rs   .zcot.taylor_termrcCsL|jd|dt}|r|jr|tj|||dS|tj|||dSNrr)r5rrr rrrrrr/r/r0rs zcot._eval_nseriescCrrrrr/r/r0rrzcot._eval_conjugateTcKsv|jdd|i|\}}|r3ddlm}m}td||d|}td| ||d||fS||tjfSrrrr/r/r0rBs "zcot.as_real_imagcKsfddlm}tj}t|t|fr||jdt }t | |t ||}}|||||Srr)r9rrrrrrr/r/r0rs zcot._eval_rewrite_as_expcKsHt|tr"tj}|jd}| || |||| ||SdSrrrr/r/r0r s  *zcot._eval_rewrite_as_PowcKstd|dt|dSrrrr/r/r0rUrzcot._eval_rewrite_as_sincKst|t|tdddSrrrr/r/r0rrzcot._eval_rewrite_as_coscKrr]rrrr/r/r0rrzcot._eval_rewrite_as_sincoscKrrrrr/r/r0rrzcot._eval_rewrite_as_tancKrr])rrrr)r9rrrrr/r/r0rrzcot._eval_rewrite_as_seccKrr])rrrr)r9rrrrr/r/r0r rzcot._eval_rewrite_as_csccKrr]rrr/r/r0r%rzcot._eval_rewrite_as_powcKrr]rrr/r/r0r+rzcot._eval_rewrite_as_sqrtc Csddlm}ddlm}|jd}||d}d|t}|jr7||td |} |j r4d| S| S|t j urK|j |d||jrGdndd}|t jt jfvrZ|t jt jS|jrb||S|S) Nrrrrerqrrrrrr/r/r0r1s     zcot._eval_as_leading_termcCrrr rr/r/r0r!@rzcot._eval_is_extended_realc s@|jd}d}|jrit|j}g}|jD]}t|dd}||qtdfddt|D}ddg}t|ddD]} ||| dt| |d|| d d7<q=|d|d  t t ||S|j r|j d d \} } | jr| d krtj} td d d} | | | }t|t| | t| fgSt|S)NrFrrcrr/rrXrr/r0r\Nr]z)cot._eval_expand_trig..rrer7rqTrrr)r5rSrrr rhr+rr*rrrgrPrsr rr.rrIr r)r9rCrr{rCXryrr|rrgrrrrr/rr0r Cs,    4   zcot._eval_expand_trigcCs0|jd}|jr|tjdurdS|jrdSdSr<)r5rrrjrr$r/r/r0r%]s zcot._eval_is_finitecC*|jd}|jr|tjdurdSdSdSr<r5rrrjr$r/r/r0rd zcot._eval_is_realcCrr<rr$r/r/r0r0irzcot._eval_is_complexcCs0t|jd\}}|r|jr|tjjSdSdSr)rpr5r7rrirj)r9r)pimultr/r/r0r+ns  zcot._eval_is_zerocCs6|jd}|||}||kr|tjrtjSt|Sr)r5rrrjrrcr)r9oldnewrargnewr/r/r0 _eval_subsss  zcot._eval_subsr]r2r3r\r)!r^r_r`rarrrr4rr5rrrrrBrrrUrrrrrrrrr!r r%rr0r+rr/r/r/r0r:s> %   p      rc@seZdZdZdZejfZdZdZ e ddZ ddZ ddZ d d Zd d Zd.ddZddZddZddZddZddZddZddZddZd/d!d"Zd#d$Zd%d&Zd0d(d)Zd*d+Zd1d,d-ZdS)2ReciprocalTrigonometricFunctionz@Base class for reciprocal functions of trigonometric functions. NcCs>|r|jr || S|jr||  St|}|durYd|jsY|jrY|j}|jd|}||kr>|dt}|| Sd||krYd|t}|jrQ||S|jrY|| St |dri| |kri|j dS|j |}|duru|Stdd|| fDrd|tStdd|| fDrd|tSd|S)Nrerqrrcs|]}t|tVqdSr])r,rrXr/r/r0 z7ReciprocalTrigonometricFunction.eval..csr r])r,rrXr/r/r0r r )r_is_even_is_oddr>rjrrKr|rhasattrrr5_reciprocal_ofranyrrr)rrr?rKr|rtr/r/r0rs@         z$ReciprocalTrigonometricFunction.evalcOs$||jd}t|||i|Sr)rr5getattr)r9 method_namer5ror/r/r0_call_reciprocalsz0ReciprocalTrigonometricFunction._call_reciprocalcOs,|j|g|Ri|}|durd|S|Sr)r)r9rr5rrr/r/r0_calculate_reciprocalsz5ReciprocalTrigonometricFunction._calculate_reciprocalcCs2|||}|dur|||krd|SdSdSr)rr)r9rrrr/r/r0_rewrite_reciprocals z3ReciprocalTrigonometricFunction._rewrite_reciprocalcCst|jd}|||Sr)r r5rr)r9rVrWr/r/r0r[sz'ReciprocalTrigonometricFunction._periodrqcCs|d| |dS)Nrrerrr/r/r0rz%ReciprocalTrigonometricFunction.fdiffcK |d|S)Nrrrr/r/r0rrz4ReciprocalTrigonometricFunction._eval_rewrite_as_expcKr)Nrrrr/r/r0rrz4ReciprocalTrigonometricFunction._eval_rewrite_as_PowcKr)NrUrrr/r/r0rUrz4ReciprocalTrigonometricFunction._eval_rewrite_as_sincKr)Nrrrr/r/r0rrz4ReciprocalTrigonometricFunction._eval_rewrite_as_coscKr)Nrrrr/r/r0rrz4ReciprocalTrigonometricFunction._eval_rewrite_as_tancKr)Nrrrr/r/r0rrz4ReciprocalTrigonometricFunction._eval_rewrite_as_powcKr)Nrrrr/r/r0rrz5ReciprocalTrigonometricFunction._eval_rewrite_as_sqrtcCrrrrr/r/r0rrz/ReciprocalTrigonometricFunction._eval_conjugateTcKs"d||jdj|fi|Sr)rr5rB)r9rArCr/r/r0rBsz,ReciprocalTrigonometricFunction.as_real_imagcKs|jdi|S)Nr )r r)r9rCr/r/r0r rz1ReciprocalTrigonometricFunction._eval_expand_trigcCs||jdSr)rr5r!rr/r/r0r!rz6ReciprocalTrigonometricFunction._eval_is_extended_realrcCsd||jd|Sr)rr5r)r9r{rrr/r/r0rsz5ReciprocalTrigonometricFunction._eval_as_leading_termcCsd||jdjSr)rr5rrr/r/r0r%rz/ReciprocalTrigonometricFunction._eval_is_finitecCsd||jd|||Sr)rr5rr9r{rrrr/r/r0rsz-ReciprocalTrigonometricFunction._eval_nseriesr2r\rr3)r^r_r`rarrrcrdr r r4rrrrr[rrrrUrrrrrrBr r!rr%rr/r/r/r0r{s6 $   rc@~eZdZdZeZdZdddZddZdd Z d d Z d d Z ddZ ddZ dddZddZeeddZdddZdS)ra The secant function. Returns the secant of x (measured in radians). Explanation =========== See :class:`sin` for notes about automatic evaluation. Examples ======== >>> from sympy import sec >>> from sympy.abc import x >>> sec(x**2).diff(x) 2*x*tan(x**2)*sec(x**2) >>> sec(1).diff(x) 0 See Also ======== sin, csc, cos, tan, cot asin, acsc, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions .. [2] http://dlmf.nist.gov/4.14 .. [3] http://functions.wolfram.com/ElementaryFunctions/Sec TNcCrVr]r[rr/r/r0rrWz sec.periodcKs t|dd}|d|dSrr)r9rr cot_half_sqr/r/r0rszsec._eval_rewrite_as_cotcKrrrrr/r/r0r!rzsec._eval_rewrite_as_coscKt|t|t|Sr]rrr/r/r0r$rzsec._eval_rewrite_as_sincoscKrr)rrrrr/r/r0rU'rzsec._eval_rewrite_as_sincKrr)rrrrr/r/r0r*rzsec._eval_rewrite_as_tancKttd|ddSr)rrrr/r/r0r-rzsec._eval_rewrite_as_cscrqcCs.|dkrt|jdt|jdSt||r)rr5rrrr/r/r0r0s z sec.fdiffcCrr<)r5r/rrrirjr$r/r/r0r06rzsec._eval_is_complexcGsX|dks |ddkr tjSt|}|d}tj|td|td||d|Sr)rrJrrrrrr{rkr/r/r0r<s .zsec.taylor_termrc Csddlm}ddlm}|jd}||d}|tdt}|jr8||ttd |} t j || S|t j urL|j |d||jrHdndd}|t jt jfvr[|t jt jS|jrc||S|S)Nrrrrerrrrrrr r5rrrrjrrrrcrrrrrr4rr/r/r0rHs    zsec._eval_as_leading_termr]r2r)r^r_r`rarrr rrrrrUrrrr0r5rrrr/r/r/r0rs"#    rc@r)ra The cosecant function. Returns the cosecant of x (measured in radians). Explanation =========== See :func:`sin` for notes about automatic evaluation. Examples ======== >>> from sympy import csc >>> from sympy.abc import x >>> csc(x**2).diff(x) -2*x*cot(x**2)*csc(x**2) >>> csc(1).diff(x) 0 See Also ======== sin, cos, sec, tan, cot asin, acsc, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions .. [2] http://dlmf.nist.gov/4.14 .. [3] http://functions.wolfram.com/ElementaryFunctions/Csc TNcCrVr]rrr/r/r0rrWz