ArcTanDegrees
gives the arc tangent in degrees of the complex number 
.
Details
- ArcTanDegrees, along with other inverse trigonometric and trigonometric functions, is studied in high-school geometry courses and is also used in many scientific disciplines.
- All results are given in degrees.
- For real
, the results are always in the range 
to 
. - ArcTanDegrees[z] returns the angle
in degrees for which the ratio of the opposite side to the adjacent side of a right triangle is 
. - For certain special arguments, ArcTanDegrees automatically evaluates to exact values.
- ArcTanDegrees can be evaluated to arbitrary numerical precision.
- ArcTanDegrees automatically threads over lists.
- ArcTanDegrees[z] has branch cut discontinuities in the complex
plane running from 
to 
and 
to 
. - ArcTanDegrees can be used with Interval, CenteredInterval and Around objects.
- Mathematical function, suitable for both symbolic and numerical manipulation.
Examples
open all close allBasic Examples (7)
ArcTanDegrees[1]Calculate the angle BAC of this right triangle:
β = ArcTanDegrees[4 / 6]The numerical value of this angle:
%//NSolve an inverse trigonometric equation:
Solve[ArcTanDegrees[x] == 60, x]Solve an inverse trigonometric inequality:
Reduce[ArcTanDegrees[x] > 60, x]Apply ArcTanDegrees to the following list:
ArcTanDegrees[{0, 2 - Sqrt[3], (1/Sqrt[3]), 1, Sqrt[3], 2 + Sqrt[3]}]Plot over a subset of the reals:
Plot[ArcTanDegrees[x], {x, -5, 5}]Series[ArcTanDegrees[x], {x, 0, 15}]Scope (40)
Numerical Evaluation (6)
ArcTanDegrees[0.5]N[ArcTanDegrees[1 / 2], 50]The precision of the output tracks the precision of the input:
ArcTanDegrees[0.50000000000000000000000000000000000000000]Evaluate for complex arguments:
ArcTanDegrees[2.5 + I]Evaluate ArcTanDegrees efficiently at high precision:
ArcTanDegrees[0.4`500]//TimingArcTanDegrees[0.4`100000];//TimingCompute worst-case guaranteed intervals using Interval and CenteredInterval objects:
ArcTanDegrees[Interval[{-1, 3}]]ArcTanDegrees[CenteredInterval[1, 1 / 100]]ArcTanDegrees[CenteredInterval[2 + 3I, (1 + I) / 100]]Or compute average-case statistical intervals using Around:
ArcTanDegrees[Around[.9, 0.01]]Compute the elementwise values of an array:
ArcTanDegrees[{{1, Sqrt[3]}, {0, Sqrt[3]}}]Or compute the matrix ArcTanDegrees function using MatrixFunction:
MatrixFunction[ArcTanDegrees[#]&, {{1, Sqrt[3]}, {0, Sqrt[3]}}]Specific Values (5)
Values of ArcTanDegrees at fixed points:
Table[ArcTanDegrees[n], {n, -1, 1}]Simple exact values are generated automatically:
ArcTanDegrees[(1/Sqrt[3])]ArcTanDegrees[Infinity]ArcTanDegrees[ComplexInfinity]Zero of ArcTanDegrees:
ArcTanDegrees[0]Find the value of 
satisfying equation 
:
f[x_] := ArcTanDegrees[x] - 60;sol = Solve[f[x] == 0, x]xzero = x /. First[sol]Plot[f[x], {x, -2, 2}, Rule[...]]Visualization (4)
Plot the ArcTanDegrees function:
Plot[ArcTanDegrees[x], {x, -7, 7}]Plot over a subset of the complexes:
ComplexPlot3D[ArcTanDegrees[z ^ 2], {z, -1 / 2 - I, 1 / 2 + I}, Rule[...]]Plot the real part of ArcTanDegrees:
