ArcCosDegrees
gives the arc cosine in degrees of the complex number 
.
Details
- ArcCosDegrees, 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
between 
and 
, the results are always in the range 
to 
. - ArcCosDegrees[z] returns the angle
in degrees for which the ratio of the adjacent side to the hypotenuse of a right triangle is 
. - For certain special arguments, ArcCosDegrees automatically evaluates to exact values.
- ArcCosDegrees can be evaluated to arbitrary numerical precision.
- ArcCosDegrees automatically threads over lists.
- ArcCosDegrees[z] has branch cut discontinuities in the complex
plane running from 
to 
and 
to 
. - ArcCosDegrees 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)
ArcCosDegrees[0]Calculate the angle BAC of this right triangle:
β = ArcCosDegrees[3 / 8]The numerical value of this angle:
%//NSolve an inverse trigonometric equation:
Solve[ArcCosDegrees[x] == 60, x]Solve an inverse trigonometric inequality:
Reduce[ArcCosDegrees[x] > 60, x]Apply ArcCosDegrees to the following list:
ArcCosDegrees[{0, 1 / 2, 1 / Sqrt[2], Sqrt[3] / 2, 1}]Plot over a subset of the reals:
Plot[ArcCosDegrees[x], {x, -1, 1}]Series[ArcCosDegrees[x], {x, 0, 10}]Scope (38)
Numerical Evaluation (6)
ArcCosDegrees[0.4]ArcCosDegrees[1.5]N[ArcCosDegrees[1 / 3], 50]The precision of the output tracks the precision of the input:
ArcCosDegrees[0.33333333333333333333333333333333333333333]Evaluate for complex arguments:
ArcCosDegrees[2.5 + I]Evaluate ArcCosDegrees efficiently at high precision:
ArcCosDegrees[0.4`500]//TimingArcCosDegrees[0.4`100000];//TimingCompute worst-case guaranteed intervals using Interval and CenteredInterval objects:
ArcCosDegrees[Interval[{1 / 3, 1 / 2}]]ArcCosDegrees[CenteredInterval[1 / 2, 1 / 100]]ArcCosDegrees[CenteredInterval[2 + 3I, (1 + I) / 100]]Or compute average-case statistical intervals using Around:
ArcCosDegrees[Around[.9, 0.01]]Compute the elementwise values of an array:
ArcCosDegrees[{{1, 1 / Sqrt[2]}, {0, 1 / Sqrt[2]}}]Or compute the matrix ArcCosDegrees function using MatrixFunction:
MatrixFunction[ArcCosDegrees[#]&, {{1, 1 / Sqrt[2]}, {0, 1 / Sqrt[2]}}]Specific Values (5)
Values of ArcCosDegrees at fixed points:
Table[ArcCosDegrees[(n /2)], {n, -2, 2}]Simple exact values are generated automatically:
ArcCosDegrees[(1/Sqrt[2])]ArcCosDegrees[Infinity]ArcCosDegrees[ComplexInfinity]Zero of ArcCosDegrees:
ArcCosDegrees[1]Find the value of 
satisfying equation 
:
f[x_] := ArcCosDegrees[x] - 60;sol = Solve[f[x] == 0, x]xzero = x /. First[sol]Plot[f[x], {x, -1, 1}, Rule[...]]Visualization (4)
Plot the ArcCosDegrees function:
Plot[ArcCosDegrees[x], {x, -1, 1}]Plot over a subset of the complexes:
ComplexPlot3D[ArcCosDegrees[z], {z, -4 - 2I, 4 + 2I}, Rule[...]]Plot the real part of ArcCosDegrees:
ComplexContourPlot[Re[ArcCosDegrees[z]], {z, -3 - 3I, 3 + 3I}, ...]Plot the imaginary part of ArcCosDegrees:
ComplexContourPlot[Im[ArcCosDegrees[z]], {z, -3 - 3I, 3 + 3I}, ...]Polar plot with ArcCosDegrees:
