Additional Trigonometric functions (real arguments only)#
Sine, \(x\) in degrees, \(\mathrm{sind}(x)\)#
- math53.sind(x)#
Returns the sine of \(x\), with \(x\) in degrees, \(\mathrm{sind}(x)\). See also Wikipedia [1350], MathWorld [947], NIST [517].
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Sind(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Sind('0.51') ereal('5.3518479027559984754E-1')
Inverse sine, input in degrees, \(\mathrm{asind}(x)\)#
Cosine, \(x\) in degrees, \(\mathrm{cosd}(x)\)#
- math53.cosd(x)#
Returns the cosine of \(x\), with \(x\) in degrees, \(\mathrm{cosd}(x)\). See also Wikipedia [1353], MathWorld [932], NIST [517].
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Cosd(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Cosd('0.51') ereal('5.3518479027559984754E-1')
Inverse cosine, input in degrees, \(\mathrm{acosd}(x)\)#
Tangent, with \(x\) in degrees, \(\mathrm{tand}(x)\)#
- math53.tand(x)#
Returns the tangent of \(x\), with \(x\) in degrees, \(\mathrm{tand}tan(x)\). See also Wikipedia [1353], MathWorld [948], NIST [517].
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Tand(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Tand('0.51') ereal('5.3518479027559984754E-1')
Inverse tangent, input in degrees, \(\mathrm{atand}(x)\)#
Cotangent, with \(x\) in degrees, \(\mathrm{cotd}(x)\)#
- math53.cotd(x)#
Returns the cotangent of \(x\), with \(x\) in degrees, \(\mathrm{cotd}(x)\). See also Wikipedia [1353], MathWorld [933], NIST [517].
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Cotd(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Cotd('0.51') ereal('5.3518479027559984754E-1')
Inverse cotangent, input in degrees, \(\mathrm{acotd}(x)\)#
Continuous inverse cotangent, \(\mathrm{acotc}(x)\)#
- ctx.acotc(x)#
where
ctxismath53,mathc53,ctxcpporctxflint.Returns the continuous inverse cotangent of \(x\), \(\mathrm{acotc}(x) = \pi/2 - \mathrm{atan}(x)\). See also Wikipedia [1366], MathWorld [942], NIST [512], Mpmath [596].
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Acotc(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Acotc('0.51') ereal('5.3518479027559984754E-1')
Coversine, \(\mathrm{covers}(x) = 1 - \sin(x)\)#
- math53.covers(x)#
Returns the \(\mathrm{covers}(x) = 1 - \sin(x)\).
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Covers(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Covers('0.51') ereal('5.3518479027559984754E-1')
Versint function \(\mathrm{versint}(x) = x - \sin(x)\)#
- math53.versint(x)#
Returns \(\displaystyle \mathrm{versint}(x) = \int_0^x \mathrm{vers}(t) \mathrm{d}t = x - \sin(x)\), accurate also near 0.
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Versint(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Versint('0.51') ereal('5.3518479027559984754E-1')
Versine function \(\mathrm{vers}(x) = 1 - \cos(x)\)#
- math53.versine(x)#
Returns the versine function \(\mathrm{vers}(x) = 1 - \cos(x)\).
See also: https://en.wikipedia.org/wiki/Versine#Haversine
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Versine(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Versine('0.51') ereal('5.3518479027559984754E-1')
Haversine function \(\mathrm{hav}(x) = (1 - \cos(x))/2\)#
- math53.haversine(x)#
Returns the haversine function \(\mathrm{hav}(x) = (1 - \cos(x))/2 = \sin^2(x/2)\). See also MathWorld [1057], Wikipedia [1452].
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Haversine(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Haversine('0.51') ereal('5.3518479027559984754E-1')
Integral of cos powers, \(\mathrm{cosint}(n,x)\)#
- math53.cosint(n, x)#
Returns \(\displaystyle \mathrm{IC}_n(x) = \int_0^x \cos^n(t) \, \mathrm{d}t\), the integral of the nth cos power, for \(n \ge 0\).
