AcosEdit
Acos, commonly written as arccos or acos, is the inverse of the cosine function on a restricted domain. It takes a real number x in the interval [-1, 1] and returns an angle y in the interval [0, π], such that cos(y) = x. This function is essential in solving triangles, recovering angles from dot products, and in a wide range of applications across science and engineering. For a concise mathematical reference, acos is the inverse of the cosine function on the standard principal domain, and it appears in both pure and applied contexts as a fundamental building block of trigonometry. It is often denoted as arccos or acos, with the symbol acos frequently used in programming and numerical work arccos; see also acos for other naming conventions.
Formally, acos is defined by the relation cos(y) = x with y ∈ [0, π]. Therefore, the compositions cos(acos(x)) = x for x ∈ [-1, 1], and acos(cos(θ)) = θ for θ ∈ [0, π]. The function is continuous on [-1, 1] and differentiable on (-1, 1), with a strictly decreasing behavior on that interval. These properties reflect its role as the inverse of the cosine function when the latter is restricted to the interval 0, π on the unit circle. For a geometric intuition, think of the unit circle where a point with abscissa x = cos(y) corresponds to the angle y measured from the positive x-axis; then acos(x) returns that angle in the range [0, π] unit circle.
Acos is often discussed alongside the other inverse trigonometric functions, such as arcsin and arctan. A key identity that ties these functions together is arccos(x) + arcsin(x) = π/2 for x in [-1, 1], which follows from the fundamental relationships among sine and cosine on the unit circle. In many texts and software libraries, acos is implemented as the principal inverse and is related to the derivatives and antiderivatives that arise in calculus inverse function; see also the derivative formula below and the connections to arcsin and arctan.
Definition and notation
- Domain and range: acos maps the interval [-1, 1] to the interval [0, π]. In formulae: acos: [-1, 1] → [0, π], with cos(acos(x)) = x and acos(cos(y)) = y for y ∈ [0, π].
- Notation: acos is the common programming name, while arccos denotes the mathematical function. Some sources also write arccos as arc cos or arccosine; all refer to the same inverse function of cosine on the principal domain.
- Basic identities: acos(x) ∈ [0, π], and for x ∈ [-1, 1], cos(acosh(x)) is not a standard expression because acosh is a different function; instead, the relevant identity is arccos(x) + arcsin(x) = π/2, linking to the corresponding inverse trigonometric functions.
Geometry and the unit circle
The geometric meaning of acos is the angle in the upper half of the unit circle whose cosine is x. Since cos(θ) decreases monotonically from 1 to -1 as θ runs from 0 to π, there is a unique θ in [0, π] with cos(θ) = x. This interpretation connects acos to the unit circle and to the broader study of trigonometric functions as rotations and projections in the plane. The angle produced by acos is always nonnegative and does not exceed π, which makes it particularly suitable for problems involving angles between vectors and directions in two- and three-dimensional space unit circle.
Calculus and identities
- Derivative: for x ∈ (-1, 1), d/dx [acos(x)] = -1 / sqrt(1 - x^2). The derivative reflects the vertical tangent behavior as x approaches the endpoints ±1 and explains why numerical evaluation near those endpoints requires care derivative.
- Second derivative: d^2/dx^2 [acos(x)] = -x / (1 - x^2)^(3/2), valid for x ∈ (-1, 1). This emphasizes the curvature change near the endpoints.
- Antiderivative and series: acos has representations through integrals and, near x = 0, can be expanded in a series that involves powers of x; these tools are standard in analysis and numerical methods Taylor series.
- Inverse relationship: acos(x) and arcsin(x) are complementary on [-1, 1], as noted above. This reflects the fundamental link between the cosine and sine functions on the unit circle arcsin.
Computation and applications
- Computational use: in algorithms, acos is used to recover angles from dot products or from normalized vectors. Given two vectors u and v, the cosine of the angle between them is (u · v) / (||u|| ||v||); the angle itself is θ = acos((u · v) / (||u|| ||v||)) dot product.
- Geometry and graphics: in computer graphics and computer-aided design, acos helps determine angles between surfaces, normals, and directions, contributing to shading, lighting, and orientation computations. It also appears in navigation, physics simulations, and robotics for angle measurements derived from projections.
- Relationship to other inverses: in trigonometric problem solving, acos is often used together with arccosine identities and with the inverse function framework to translate between angle-valued and ratio-valued descriptions.
- Numerical considerations: near x = ±1, the derivative becomes large, so special care is taken in floating-point implementations to maintain accuracy and stability. Polynomial approximations and table-driven methods are common in high-performance libraries.