It is convenient to express corner size in circle shares in science and technology. It in most cases much more simplifies calculations. The corner expressed in circle shares is called a corner in radians. The full circle occupies two p a radian. The angle at top of the sector of a sphere is called space. The space angle is expressed in steradians. Diameter of the basis of a space angle in one steradian is equal to diameter of the sphere from which its sector is cut out.
Splitting a circle into 360 degrees was thought up by ancient Babylonians. Number 60 as a basis of a numeral system is convenient to what includes as decimal, so a duodecimal (dyuzhinny) and ternary basis. The cuneiform alphabet of Babylon contained several hundreds of syllabic signs and to allocate from them 60 under 60-richny figures possibly was.
Emergence of radians
With development of mathematics and science in general, it turned out that in many cases it is more convenient to express the size of a corner in shares of the circle which is "taken away" by a corner – radians. And them, in turn, "to tie" to number p = 3.1415926 …, expressing circle length relation to its diameter.
Number p – irrational, that is a recurring acyclic decimal decimal. It is impossible to express it as the relation of integers, already billions and trillions of signs after a comma without any signs of repetition of the sequence are counted today. In what then convenience? In expression of trigonometrical functions (a sine, for example) small corners. If to take a small corner in radians, then its value with big degree of accuracy will be equal to its sine. At scientific and, especially, technical calculations, became possible to replace the difficult trigonometrical equations in work as simple actions of arithmetics.
Flat corners in radians
In science and technology also instead of diameter of a circle it is more convenient to bowl of everything to use its radius therefore scientists agreed to consider that the full circle by 360 degrees is a corner in two pi a radian (6.2831852 … a radian). Thus, one radian contains about 57.3 angular degrees, or 57 degrees of 18 minutes of an arch of a circle. For simple calculations it is useful to remember that to 5 degrees there correspond 1/36 part p, and to 10 degrees – 1/18 p. Then the values of the most common corners expressed in radians through p are easily calculated in mind: or dozens of a corner in degrees we substitute value of the five in numerator 1/36 or 1/18 respectively, we divide, and we multiply the received fraction on p. For example, we need to know how many the radian will be in 15 angular degrees. In number 15 three five, so will turn out fraction 3/36 = 1/12. That is, the corner in 15 degrees will be equal 1/12 radians. The received values for the most often applied corners can be tabulated. But more evident and more convenient happens to use the circular angular chart like shown on the left part of the drawing.
Spherical corners
Corners are not only flat. Spherical (or spherical) the sector of the sphere of radius of R is unambiguously described by a corner at its top fi. Such angles are called space and expressed in steradians. A space angle in 1 steradian is the corner at top of the round spherical sector with a diameter of the basis (bottom) equal to diameter of a circle of R, as shown in the drawing on the right. However it is necessary to remember that there are no "stegradus" in a scientific and technical lexicon. If it is necessary to express a space angle in degrees, then and write: "the space angle in so many degrees", "an object was observed at a space angle in so many degrees". Sometimes, but it is rare, instead of expression "space angle" is written "spherical" or "a spherical corner". Anyway, if space, spherical, spherical angles and, except them – flat are mentioned in the text or the speech, they in order to avoid confusion need to be separated accurately from each other. Therefore in such cases it is accepted just not to use "corner", and to concretize: if it is about flat coal, it is called an arch corner. If it is necessary to give technical values of corners, they need also to be concretized. For example: "Angular distance on the heavenly sphere between stars And yes the B makes 13 degrees of 47 minutes of an arch"; "The object observed under a course corner in 123 degrees was visible at a space angle approximately in 2 degrees".