csc.periodcKrrrrr/r/r0rUrzcsc._eval_rewrite_as_sincKr!r]rrr/r/r0rrzcsc._eval_rewrite_as_sincoscKs t|d}d|dd|Srrrr/r/r0rs zcsc._eval_rewrite_as_cotcKrr)rrrrr/r/r0rrzcsc._eval_rewrite_as_coscKr"rrrr/r/r0rrzcsc._eval_rewrite_as_seccKrr)rrrrr/r/r0rrzcsc._eval_rewrite_as_tanrqcCs0|dkrt|jd t|jdSt||r)rr5rrrr/r/r0rs z csc.fdiffcCrr<rr$r/r/r0r0rzcsc._eval_is_complexcGs|dkr dt|S|dks|ddkrtjSt|}|dd}tj|dddd|ddtd||d|dtd|Sr)rrrJrrrr#r/r/r0rs  $  zcsc.taylor_termrc Csddlm}ddlm}|jd}||d}|t}|jr0||t |} t j || S|t j urD|j |d||jr@dndd}|t jt jfvrS|t jt jS|jr[||S|S)Nrrrrrrr%rr/r/r0rs    zcsc._eval_as_leading_termr]r2r)r^r_r`rarrr rrUrrrrrrr0r5rrrr/r/r/r0rXs"#    rc@s\eZdZdZejfZdddZeddZ ddd Z d d Z d d Z ddZ ddZeZdS)ra Represents an unnormalized sinc function: .. math:: \operatorname{sinc}(x) = \begin{cases} \frac{\sin x}{x} & \qquad x \neq 0 \\ 1 & \qquad x = 0 \end{cases} Examples ======== >>> from sympy import sinc, oo, jn >>> from sympy.abc import x >>> sinc(x) sinc(x) * Automated Evaluation >>> sinc(0) 1 >>> sinc(oo) 0 * Differentiation >>> sinc(x).diff() cos(x)/x - sin(x)/x**2 * Series Expansion >>> sinc(x).series() 1 - x**2/6 + x**4/120 + O(x**6) * As zero'th order spherical Bessel Function >>> sinc(x).rewrite(jn) jn(0, x) See also ======== sin References ========== .. [1] https://en.wikipedia.org/wiki/Sinc_function rqcCs8|jd}|dkrt||t||dSt||r)r5rrr)r9rr{r/r/r0rs  z sinc.fdiffcCs|jrtjS|jr|tjtjfvrtjS|tjurtjS|tjur$tjS| r-|| St |}|durQ|j rBt |jr@tjSdSd|j rStj |tj|SdSdSr)r7rrrrrrrJrrcrr>rjr rri)rrr?r/r/r0rs*     z sinc.evalrcCs |jd}t|||||Sr)r5rrrr/r/r0rs zsinc._eval_nseriescKsddlm}|d|S)Nr)jn)sympy.functions.special.besselr&)r9rrr&r/r/r0_eval_rewrite_as_jns  zsinc._eval_rewrite_as_jncKs&tt||t|tjftjtjfSr])r'rrrrJrrtruerr/r/r0rU!&zsinc._eval_rewrite_as_sincCsP|jdjrdSt|jd\}}|jrt|j|jgS|jr$|jr&dSdSdS)NrTF)r5 is_infiniterpr7rrj is_nonzerorr(r/r/r0r+$s  zsinc._eval_is_zerocCr-r)r5rHrrr/r/r0r-szsinc._eval_is_realNr2r3)r^r_r`rarrcrdrr4rrr(rUr+rr%r/r/r/r0rs4    rc@sTeZdZdZejejejejfZ e e ddZ e e ddZ e e ddZdS) InverseTrigonometricFunctionz/Base class for inverse trigonometric functions.c CsBitddtdtddtddtdtdtdtddtdtdtdtddtdtdtddttddtdtdtddttddtjtdtdtddtdttjtddtdtdtddttddttjtddttddtdddtddtddt dtdddttddtddtddtd td dtddt d tddtdtd dtdtdt d tddtddttdd dtdtdttdd iS) Nrrer7rqrrr;r=rG)r$rrrrir/r/r/r0 _asin_table=sP &       "z(InverseTrigonometricFunction._asin_tablecCstddtddtdtdtdtdtddtddtdt ddtdttddtddtdtdtddtdttddtddtddtdtddtddttdddtdtdd tdt ddtdttddi S) Nrr;rqrerrr=rGrr$rrr/r/r/r0 _atan_table[s "z(InverseTrigonometricFunction._atan_tablecCsidtddtdtdtdtddtddtddttddtddtdtddtddttdddttddtddttdddtdtddtdtddtdtdtdtddtdttdddtdtdttdddtdtdtddttddtdd ttd dtdtdtd tdtdttdd tdtd ttd d S) Nrerr7rrqrr;r=rGr/r/r/r/r0 _acsc_tablepsF ""(     z(InverseTrigonometricFunction._acsc_tableN)r^r_r`rarrrrrJrcrdr5rr.r0r3r/r/r/r0r-9s  r-c@seZdZdZd%ddZddZddZd