Additional Trigonometric functions (real arguments only)

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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 xreal
>>> xreal.Sind(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Sind('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Sind(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Sind('0.51')
Gpr('5.3518479027559984754E-1')

Inverse sine, input in degrees, \(\mathrm{asind}(x)\)#

ctx.asind(x)#

where ctx is math53, mathc53, ctxcpp, ctxflint.

Returns the inverse sine of \(x\), \(\mathrm{asin}(x)\). See also Wikipedia [1366], MathWorld [944], NIST [512], Ehrhardt [309] (4.2.13), Mpmath [604].

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 xreal
>>> xreal.Cosd(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Cosd('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Cosd(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Cosd('0.51')
Gpr('5.3518479027559984754E-1')

Inverse cosine, input in degrees, \(\mathrm{acosd}(x)\)#

ctx.acosd(x)#

where ctx is math53, mathc53, ctxcpp, ctxflint.

Returns the inverse cosine of \(x\), \(\mathrm{acos}(x)\). See also Wikipedia [1366], MathWorld [941], NIST [512], Ehrhardt [309] (4.2.3), Flint [809], Flint [799], Mpmath [595].

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 xreal
>>> xreal.Tand(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Tand('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Tand(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Tand('0.51')
Gpr('5.3518479027559984754E-1')

Inverse tangent, input in degrees, \(\mathrm{atand}(x)\)#

ctx.atand(x)#

where ctx is math53, mathc53, ctxcpp or ctxflint.

Returns the inverse tangent of \(x\), \(\mathrm{atan}(x)\). See also Wikipedia [1366], MathWorld [945], NIST [512], Ehrhardt [309] (4.2.15), Flint [809], Flint [799], Mpmath [605].

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 xreal
>>> xreal.Cotd(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Cotd('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Cotd(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Cotd('0.51')
Gpr('5.3518479027559984754E-1')

Inverse cotangent, input in degrees, \(\mathrm{acotd}(x)\)#

ctx.acotd(x)#

Returns the inverse cotangent of \(x\), \(\mathrm{acot}(x)\). See also Wikipedia [1366], MathWorld [942], NIST [512], Ehrhardt [309] (4.2.5).

Continuous inverse cotangent, \(\mathrm{acotc}(x)\)#

ctx.acotc(x)#

where ctx is math53, mathc53, ctxcpp or ctxflint.

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 xreal
>>> xreal.Acotc(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Acotc('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Acotc(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Acotc('0.51')
Gpr('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 xreal
>>> xreal.Covers(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Covers('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Covers(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Covers('0.51')
Gpr('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 xreal
>>> xreal.Versint(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Versint('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Versint(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Versint('0.51')
Gpr('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 xreal
>>> xreal.Versine(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Versine('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Versine(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Versine('0.51')
Gpr('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 xreal
>>> xreal.Haversine(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Haversine('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Haversine(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Haversine('0.51')
Gpr('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 xreal
>>> xreal.CosInt(3, 4)
xreal('5.2359877559829887307E-1')
>>> xreal.CosInt(3, 12)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.CosInt(3, 4)
Gpr('5.2359877559829887307E-1')
>>> Gpr.CosInt(3, 12)
Gpr('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 xreal
>>> xreal.SinInt(3, 4)
xreal('5.2359877559829887307E-1')
>>> xreal.SinInt(3, 12)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.SinInt(3, 4)
Gpr('5.2359877559829887307E-1')
>>> Gpr.SinInt(3, 12)
Gpr('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 xreal
>>> xreal.Kepler(3, 0.44)
xreal('5.2359877559829887307E-1')
>>> xreal.Kepler(3, 0.14404)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Kepler(3, 0.44)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Kepler(3, 0.14404)
Gpr('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 xreal
>>> xreal.Fibfun(3, 4)
xreal('5.2359877559829887307E-1')
>>> xreal.Fibfun(3, 12)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Fibfun(3, 4)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Fibfun(3, 12)
Gpr('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 xreal
>>> xreal.Sinhc(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Sinhc('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Sinhc(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Sinhc('0.51')
Gpr('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 xreal
>>> xreal.Sinhmx(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Sinhmx('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Sinhmx(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Sinhmx('0.51')
Gpr('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 xreal
>>> xreal.Coshm1(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Coshm1('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Coshm1(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Coshm1('0.51')
Gpr('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 xreal
>>> xreal.LangevinL(0.2)
xreal('5.2359877559829887307E-1')
>>> xreal.LangevinL(0.21)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.LangevinL(0.2)
Gpr('5.2359877559829887307E-1')
>>> Gpr.LangevinL(0.21)
Gpr('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 xreal
>>> xreal.Logcosh(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Logcosh('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Logcosh(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Logcosh('0.51')
Gpr('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 xreal
>>> xreal.Logsinh(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Logsinh('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Logsinh(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Logsinh('0.51')
Gpr('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 xreal
>>> xreal.Acos1m(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Acos1m('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Acos1m(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Acos1m('0.51')
Gpr('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 xreal
>>> xreal.Gudermann(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Gudermann('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Gudermann(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Gudermann('0.51')
Gpr('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 xreal
>>> xreal.ArcHaversine(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.ArcHaversine('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.ArcHaversine(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.ArcHaversine('0.51')
Gpr('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 xreal
>>> xreal.Acosh1p(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.Acosh1p('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.Acosh1p(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.Acosh1p('0.51')
Gpr('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 xreal
>>> xreal.ArcGd(0.5)
xreal('5.2359877559829887307E-1')
>>> xreal.ArcGd('0.51')
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.ArcGd(0.5)
Gpr('5.2359877559829887307E-1')
>>> Gpr.ArcGd('0.51')
Gpr('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 xreal
>>> xreal.LangevinLInv(0.2)
xreal('5.2359877559829887307E-1')
>>> xreal.LangevinLInv(0.21)
xreal('5.3518479027559984754E-1')

An example in Visual Basic

>>> from xlcalcnet import Gpr
>>> Gpr.LangevinLInv(0.2)
Gpr('5.2359877559829887307E-1')
>>> Gpr.LangevinLInv(0.21)
Gpr('5.3518479027559984754E-1')
Contents