Table[PolarPlot[ArcCosDegrees[k ϕ], {ϕ, -π, π}, ...], {k, 1, 4}]Function Properties (10)
ArcCosDegrees is defined for all real values from the interval 
:
FunctionDomain[ArcCosDegrees[x], x]Complex domain is the whole plane:
FunctionDomain[ArcCosDegrees[z], z, Complexes]ArcCosDegrees achieves all real values from the interval 
:
FunctionRange[ArcCosDegrees[x], x, y]FunctionRange[ArcCosDegrees[x], x, y, Complexes]ArcCosDegrees is not an analytic function:
FunctionAnalytic[ArcCosDegrees[x], x]FunctionMeromorphic[ArcCosDegrees[x], x]ArcCosDegrees is neither non-decreasing nor non-increasing:
FunctionMonotonicity[ArcCosDegrees[x], x]It is monotonic over its real domain:
FunctionMonotonicity[{ArcCosDegrees[x], -1 ≤ x ≤ 1}, x]ArcCosDegrees is injective:
FunctionInjective[ArcCosDegrees[x], x]Plot[{ArcCosDegrees[x], 90}, {x, -1, 1}]ArcCosDegrees is not surjective:
FunctionSurjective[ArcCosDegrees[x], x]Plot[{ArcCosDegrees[x], 250}, {x, -1, 1}]ArcCosDegrees is non-negative over its real domain:
FunctionSign[{ArcCosDegrees[x], -1 ≤ x ≤ 1}, x]ArcCosDegrees has both singularity and discontinuity in (-∞,-1] and [1,∞):
FunctionSingularities[ArcCosDegrees[x], x]FunctionDiscontinuities[ArcCosDegrees[x], x]ArcCosDegrees is neither convex nor concave:
FunctionConvexity[ArcCosDegrees[x], x]ArcCosDegrees is convex for x in [-1,0]:
FunctionConvexity[{ArcCosDegrees[x], -1 <= x <= 0}, x]Plot[ArcCosDegrees[x], {x, -1, 0}]TraditionalForm formatting:
ArcCosDegrees[α]//TraditionalFormDifferentiation (3)
derivative:Integration (2)
Indefinite integral of ArcCosDegrees:
Integrate[ArcCosDegrees[x], x]Definite integral of ArcCosDegrees over the entire real domain:
Integrate[ArcCosDegrees[x], {x, -1, 1}]Series Expansions (5)
Find the Taylor expansion using Series:
Series[ArcCosDegrees[x], {x, 0, 7}]Plot the first three approximations for ArcCosDegrees around 
:
terms = Normal@Table[Series[ArcCosDegrees[x], {x, 0, m}], {m, 1, 5, 2}];
Plot[{ArcCosDegrees[x], terms}, {x, -1, 1}, PlotRange -> {{-1, 1}, All}]Asymptotic expansion at Infinity:
Series[ArcCosDegrees[x], {x, ∞, 5}]Asymptotic expansion at a singular point:
Series[ArcCosDegrees[x], {x, -1, 5}, Assumptions -> x > 1]//SimplifyFind the series expansion at branch points and branch cuts:
Series[ArcCosDegrees[x], {x, 1, 1}]Series[ArcCosDegrees[x], {x, -2, 1}]ArcCosDegrees can be applied to power series:
ArcCosDegrees[x + (x^2/2) + (x^3/3) + O[x]^4]Function Identities and Simplifications (2)
Simplify expressions involving ArcCosDegrees:
FullSimplify[ArcCosDegrees[Sqrt[1 - x^2]], 0 ≤ x ≤ 1]Use TrigToExp to express through logarithms and square roots:
TrigToExp@ArcCosDegrees[z]Function Representations (1)
Represent using ArcSecDegrees:
ArcSecDegrees[1 / x]//FullSimplifyApplications (8)
Solve an inverse trigonometric equation:
Solve[ArcCosDegrees[z]^2 + 3 ArcCosDegrees[z] == 2, z]Solve an inverse trigonometric equation with a parameter:
