Angle is widely used in geometry and trigonometry.
Euclid, the father of geometry, once defined an angle as the relative inclination of two non-parallel straight lines in a plane. Proclos thinks that angle may be a trait, a quantifiable quantity, or a relationship. Oldham thinks that an angle is a deviation from a straight line, and Cabus of Antioch thinks that an angle is a space between two intersecting straight lines. Euclid thinks that an angle is a relationship, but his definitions of right angle, acute angle or obtuse angle are quantitative.
Basic introduction Chinese name: included angle mbth: included angle belongs to: mathematical term pinyin: jiá ji m: o? Applied subjects: mathematical terms: radian representation, angle measurement, angle type and angle combination. In Riemannian geometry and astronomy, the method angle is usually represented by three letters: the letters of two points are written on both sides and the letters of vertices are written in the middle. The angle in the figure is represented by ∠AOB. But if there is no confusion, it will be directly represented by the letter of the vertex, such as angle ∠ O. In mathematical formulas, Greek letters (α, β, γ, θ, φ, ...) are generally used to represent the size of the angle. In order to avoid confusion, the symbol π is generally not used to represent angles. Angle measurement draws an arc centered on the endpoint of the angle. Since the radius of the arc is proportional to the length of the arc and the angle is proportional to the length, the size of the circle will not affect the measurement of the angle. Radian: The length of the arc cut by the angle on the circle is divided by the radius of the circle, which is generally recorded as rad. The radian is the unit of measurement of the angle stipulated in the international system of units, but it is not the legal unit of measurement in China, and the angle is the legal unit of measurement in China. In addition, radian is also widely used in mathematics and trigonometry. Angle: the length of an arc divided by the angle on the circle divided by the circumference of the circle and then multiplied by 360, which is generally marked as 0 and read as "degree". It used to be divided into 60 minutes or 3600 seconds. Angle has important applications in astronomy and global positioning system. Gradient: it is the result of dividing the length of the arc cut by the angle on the circle by the circumference of the circle and multiplying it by 400. The following are some other units of measurement, corresponding to different values of n, revolutions or revolutions (n = 1): refers to a complete revolution, which will be abbreviated as cyc, rev or rot according to different applications, but in the unit of RPM, only one letter R is used. Right angle (n =4): it is 1/4 revolutions, which is the angular unit used in geometry. Right angle = 90 = π/2 radian = 1/4 turns = 100 Geller. In German, it used to mean right angle. The time angle (n =24):) is often used in astronomy, and it is 1/24 cycles. This system uses one day as a cycle (such as the relative position of stars), and its subunits under the sexagesimal are called "time minute angle" and "time second angle". The minutes and seconds of these two units and angles are different, and the former is fifteen times as big as the latter. 1 hour angle =15 = π/12 rad =1/6 quad. = 1/24 rpm ≈ 16.667 grad. Meter level (n =6000-6400): This unit refers to an angle, and its unit is approximately equal to milliradians. There are many different definitions, ranging from 0.05625 degrees to 0.06 degrees (3.375 to 3.6 minutes), and milliradian is about 0.05729578 degrees (3.43775 minutes). In the countries of the North Atlantic Treaty Organization, the position of the meter is defined as a circle 1/6400. Its value is approximately equal to the angle (2π/6400 = 0.00098 1 7 ... ≒1000) with an arc length of one meter and a radius of one kilometer, and the angle minute (n =2 1 600). The ocean was once defined as the arc length of a dime on the great circle of the earth. Angular seconds (n = 1, 296,000): defined as 1/60 for one minute, it will be expressed as ",for example, 3 7' 30" is equal to 3+7/60+30/3600 degrees, that is, 3. 125 degrees. Angle type zero angle
The angle is equal to 0, or an acute angle of a straight line.
Angle greater than 0 and less than 90, or radian greater than 0 and less than {\displaystyle \pi /2}. right angle
The angle is equal to 90, or the angle with radian of {\displaystyle \pi /2}. blunt angle
The angle is greater than 90 and less than 180, or the radian is greater than {\displaystyle \pi /2} and less than {\displaystyle \pi}. flat angle
The angle is equal to 180, or the angle with radian of {\displaystyle \pi}. Dominant angle or inverse angle
The angle is greater than 180 and less than 360, or the radian is greater than {\displaystyle \pi} and less than {\displaystyle 2\pi}. perigon
The angle is equal to 360, or the angle with radian of {\displaystyle 2\pi}. There are three special combinations of angles, and the sum of their degrees is a special value: complementary angle: when the sum of the degrees of two angles is equal to 90, that is, a right angle, these two angles are complementary angles. If two adjacent corners are complementary, then two unused edges will form a right angle. In Euclidean geometry, two non-right angles are complementary angles. If Angle A and Angle B are complementary angles, the following mathematical formula holds: (Tangent of one angle is equal to cotangent of other angles, and secant of one angle is equal to cotangent of other angles) Complementary angle: When the sum of degrees of two angles is equal to 180, that is, a right angle, these two angles are complementary angles. If two adjacent corners are complementary, then two unused edges will form a straight line. However, two non-adjacent angles can also be complementary angles. For example, in a parallelogram, any two adjacent angles are complementary angles. The diagonal of a quadrilateral inscribed with a circle is also a complementary angle. If point P is a point outside circle O, the passing point P is tangent to the circle, and the tangent points are at point T and point Q respectively, then ∠TPQ and ∠TOQ are complementary angles. The sine of two complementary angles is equal, and the other chords and tangents (if defined) are equal in size, but different in sign. In Euclidean geometry, the sum of two angles of a triangle is the complementary angle of the third triangle. In Riemannian geometry, metric tensor is used to define the included angle between two tangents, where U and V are tangent vectors and Geij is the component of metric tensor G. In astronomy, from the geographical point of view, any position on the earth can be represented by a geographical coordinate system. This system represents the longitude and latitude of the position, both of which are expressed by the angle connecting this point to the center of the earth. Longitude is based on Greenwich meridian and latitude is based on equator. In astronomy, a point on the celestial sphere can be represented by any celestial coordinate system, but its datum varies with different coordinate systems. When astronomy measures the angular distance between two stars, it will assume that the two stars are connected with the earth by two straight lines, and then measure the included angle between the two straight lines, which is the angular distance. Astronomers also use angular diameter to measure the apparent size of objects. For example, the angular diameter of a full moon is about 0.5. The small angle formula can convert the above angle measurement into the ratio of distance and size.