ComplexContourPlot[Re[ArcTanDegrees[z]], {z, -3 - 3I, 3 + 3I}, IconizedObject[«PlotOptions»]]Plot the imaginary part of ArcTanDegrees:
ComplexContourPlot[Im[ArcTanDegrees[z]], {z, -3 - 3I, 3 + 3I}, IconizedObject[«PlotOptions»]]Polar plot with ArcTanDegrees:
Table[PolarPlot[ArcTanDegrees[k ϕ], {ϕ, -π, π}, ...], {k, 1, 4}]Function Properties (12)
ArcTanDegrees is defined for all real values:
FunctionDomain[ArcTanDegrees[x], x]FunctionDomain[ArcTanDegrees[z], z, Complexes]ArcTanDegrees achieves all real values from the interval 
:
FunctionRange[ArcTanDegrees[x], x, y]FunctionRange[ArcTanDegrees[x], x, y, Complexes]ArcTanDegrees is an odd function:
ArcTanDegrees[-x]ArcTanDegrees has the mirror property ![tan^(-1)(TemplateBox[{x}, Conjugate])=TemplateBox[{{{tan, ^, {(, {-, 1}, )}}, (, x, )}}, Conjugate]](Files/ArcTanDegrees.en/16.png "tan^(-1)(TemplateBox[{x}, Conjugate])=TemplateBox[{{{tan, ^, {(, {-, 1}, )}}, (, x, )}}, Conjugate]"){kind=small}
:
FullSimplify[ArcTanDegrees[Conjugate[x]] == Conjugate[ArcTanDegrees[x]]]ArcTanDegrees is an analytic function of 
over the reals:
FunctionAnalytic[ArcTanDegrees[x], x]It is neither analytic nor meromorphic over the complex plane:
FunctionAnalytic[ArcTanDegrees[x], x, Complexes]FunctionMeromorphic[ArcTanDegrees[x], x]ArcTanDegrees is an increasing function:
FunctionMonotonicity[ArcTanDegrees[x], x, StrictInequalities -> True]ArcTanDegrees is injective:
FunctionInjective[ArcTanDegrees[x], x]Plot[{ArcTanDegrees[x], 20}, {x, -5, 5}]ArcTanDegrees is not surjective:
FunctionSurjective[ArcTanDegrees[x], x]Plot[{ArcTanDegrees[x], 150}, {x, -5, 5}]ArcTanDegrees is neither non-negative nor non-positive:
FunctionSign[ArcTanDegrees[x], x]ArcTanDegrees has no singularities or discontinuities:
FunctionSingularities[ArcTanDegrees[x], x]FunctionDiscontinuities[ArcTanDegrees[x], x]ArcTanDegrees is neither convex nor concave:
FunctionConvexity[ArcTanDegrees[x], x]ArcSind is convex for x in [-10,0]:
FunctionConvexity[{ArcTanDegrees[x], -10 <= x <= 0}, x]Plot[ArcTanDegrees[x], {x, -10, 10}]TraditionalForm formatting:
ArcTanDegrees[α]//TraditionalFormDifferentiation (3)
derivative:Integration (2)
Indefinite integral of ArcTanDegrees:
Integrate[ArcTanDegrees[x], x]Definite integral of ArcTanDegrees over an interval centered at the origin is 0:
Integrate[ArcTanDegrees[x], {x, -1, 1}]Series Expansions (5)
Taylor expansion for ArcTanDegrees:
Series[ArcTanDegrees[x], {x, 0, 7}]Plot the first three approximations for ArcTanDegrees around 
:
terms = Normal@Table[Series[ArcTanDegrees[x], {x, 0, m}], {m, 1, 5, 2}];
Plot[{ArcTanDegrees[x], terms}, {x, -1, 1}, PlotRange -> {{-1, 1}, All}]Asymptotic expansions at Infinity:
Series[ArcTanDegrees[x], {x, ∞, 5}]Asymptotic expansion at one of the singular points:
Series[ArcTanDegrees[x], {x, I, 3}]//FullSimplifyFind series expansions at branch points and branch cuts:
Series[ArcTanDegrees[x], {x, I, 1}]Series[ArcTanDegrees[x], {x, 2I, 1}]ArcTanDegrees can be applied to a power series:
