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acos

Compute the principal value of the arc cosine of x, return in radians.

  • For x in range [-1, 1], return value in range [0, pi]
  • Otherwise, throw a DomainError
Base.acos — Method
acos(x::T) where {T <: Number} -> float(T)

Compute the inverse cosine of x, where the output is in radians

Return a T(NaN) if isnan(x).

source

Methods

julia> methods(acos, (Any,), [Base, Base.Math, Base.MathConstants, Base.MPFR])# 7 methods for generic function "acos" from Base:
 [1] acos(a::ComplexF16)
     @ math.jl:1527
 [2] acos(::Missing)
     @ math.jl:1548
 [3] acos(x::BigFloat)
     @ mpfr.jl:946
 [4] acos(a::Float16)
     @ math.jl:1526
 [5] acos(z::Complex)
     @ complex.jl:952
 [6] acos(x::T) where T<:Union{Float32, Float64}
     @ special/trig.jl:673
 [7] acos(x::Real)
     @ math.jl:1543

Examples

julia> using UnicodePlots
julia> lineplot(-1, 1, acos; ylim=(0, pi))                ┌────────────────────────────────────────┐
        3.14159 │⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ acos(x)
                │⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
                │⠀⠈⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
                │⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
                │⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
                │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
                │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
   f(x)         │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
                │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
                │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀│ 
                │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀│ 
                │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀│ 
                │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⡀⠀│ 
                │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆│ 
              0 │⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸│ 
                └────────────────────────────────────────┘
                ⠀-1⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀1⠀
                ⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀x⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀

Real Numbers

julia> acos(1.0)
0.0

julia> acos(0.0)
1.5707963267948966

julia> acos(-1.0)
3.141592653589793

julia> acos(2.0)
ERROR: DomainError with 2.0:
acos(x) not defined for |x| > 1
Stacktrace:
[...]

Complex

julia> acos(2.0+0im)
0.0 - 1.3169578969248166im

julia> acos(2.0-0im)
0.0 + 1.3169578969248166im

julia> acos(-2.0+0im)
3.141592653589793 - 1.3169578969248166im

julia> acos(-2.0-0im)
3.141592653589793 + 1.3169578969248166im

Tips

  • For real number input, the input range is limited to [0, 1]. For larger ranges, use complex inputs.
  • The output is expressed in radians, and needs to be converted if a degrees is required: degrees = radians * 180 / pi

See Also

cos, acosd, acosh

Extended Inputs

Matrix

With Array like input:

Base.acos — Method
acos(A::AbstractMatrix)

Compute the inverse matrix cosine of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the inverse cosine. Otherwise, the inverse cosine is determined by using log and sqrt. For the theory and logarithmic formulas used to compute this function, see [AH16_1].

Examples

julia> acos(cos([0.5 0.1; -0.2 0.3]))
2×2 Matrix{ComplexF64}:
  0.5-8.32667e-17im  0.1+0.0im
 -0.2+2.63678e-16im  0.3-3.46945e-16im
julia> methods(acos, (Any,), [LinearAlgebra])# 5 methods for generic function "acos" from Base:
 [1] acos(D::Diagonal)
     @ /opt/hostedtoolcache/julia/1.12.2/x64/share/julia/stdlib/v1.12/LinearAlgebra/src/diagonal.jl:879
 [2] acos(J::UniformScaling)
     @ /opt/hostedtoolcache/julia/1.12.2/x64/share/julia/stdlib/v1.12/LinearAlgebra/src/uniformscaling.jl:176
 [3] acos(A::Hermitian{var"#s4811", S} where {var"#s4811"<:Complex, S<:(AbstractMatrix{<:var"#s4811"})})
     @ /opt/hostedtoolcache/julia/1.12.2/x64/share/julia/stdlib/v1.12/LinearAlgebra/src/symmetric.jl:929
 [4] acos(A::Union{Hermitian{T, S} where S, SymTridiagonal{T, V} where V<:AbstractVector{T}, Symmetric{T, S} where S} where T<:Real)
     @ /opt/hostedtoolcache/julia/1.12.2/x64/share/julia/stdlib/v1.12/LinearAlgebra/src/symmetric.jl:921
 [5] acos(A::AbstractMatrix)
     @ /opt/hostedtoolcache/julia/1.12.2/x64/share/julia/stdlib/v1.12/LinearAlgebra/src/dense.jl:1347
julia> acos([-1 0; 0 -1])
2×2 Matrix{Float64}:
 3.14159  0.0
 0.0      3.14159

Tech Notes

  • acos(::Real): by pure julia
  • acos(::BigFloat): by MPFR

Version History

Introduced in Julia v1.0 (2018)