Reduce[ArcCosDegrees[Cos[z]] == w, z]Get the zeros of ArcCosDegrees:
Solve[ArcCosDegrees[α x + β] == 0, x]Use Reduce to solve inequalities involving ArcCosDegrees:
Reduce[ArcCosDegrees[x] > 60, x]Numerically find a root of a transcendental equation:
FindRoot[ArcCosDegrees[z] - 1 / 3 ArcCosDegrees[z + 1 / 3] == 50, {z, 1 / 2, 2}]//ChopPlot the function to check if the solution is correct:
Plot[ArcCosDegrees[z] - 1 / 3 ArcCosDegrees[z + 1 / 3] - 50, {z, 0, 1}]Plot the real and imaginary part of ArcCosDegrees:
ReImPlot[ArcCosDegrees[x], {x, -3, 3}]Plot the Riemann surface of ArcCosDegrees:
ParametricPlot3D[Evaluate[{Re[CosDegrees[u + I v]], Im[CosDegrees[u + I v]], u}], {u, 0, 720}, {v, -450, 450}]Find the angle between two vectors:
ArcCosDegrees[Normalize[{1, 1}].Normalize[{-1, 1}]]Graphics[{Arrow[{{0, 0}, {1, 1}}], Arrow[{{0, 0}, {-1, 1}}]}, PlotRange -> All]Properties & Relations (5)
Compose with the inverse function:
{CosDegrees[ArcCosDegrees[z]], ArcCosDegrees[CosDegrees[z]]}Use PowerExpand to disregard multivaluedness of the ArcCosDegrees:
PowerExpand[%]Alternatively, evaluate under additional assumptions:
Refine[ArcCosDegrees[CosDegrees[z]], 0 < z < 90]This shows the branch cuts of the ArcCosDegrees function:
Plot3D[Im[ArcCosDegrees[x - I y]], {x, -2, 2}, {y, -2, 2}]ArcCosDegrees gives the angle in degrees, while ArcCos gives the same angle in radians:
ArcCosDegrees[1 / 2]ArcCos[1 / 2]FunctionExpand applied to ArcCosDegrees generates expressions in trigonometric functions in radians:
FunctionExpand[ArcCosDegrees[x]]FunctionExpand[ArcCosDegrees[x ^ 2]ArcCosDegrees[120 - x / 2]]ExpToTrig applied to the outputs of TrigToExp will generate trigonometric functions in radians:
ArcCosDegrees[z]//TrigToExpExpToTrig[%]Possible Issues (3)
ArcCosDegrees[CosDegrees[30.]]CosDegrees[ArcCosDegrees[30.]]On branch cuts, machine-precision inputs can give numerically wrong answers:
ArcCosDegrees[-2 + I((180 + 1) ^ 2 - 180 ^ 2 - 2180 - 1 - Exp[-180 ^ 4])]//N
ArcCosDegrees[-2 + I((180 + 1) ^ 2 - 180 ^ 2 - 2180 - 1 - Exp[-180 ^ 4])]//N[#, 20]&The precision of the output can be much less than the precision of the input:
ArcCosDegrees[0.99999999999999999999999999999999999991]Neat Examples (3)
Solve trigonometric equations involving ArcCosDegrees:
Reduce[ArcCosDegrees[z] + ArcCosDegrees[z - 1 / 2] == 90, z]//QuietNumerical value of this angle in degrees:
%//NPlot a specific ArcCosDegrees:
Plot3D[Im[ArcCosDegrees[(x + I y) ^ 3]], {x, -2, 2}, {y, -2, 2}, ...]Plot ArcCosDegrees at integer points:
ArrayPlot[Table[FractionalPart[Abs[ArcCosDegrees[x y]]], {x, -40, 40}, {y, -40, 40}]]Text
Wolfram Research (2024), ArcCosDegrees, Wolfram Language function, https://reference.wolfram.com/language/ref/ArcCosDegrees.html.
CMS
Wolfram Language. 2024. "ArcCosDegrees." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ArcCosDegrees.html.
APA
Wolfram Language. (2024). ArcCosDegrees. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ArcCosDegrees.html