d Zed d Z e e d dZ d&ddZ d'ddZddZddZddZeZddZddZdd Zd!d"Zd%d#d$ZdS)(rab The inverse sine function. Returns the arcsine of x in radians. Explanation =========== ``asin(x)`` will evaluate automatically in the cases $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the result is a rational multiple of $\pi$ (see the ``eval`` class method). A purely imaginary argument will lead to an asinh expression. Examples ======== >>> from sympy import asin, oo >>> asin(1) pi/2 >>> asin(-1) -pi/2 >>> asin(-oo) oo*I >>> asin(oo) -oo*I See Also ======== sin, csc, cos, sec, tan, cot acsc, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] http://dlmf.nist.gov/4.23 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSin rqcCs,|dkrdtd|jddSt||Nrqrrer$r5rrr/r/r0r z asin.fdiffcC2|j|j}|j|jkr|jdjrdSdS|jSr3r4r5r6r8r/r/r0r;   zasin._eval_is_rationalcC|o |jdjSr)r!r5 is_positiverr/r/r0_eval_is_positiverzasin._eval_is_positivecCr:r)r!r5rrr/r/r0_eval_is_negativerzasin._eval_is_negativecCs|jr:|tjur tjS|tjurtjtjS|tjur!tjtjS|jr'tjS|tjur0t dS|tj ur:t dS|tj urBtj S| rL||  S|j r[|}||vr[||St|}|durpddlm}tj||S|jrvtjSt|tr|jd}|jr|dt ;}|t krt |}|t dkrt |}|t dkrt |}|St|tr|jd}|jrt dt|SdSdS)Nrer)asinh)rrrrrr.r7rJrrrrrcr is_numberr.r1rr>r,rr5 is_comparablerr)rr asin_tablerr>angr/r/r0rsX                  z asin.evalcGs|dks |ddkr tjSt|}t|dkr1|dkr1|d}||dd||d|dS|dd}ttj|}t|}|||||Sr)rrJrrrrirrr{rr|r$Rrr/r/r0r s$  zasin.taylor_termNrcCs|jd}||d}|jr||S|tj tjtjfvr-|t j |||d S|dkr7| ||}t |dkrM|jrM|tjkrMt ||St |dkrb|jrb|tjkrbt||S||SNrrr)r5rrr7rrrrrcrr!rrIrrrrrr4r9r{rrrrr/r/r0r s    zasin._eval_as_leading_termcCsddlm}|jd|d}|tjurtddd}ttj|dt  |dd|}tj|jd} | |} | | | } | |dsX|dkrN|dSt d|t|Sttj| j|||d} | t| } ||| ||||S|tjurtddd}ttj|dt  |dd|}tj|jd} | |} | | | } | |ds|dkr|dSt d|t|Sttj| j|||d} | t| } ||| ||||Stj||||d} |tjur| S|dkr|jd||}t|dkr+|jr+|tjkr+t | St|dkr@|jr@|tjkr@t | S| S NrOrTpositivererqr)sympy.series.orderrJr5rrrrrrrr!nseriesris_meromorphicrr$rremoveOrIpowsimprrrcrrrr9r{rrrrJarg0rserarg1rWrXres1resr/r/r0r sD   &   $&  &   &&  " "zasin._eval_nseriescKtdt|Srrrrr/r/r0_eval_rewrite_as_acosF rzasin._eval_rewrite_as_acoscKs dt|dtd|dSr)rr$rr/r/r0_eval_rewrite_as_atanI  zasin._eval_rewrite_as_atancKs&tj ttj|td|dSrrr.r!r$rr/r/r0_eval_rewrite_as_logL r*zasin._eval_rewrite_as_logcKs dtdtd|d|Sr)rr$rr/r/r0_eval_rewrite_as_acotP r\zasin._eval_rewrite_as_acotcKtdtd|Srrrrr/r/r0_eval_rewrite_as_asecS rzasin._eval_rewrite_as_aseccK td|Sr)rrr/r/r0_eval_rewrite_as_acscV rzasin._eval_rewrite_as_acsccC|jd}|jodt|jSNrrqr5rHrRis_nonnegativer9r{r/r/r0r!Y  zasin._eval_is_extended_realcCrrrrr/r/r0r] rz asin.inverser2rr3)r^r_r`rarr;r<r=r4rr5rrrrrZr[r^_eval_rewrite_as_tractabler_rbrdr!rr/r/r/r0rs, * 6   &rc@seZdZdZd%ddZddZeddZee d d Z d&d dZ ddZ ddZ d'ddZddZeZddZddZd%ddZddZdd Zd!d"Zd#d$Zd S)(ra The inverse cosine function. Returns the arc cosine of