See also Ehrhardt [309] (3.10.13).
An example in Python
>>> from xlcalcnet import ereal >>> ereal.CosInt(3, 4) ereal('5.2359877559829887307E-1') >>> ereal.CosInt(3, 12) ereal('5.3518479027559984754E-1')
Integral of sin powers, \(\mathrm{sinint}(n,x)\)#
- math53.sinint(n, x)#
Returns sinint(n,x) = integral(sin(t)^n, t=0..x), n >= 0
Returns \(\displaystyle \mathrm{IS}_n(x) = \int_0^x \sin^n(t) \, \mathrm{d}t\), the integral of the nth sin power, for \(n \ge 0\).
See also Ehrhardt [309] (3.10.14).
An example in Python
>>> from xlcalcnet import ereal >>> ereal.SinInt(3, 4) ereal('5.2359877559829887307E-1') >>> ereal.SinInt(3, 12) ereal('5.3518479027559984754E-1')
Solutions of Kepler’s equation, \(\mathrm{kepler}(M,e)\)#
- math53.kepler(M, e)#
Returns the solutions (eccentric anomaly \(x\)) of Kepler’s equation from the mean anomaly \(M\) and the eccentricity \(e\), more precisely the solutions \(x\) of
\[\begin{split}M =\begin{cases} x - e \sin(x), & e<1,\\ x+x^3/3, & e=1 \text{ (Barker's equation)},\\ e \sinh(x)-x, & e>1. \end{cases}\end{split}\]See also Ehrhardt [309] (3.10.23).
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Kepler(3, 0.44) ereal('5.2359877559829887307E-1') >>> ereal.Kepler(3, 0.14404) ereal('5.3518479027559984754E-1')
Fibonacci function, \(F_{\nu}(x)\), of real index \(\nu\)#
- math53.fibfun(v, x)#
Returns \(\displaystyle F_{\nu}(x) = \frac{2^{-\nu}(x+\sqrt{x^2+4})^{\nu} - \cos(\pi\nu) \cdot 2^{\nu} (x+\sqrt{x^2+4})^{-\nu} }{\sqrt{x^2+4}}\), the general Fibonacci function of real index \(\nu\).
See also Ehrhardt [309] (3.10.12) , https://en.wikipedia.org/wiki/Fibonacci_polynomials.
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Fibfun(3, 4) ereal('5.2359877559829887307E-1') >>> ereal.Fibfun(3, 12) ereal('5.3518479027559984754E-1')
Cardinal hyperbolic sine, \(\mathrm{sinhc}(x) = \sinh(x)/x\)#
- math53.sinhc(x)#
Returns \(\mathrm{sinhc}(x) = \sinh(x)/x\), accurate also for \(x\) near 0.
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Sinhc(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Sinhc('0.51') ereal('5.3518479027559984754E-1')
Auxiliary function, \(\mathrm{sinhmx}(x) = \sinh(x)-x\)#
- math53.sinhmx(x)#
Returns sinh(x)-x, accurate also for \(x\) near 0.
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Sinhmx(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Sinhmx('0.51') ereal('5.3518479027559984754E-1')
Auxiliary function, \(\mathrm{coshm1}(x) = \cosh(x)-1\)#
- math53.coshm1(x)#
Returns \(\cosh(x)-1\), accurate also for \(x\) near 0.
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Coshm1(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Coshm1('0.51') ereal('5.3518479027559984754E-1')
Langevin function, \(L(x)\)#
- math53.langevin_l(x)#
Returns the Langevin function \(L(x)\), defined as \(L(x) = \coth(x) - 1/x\) for \(x \ne 0\), and \(L(0) = 0\) for \(x = 0\).
