Angle

In any case, the resulting angle lies in a plane (spanned by the two rays or perpendicular to the line of plane-plane intersection).

[2][3][a] Angle of rotation is a measure conventionally defined as the ratio of a circular arc length to its radius, and may be a negative number.

In the case of an ordinary angle, the arc is centered at the vertex and delimited by the sides.

[8] In mathematical expressions, it is common to use Greek letters (α, β, γ, θ, φ, . . . )

However, in many geometrical situations, it is evident from the context that the positive angle less than or equal to 180 degrees is meant, and in these cases, no ambiguity arises.

According to a historical note,[15] when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal.

The adjective complementary is from the Latin complementum, associated with the verb complere, "to fill up".

The prefix "co-" in the names of some trigonometric ratios refers to the word "complementary".

The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other.

In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent.

In other contexts, such as identifying a point on a spiral curve or describing an object's cumulative rotation in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.

The ratio of the length s of the arc by the radius r of the circle is the number of radians in the angle:[25]

Conventionally, in mathematics and the SI, the radian is treated as being equal to the dimensionless unit 1, thus being normally omitted.

The angle expressed by another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form ⁠k/2π⁠, where k is the measure of a complete turn expressed in the chosen unit (for example, k = 360° for degrees or 400 grad for gradians):

[27] Most units of angular measurement are defined such that one turn (i.e., the angle subtended by the circumference of a circle at its centre) is equal to n units, for some whole number n. Two exceptions are the radian (and its decimal submultiples) and the diameter part.

[34] A similar calculation using the area of a circular sector θ = 2A/r2 gives 1 radian as 1 m2/m2 = 1.

[37][38] Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations".

[40] Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics".

[42] In 1993 the American Association of Physics Teachers Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of angle measure (rad), angular speed (rad/s), angular acceleration (rad/s2), and torsional stiffness (N⋅m/rad), and not in the quantities of torque (N⋅m) and angular momentum (kg⋅m2/s).

The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for the area of a circle, πr2.

According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations".

[46][50] Current SI can be considered relative to this framework as a natural unit system where the equation η = 1 is assumed to hold, or similarly, 1 rad = 1.

[51] It is frequently helpful to impose a convention that allows positive and negative angular values to represent orientations and/or rotations in opposite directions or "sense" relative to some reference.

In a two-dimensional Cartesian coordinate system, an angle is typically defined by its two sides, with its vertex at the origin.

In a complex inner product space, the expression for the cosine above may give non-real values, so it is replaced with

This comparison of the two series corresponding to functions of angles was described by Leonhard Euler in Introduction to the Analysis of the Infinite (1748).

In geography, the location of any point on the Earth can be identified using a geographic coordinate system.

This system specifies the latitude and longitude of any location in terms of angles subtended at the center of the Earth, using the equator and (usually) the Greenwich meridian as references.

For example, the full moon has an angular diameter of approximately 0.5° when viewed from Earth.

In astronomy, right ascension and declination are usually measured in angular units, expressed in terms of time, based on a 24-hour day.