x (measured in radians). Examples ======== ``acos(x)`` will evaluate automatically in the cases $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the result is a rational multiple of $\pi$ (see the eval class method). ``acos(zoo)`` evaluates to ``zoo`` (see note in :class:`sympy.functions.elementary.trigonometric.asec`) A purely imaginary argument will be rewritten to asinh. Examples ======== >>> from sympy import acos, oo >>> acos(1) 0 >>> acos(0) pi/2 >>> acos(oo) oo*I See Also ======== sin, csc, cos, sec, tan, cot asin, acsc, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] http://dlmf.nist.gov/4.23 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcCos rqcCs,|dkrdtd|jddSt||Nrqrrrer5rr/r/r0r r6z acos.fdiffcCr7r3r8r8r/r/r0r; r9zacos._eval_is_rationalcCsP|jr7|tjur tjS|tjurtjtjS|tjur!tjtjS|jr(tdS|tjur0tj S|tj ur7tS|tj ur?tj S|j r`| }||vrRtd||S| |vr`td|| St|}|durptdt|St|tr|jd}|jr|dt;}|tkrdt|}|St|tr|jd}|jrtdt|SdSdSNrer)rrrrr.rr7rrrrJrrcr?r.r1rr,rr5r@r)rrrArrBr/r/r0r sJ               z acos.evalcGs|dkrtdS|dks|ddkrtjSt|}t|dkr9|dkr9|d}||dd||d|dS|dd}ttj|}t|}| ||||Sr)rrrJrrrrirrCr/r/r0r s$  zacos.taylor_termNrcCs|jd}||d}|dkrtdttj||S|tj tjfvr3|t j |||dS|dkr=| ||}t |dkrT|j rT|tjkrTdt||St |dkrh|j rh|tjkrh|| S||SNrrqrerF)r5rrr$rrrrrcrr!rrrrrrr4rGr/r/r0r s    zacos._eval_as_leading_termcCrerfrgrir/r/r0r! rjzacos._eval_is_extended_realcCs|Sr])r!rr/r/r0_eval_is_nonnegative szacos._eval_is_nonnegativecCstddlm}|jd|d}|tjur~tddd}ttj|dt  |dd|}tj|jd} | |} | | | } | |dsT|dkrN|dS|t |St tj| j|||d} | t | } ||| ||||S|tjurtddd}ttj|dt  |dd|}tj|jd} | |} | | | } | |ds|dkr|dSt|t |St tj| j|||d} | t | } ||| ||||Stj||||d} |tjur| S|dkr |jd||}t|dkr$|jr$|tjkr$dt| St|dkr8|jr8|tjkr8| S| SrH)rMrJr5rrrrrrrr!rNrrOr$rrPrIrQrrrrcrrrrRr/r/r0r sD   &   &  &    &  " "zacos._eval_nseriescKs,tdtjttj|td|dSrrrr.r!r$rr/r/r0r^ s zacos._eval_rewrite_as_logcKrXrrrrr/r/r0_eval_rewrite_as_asin rzacos._eval_rewrite_as_asincKs8ttd|d|tdd|td|dSr)rr$rrr/r/r0r[! 8zacos._eval_rewrite_as_atancCrrr rr/r/r0r$ rz acos.inversecKs(tddtdtd|d|Sr)rrr$rr/r/r0r_* (zacos._eval_rewrite_as_acotcKrcr)rrr/r/r0rb- rzacos._eval_rewrite_as_aseccKr`rrrrr/r/r0rd0 rzacos._eval_rewrite_as_acsccCsV|jd}||jd}|jdur|S|jr%|djr'|djr)|SdSdSdSNrFrq)r5r4rrHrhis_nonpositive)r9rrdr/r/r0r3 s   zacos._eval_conjugater2rr3)r^r_r`rarr;r4rr5rrrr!rorr^rkrrr[rr_rbrdrr/r/r/r0rd s, + +   &  rcseZdZUdZeeed<ejej fZ d*ddZ ddZ dd Z d d Z d d ZddZeddZeeddZd+ddZd,ddZddZeZfddZd*ddZd d!Zd"d#Zd$d%Zd&d'Zd(d)ZZ S)-ra  The inverse tangent function. Returns the arc tangent of x (measured in radians). Explanation =========== ``atan(x)`` will evaluate automatically in the cases $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the result is a rational multiple of $\pi$ (see the eval class method). Examples ======== >>> from sympy import atan, oo >>> atan(0) 0 >>> atan(1) pi/4 >>> atan(oo) pi/2 See Also ======== sin, csc, cos, sec, tan, cot asin, acsc, acos, asec, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] http://dlmf.nist.gov/4.23 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcTan r5rqcCs(|dkrdd|jddSt||r4r5rrr/r/r0rg  z atan.fdiffcCr7r3r8r8r/r/r0r;m r9zatan._eval_is_rationalcCrr)r5is_extended_positiverr/r/r0r<u rzatan._eval_is_positivecCrr)r5is_extended_nonnegativerr/r/r0rox rzatan._eval_is_nonnegativecCrr)r5r7rr/r/r0r+{ rzatan._eval_is_zerocCrrr rr/r/r0r~ rzatan._eval_is_realcCs|jr7|tjur tjS|tjurtdS|tjurt dS|jr$tjS|tjur-tdS|tj ur7t dS|tj urLddl m }|t dtdS| rV||  S|jre|}||vre||St|}|durzddlm}tj||S|jrtjSt|tr|jd}|jr|t;}|tdkr|t8}|St|tr|jd}|jrtdt|}|tdkr|t8}|SdSdS)Nrer7rr)atanh)rrrrrrr7rJrrrrcrrrr?r0r1rr|r.r,rr5r@rr)rrr atan_tablerr|rBr/r/r0r sX                 z atan.evalcGs@|dks |ddkr tjSt|}tj|dd|||Sr)rrJrrrr{rr/r/r0r szatan.taylor_termNrcCs|jd}||d}|jr||S|tj tjtjfvr-|t j |||d S|dkr7| ||}t |dkrPt |jrPt|tjkrP||tSt |dkrit |jrit|tjkri||tS||SrE)r5rrr7rrr.rcrr!rrIrr rrrr4rrrGr/r/r0r s   $$ zatan._eval_as_leading_termcCs|jd|d}tj||||d}|dkr|jd||}|tjur0t|dkr.|tS|St|dkrFt|j rFt |tj krF|tSt|dkr\t|j r\t |tj kr\|tS|Sr) r5rrrrrrcr rr7rrrrr9r{rrrrSrWr/r/r0r s  $$zatan._eval_nseriescKs2tjdttjtj|ttjtj|Sr)rr.r!rrrr/r/r0r^ szatan._eval_rewrite_as_logcsx|dtjurtdtd|jd|||S|dtjur3t dtd|jd|||St||||Sr) rrrrr5rrsuper _eval_aseriesr9rargs0r{r __class__r/r0r s $&zatan._eval_aseriescCrrrrr/r/r0r rz atan.inversecKs0t|d|tdtdtd|dSrr$rrrr/r/r0rr rozatan._eval_rewrite_as_asincKs(t|d|tdtd|dSrr$rrr/r/r0rZ rtzatan._eval_rewrite_as_acoscKrcrrrr/r/r0r_ rzatan._eval_rewrite_as_acotcKs$t|d|ttd|dSrr$rrr/r/r0rb s$zatan._eval_rewrite_as_aseccKs,t|d|tdttd|dSrr$rrrr/r/r0rd ,zatan._eval_rewrite_as_acscr2rr3)!r^r_r`ratTupler__annotations__rr.rdrr;r<ror+rr4rr5rrrrr^rkrrrrrZr_rbrd __classcell__r/r/rr0r< s4  &  4     rcseZdZdZejej fZd'ddZddZddZ d d Z d d Z e d dZ eeddZd(ddZd)ddZfddZddZeZd'ddZddZdd Zd!d"Zd#d$Zd%d&ZZS)*ra The inverse cotangent function. Returns the arc cotangent of x (measured in radians). Explanation =========== ``acot(x)`` will evaluate automatically in the cases $x \in \{\infty, -\infty, \tilde{\infty}, 0, 1, -1\}$ and for some instances when the result is a rational multiple of $\pi$ (see the eval class method). A purely imaginary argument will lead to an ``acoth`` expression. ``acot(x)`` has a branch cut along $(-i, i)$, hence it is discontinuous at 0. Its range for real $x$ is $(-\frac{\pi}{2}, \frac{\pi}{2}]$. Examples ======== >>> from sympy import acot, sqrt >>> acot(0) pi/2 >>> acot(1) pi/4 >>> acot(sqrt(3) - 2) -5*pi/12 See Also ======== sin, csc, cos, sec, tan, cot asin, acsc, acos, asec, atan, atan2 References ========== .. [1] http://dlmf.nist.gov/4.23 .. [2] http://functions.wolfram.com/ElementaryFunctions/ArcCot rqcCs(|dkrdd|jddSt||rlrxrr/r/r0r. ryz