See also Ehrhardt [309] (3.10.16).
https://en.wikipedia.org/wiki/Brillouin_and_Langevin_functions
An example in Python
>>> from xlcalcnet import ereal >>> ereal.LangevinL(0.2) ereal('5.2359877559829887307E-1') >>> ereal.LangevinL(0.21) ereal('5.3518479027559984754E-1')
Auxiliary function \(\log(\cosh(x))\)#
- math53.logcosh(x)#
Returns ln(cosh(x)), accurate for x ~ 0 and without overflow for large x
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Logcosh(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Logcosh('0.51') ereal('5.3518479027559984754E-1')
Auxiliary function \(\log(\sinh(x))\)#
- math53.logsinh(x)#
Returns ln(sinh(x)), x > 0, accurate for x ~ 0 and without overflow for large x
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Logsinh(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Logsinh('0.51') ereal('5.3518479027559984754E-1')
Auxiliary function, \(\mathrm{acos}(1-x)\)#
- math53.acos1m(x)#
Returns \(\mathrm{acos}(1-x)\), \(0 \le x \le 2\), accurate also for \(x\) near 0.
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Acos1m(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Acos1m('0.51') ereal('5.3518479027559984754E-1')
Gudermannian function \(\mathrm{gd}(x) = \mathrm{asin}(\mathrm{tanh}(x))\)#
- math53.gd(x)#
Returns the Gudermannian function \(\mathrm{gd}(x) = \mathrm{asin}(\mathrm{tanh}(x))\). See also Wikipedia [1451], MathWorld [1056].
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Gudermann(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Gudermann('0.51') ereal('5.3518479027559984754E-1')
Inverse haversine function \(\mathrm{archav}(x) = \mathrm{acos}(1-2x)\)#
- math53.archav(z)#
Returns the inverse haversine function \(\mathrm{archav}(x) = \mathrm{acos}(1-2x) 2 \mathrm{asin}(\sqrt{x})\), \(0 \le x \le 1\). See also MathWorld [1061], Wikipedia [1454].
An example in Python
>>> from xlcalcnet import ereal >>> ereal.ArcHaversine(0.5) ereal('5.2359877559829887307E-1') >>> ereal.ArcHaversine('0.51') ereal('5.3518479027559984754E-1')
Auxiliary function \(\mathrm{acosh}(1+x)\)#
- math53.acosh1p(z)#
Returns \(\mathrm{acosh}(1+x), x \ge 0\), accurate also for \(x\) near 0.
An example in Python
>>> from xlcalcnet import ereal >>> ereal.Acosh1p(0.5) ereal('5.2359877559829887307E-1') >>> ereal.Acosh1p('0.51') ereal('5.3518479027559984754E-1')
Inverse Gudermannian function \(\mathrm{arcgd}(x) = \mathrm{atanh}(\sin(x))\)#
- math53.arcgd(z)#
Returns the inverse Gudermannian function \(\mathrm{arcgd}(x) = \mathrm{atanh}(\sin(x)), |x| < \pi/2.\)
See also Wikipedia [1453], MathWorld [1060].
An example in Python
>>> from xlcalcnet import ereal >>> ereal.ArcGd(0.5) ereal('5.2359877559829887307E-1') >>> ereal.ArcGd('0.51') ereal('5.3518479027559984754E-1')
Inverse Langevin function, \(L^{-1}(x)\)#
- math53.langevin_l_inv(x)#
Returns the functional inverse \(L^{-1}\) of the Langevin function, i.e. \(L(L^{-1}(x))= x\), \(|x| < 1\).
See also Ehrhardt [309] (3.10.17).
https://en.wikipedia.org/wiki/Brillouin_and_Langevin_functions
An example in Python
>>> from xlcalcnet import ereal >>> ereal.LangevinLInv(0.2) ereal('5.2359877559829887307E-1') >>> ereal.LangevinLInv(0.21) ereal('5.3518479027559984754E-1')