ArcTanDegrees[SeriesData[x, 0, {1, 0, 1/3, 0, 2/15, 0, 17/315, 0, 62/2835}, 1, 10, 1]]Function Identities and Simplifications (2)
Use FullSimplify to simplify expressions with ArcTanDegrees:
FullSimplify[16 ArcTanDegrees[1 / 5] - 4 ArcTanDegrees[1 / 239]]Use TrigToExp to express ArcTanDegrees using Log:
TrigToExp[ArcTanDegrees[z]]Function Representations (1)
Represent using ArcCotDegrees:
ArcCotDegrees[1 / x]//FullSimplifyApplications (7)
Solve inverse trigonometric equations:
Solve[ArcTanDegrees[α x + β] == 4, x]Solve[ArcTanDegrees[z]^2 + 3 ArcTanDegrees[z] == 2, z]Solve an inverse trigonometric equation with a parameter:
Reduce[ArcTanDegrees[TanDegrees[z]] == w, z]Use Reduce to solve inequalities involving ArcTanDegrees:
Reduce[ArcTanDegrees[x] > 60, x]Numerically find a root of a transcendental equation:
FindRoot[ArcTanDegrees[z]^2 - 2 ArcTanDegrees[z + 1 / 3] == -3, {z, 0, 1 / 2}]Plot the function to check if the solution is correct:
Plot[ArcTanDegrees[z]^2 - 2 ArcTanDegrees[z + 1 / 3] + 3, {z, 0, 1 / 2}]Plot the real and imaginary parts of ArcTanDegrees:
ReImPlot[ArcTanDegrees[x], {x, -2, 2}]Different combinations of ArcTanDegrees with trigonometric functions:
{TanDegrees[ArcTanDegrees[z]], SecDegrees[ArcTanDegrees[1 / z]], CotDegrees[ArcTanDegrees[z ^ 2]], SinDegrees[ArcTanDegrees[z]]}Addition theorem for tangent function:
TanDegrees[ArcTanDegrees[x] + ArcTanDegrees[y]]//TrigExpand//SimplifyProperties & Relations (5)
Compositions with the inverse trigonometric functions:
{TanDegrees[ArcTanDegrees[z]], ArcTanDegrees[TanDegrees[z]]}Use PowerExpand to disregard multivaluedness of the ArcTanDegrees:
PowerExpand[%]Alternatively, evaluate under additional assumptions:
Refine[ArcTanDegrees[TanDegrees[z]], 0 < z < 90]Branch cuts of ArcTanDegrees run along the imaginary axis:
Plot3D[Re[ArcTanDegrees[x + I y]], {y, -2, 2}, {x, -2.5, 2.5}]ArcTanDegrees gives the angle in degrees, while ArcTan gives the same angle in radians:
ArcTanDegrees[1]ArcTan[1]FunctionExpand applied to ArcTanDegrees generates expressions in trigonometric functions in radians:
FunctionExpand[ArcTanDegrees[x]]FunctionExpand[ArcTanDegrees[x ^ 2]ArcTanDegrees[120 - x / 2]]ExpToTrig applied to the outputs of TrigToExp will generate trigonometric functions in radians:
ArcTanDegrees[z]//TrigToExpExpToTrig[%]Possible Issues (1)
Neat Examples (2)
Solve trigonometric equations involving ArcTanDegrees:
Reduce[ArcTanDegrees[z] + ArcTanDegrees[z - 1] == 90, z]//QuietNumerical value of this angle in degrees:
%//NPlot ArcTanDegrees at integer points:
ArrayPlot[Table[FractionalPart[Abs[ArcTanDegrees[x y]]], {x, -40, 40}, {y, -40, 40}]]Text
Wolfram Research (2024), ArcTanDegrees, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcTanDegrees.html.
CMS
Wolfram Language. 2024. "ArcTanDegrees." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ArcTanDegrees.html.
APA
Wolfram Language. (2024). ArcTanDegrees. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcTanDegrees.html