acot.fdiffcCr7r3r8r8r/r/r0r;4 r9zacot._eval_is_rationalcCrr)r5rhrr/r/r0r<< rzacot._eval_is_positivecCrr)r5rrr/r/r0r=? rzacot._eval_is_negativecCrrr rr/r/r0r!B rzacot._eval_is_extended_realcCs|jr5|tjur tjS|tjurtjS|tjurtjS|jr"tdS|tjur+tdS|tj ur5t dS|tj ur=tjS| rG||  S|j rf| }||vrftd||}|tdkrd|t8}|St|}|dur|ddlm}tj ||S|jrttjSt|tr|jd}|jr|t;}|tdkr|t8}|St|tr|jd}|jrtdt|}|tdkr|t8}|SdSdS)Nrer7r)acoth)rrrrrJrr7rrrrrcrr?r0r1rrr.rir,rr5r@rr)rrr}rBrrr/r/r0rE s\                 z acot.evalcGsP|dkrtdS|dks|ddkrtjSt|}tj|dd|||Sr)rrrJrrr~r/r/r0r{ s zacot.taylor_termNrcCs|jd}||d}|tjurd||S|dkr#|||}|tj tjtjfvr;| t j |||d St |dkr[t |jr[t|tjkr[t|tjkr[||tSt |dkr{t |jr{t|tjkr{t|tjkr{||tS||S)NrrqrF)r5rrrrcrrr.rJrr!rrIr r7rrrr4rrrGr/r/r0r s   22 zacot._eval_as_leading_termcCs|jd|d}tj||||d}|tjur|S|dkr&|jd||}|jr5t|dkr3|t S|St|dkrRt|jrRt |tj krRt |tj krR|t St|dkrot|jrot |tj krot |tj kro|t S|Sr)r5rrrrrcrr7r rrrJrrrrr/r/r0r s  22zacot._eval_nseriescs|dtjurtdtd|jd|||S|dtjur5ttddtd|jd|||Stt | ||||S)Nrrerqr) rrrrr5rrrrrrrrr/r0r s $*zacot._eval_aseriescKs.tjdtdtj|tdtj|Sr)rr.r!rr/r/r0r^ szacot._eval_rewrite_as_logcCrrrrr/r/r0r rz acot.inversecKs@|td|dtdtt|d t|d dSrrrr/r/r0rr s*zacot._eval_rewrite_as_asincKs8|td|dtt|d t|d dSrrrr/r/r0rZ rszacot._eval_rewrite_as_acoscKrcrrrr/r/r0r[ rzacot._eval_rewrite_as_atancKs0|td|dttd|d|dSrrrr/r/r0rb rozacot._eval_rewrite_as_aseccKs8|td|dtdttd|d|dSrrrr/r/r0rd rszacot._eval_rewrite_as_acscr2rr3)r^r_r`rarr.rdrr;r<r=r!r4rr5rrrrrr^rkrrrrZr[rbrdrr/r/rr0r s0*  5    rc@seZdZdZeddZdddZdddZdd d Zdd dZ ddZ ddZ e Z ddZ ddZddZddZddZd S) ra The inverse secant function. Returns the arc secant of x (measured in radians). Explanation =========== ``asec(x)`` will evaluate automatically in the cases $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the result is a rational multiple of $\pi$ (see the eval class method). ``asec(x)`` has branch cut in the interval $[-1, 1]$. For complex arguments, it can be defined [4]_ as .. math:: \operatorname{sec^{-1}}(z) = -i\frac{\log\left(\sqrt{1 - z^2} + 1\right)}{z} At ``x = 0``, for positive branch cut, the limit evaluates to ``zoo``. For negative branch cut, the limit .. math:: \lim_{z \to 0}-i\frac{\log\left(-\sqrt{1 - z^2} + 1\right)}{z} simplifies to :math:`-i\log\left(z/2 + O\left(z^3\right)\right)` which ultimately evaluates to ``zoo``. As ``acos(x) = asec(1/x)``, a similar argument can be given for ``acos(x)``. Examples ======== >>> from sympy import asec, oo >>> asec(1) 0 >>> asec(-1) pi >>> asec(0) zoo >>> asec(-oo) pi/2 See Also ======== sin, csc, cos, sec, tan, cot asin, acsc, acos, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] http://dlmf.nist.gov/4.23 .. [3] http://functions.wolfram.com/ElementaryFunctions/ArcSec .. 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Returns the arc cosecant of x (measured in radians). Explanation =========== ``acsc(x)`` will evaluate automatically in the cases $x \in \{\infty, -\infty, 0, 1, -1\}$` and for some instances when the result is a rational multiple of $\pi$ (see the ``eval`` class method). Examples ======== >>> from sympy import acsc, oo >>> acsc(1) pi/2 >>> acsc(-1) -pi/2 >>> acsc(oo) 0 >>> acsc(-oo) == acsc(oo) True >>> acsc(0) zoo See Also ======== sin, csc, cos, sec, tan, cot asin, acos, asec, atan, acot, atan2 References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] http://dlmf.nist.gov/4.23 .. 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Signs of both `y` and `x` are considered to determine the appropriate quadrant of `\operatorname{atan}(y/x)`. The range is `(-\pi, \pi]`. The complete definition reads as follows: .. math:: \operatorname{atan2}(y, x) = \begin{cases} \arctan\left(\frac y x\right) & \qquad x > 0 \\ \arctan\left(\frac y x\right) + \pi& \qquad y \ge 0, x < 0 \\ \arctan\left(\frac y x\right) - \pi& \qquad y < 0, x < 0 \\ +\frac{\pi}{2} & \qquad y > 0, x = 0 \\ -\frac{\pi}{2} & \qquad y < 0, x = 0 \\ \text{undefined} & \qquad y = 0, x = 0 \end{cases} Attention: Note the role reversal of both arguments. The `y`-coordinate is the first argument and the `x`-coordinate the second. If either `x` or `y` is complex: .. math:: \operatorname{atan2}(y, x) = -i\log\left(\frac{x + iy}{\sqrt{x^2 + y^2}}\right) Examples ======== Going counter-clock wise around the origin we find the following angles: >>> from sympy import atan2 >>> atan2(0, 1) 0 >>> atan2(1, 1) pi/4 >>> atan2(1, 0) pi/2 >>> atan2(1, -1) 3*pi/4 >>> atan2(0, -1) pi >>> atan2(-1, -1) -3*pi/4 >>> atan2(-1, 0) -pi/2 >>> atan2(-1, 1) -pi/4 which are all correct. Compare this to the results of the ordinary `\operatorname{atan}` function for the point `(x, y) = (-1, 1)` >>> from sympy import atan, S >>> atan(S(1)/-1) -pi/4 >>> atan2(1, -1) 3*pi/4 where only the `\operatorname{atan2}` function reurns what we expect. We can differentiate the function with respect to both arguments: >>> from sympy import diff >>> from sympy.abc import x, y >>> diff(atan2(y, x), x) -y/(x**2 + y**2) >>> diff(atan2(y, x), y) x/(x**2 + y**2) We can express the `\operatorname{atan2}` function in terms of complex logarithms: >>> from sympy import log >>> atan2(y, x).rewrite(log) -I*log((x + I*y)/sqrt(x**2 + y**2)) and in terms of `\operatorname(atan)`: >>> from sympy import atan >>> atan2(y, x).rewrite(atan) Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, re(x) < 0), (0, Ne(x, 0)), (nan, True)) but note that this form is undefined on the negative real axis. See Also ======== sin, csc, cos, sec, tan, cot asin, acsc, acos, asec, atan, acot References ========== .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions .. [2] https://en.wikipedia.org/wiki/Atan2 .. 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