In mathematics, twostraight line(orvector）The minimum positive angle formed by the intersection is called the angle between the two lines (or vectors),usuallyRecorded as ∠θ (Included angle)sectionThe range is {0 | 0 ≤ 0 ≤ π/2}sectionThe range is {0 | 0 ≤ 0 ≤ π}.
The Father of GeometryEuclidAn angle was defined as the relative slope of two non parallel lines in the plane.Procruz believes that angle may be a trait, a quantifiable quantity, or a relationship.Eudemos believed that an angle was a deviation from a straight line, and Antioch's Cabus believed that an angle was the space between two intersecting straight lines.Euclid believed that angle was a relationship, but his definitions of right angle, acute angle or obtuse angle were quantitative.[1]
Corners are usually represented by three letters: the letters of the points on the two sides are written on both sides, and the letters on the vertices are written in the middle.Corner is represented by ∠ AOB.However, if there is no confusion, it will also be directly represented by the letter of vertex, such as angle ∠ O.
In the mathematical formulaGreek alphabet(α, β, γ, θ, φ,...) indicates the size of the angle.To avoid confusion, symbolsπGenerally, it is not used to express angle.
Angle measurement
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Take the end point of the angle ascenter of a circledoarc。Due to theradiusAnd arc growthProportional, and the angle is of lengthproportion, so the size of the circle will not affect the angle measurement.
radian: Cut out on the circle with cornerarcThe length of a circle divided by the radius of a circle is generally recorded as rad.Radian is the measurement of angle specified in the International System of Units, but it is not the legal measurement unit in China. Angle is the legal measurement unit of angle in China.In addition, radians are widely used in mathematics and trigonometry.
angle: The result of dividing the length of the arc cut by the angle on the circle by the circumference of the circle and multiplying by 360 is generally marked with ° and read as "degree".A degree can continue to be divided into 60 "minutes" or 3600 "seconds".Angle has important applications in astronomy and global positioning system.
gradient: is the result of dividing the length of the arc cut by the corner on the circle by the circumference of the circle and multiplying by 400.
The following are some other measurement units, corresponding to different n values.[2]
Number of turns orrevolutions（n=1) : refers to a complete rotation, which can be abbreviated ascyc、revorrot, but inRPM(RPM) is represented by only one letter r.
right angle（n=4) : Yes 1/4 turn YesGeometric primitivesAngle unit used in, right angle=90 °=π/2rad=1/4 turn=100grad.In German, it was used to indicate a right angle.
Hour angle（n=24):) Commonly used in astronomy, it is 1/24 turn.This system is used for the cycle of one cycle a day (such as the relative position of stars). Its sub units under the hexadecimal system are called "time minute angle" and "time second angle". These two units are different from the angular minutes and angular seconds. The former is fifteen times the size of the latter.1 hour angle=15 °=π/12rad=1/6quad=1/24turn≈ 16.667grad.。
Meter level（n=6000 – 6400): This unit refers to an angle that is approximately equal to milliradian. There are many different definitions. Its value ranges from 0.05625 degrees to 0.06 degrees (3.375 to 3.6 angular minutes), while milliradian is approximately 0.05729578 degrees (3.43775 angular minutes).In the countries of the North Atlantic Treaty Organization, the meter level is defined as 1/6400 of the circle.Its value is approximately equal to an angle whose arc length is one meter and radius is one kilometer (2 π/6400=0.0009817... ≈ 1/1000).
Angular minute（n=21600): It is defined as 1/60 of one degree, which is 1/21600 circle, and can be expressed by '. For example, 3 ° 30' is equal to 3+30/60 degrees, that is, 3.5 degrees. Sometimes there will be decimals, for example, 3 ° 5.72 '=3+5.72/60 degrees.The sea was once defined as the arc length of a quarter on the great circle of the earth.
Angular second（n=1296000): defined as 1/60 of a quarter, it will be represented by "", for example, 3 ° 7 ′ 30 "is equal to 3+7/60+30/3600 degrees, or 3.125 degrees.
Type of angle
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Zero angle equals 0 °, or a line
The acute angle is greater than 0 ° and less than 90 °, or the radian is greater than 0 and less than { displaystyle pi/2}.
A right angle equals 90 °, or an angle whose radian is { displaystyle pi/2}.
The obtuse angle is greater than 90 ° and less than 180 °, or the radian is greater than { displaystyle pi/2} and less than { displaystyle pi}.
The flat angle is equal to 180 °, or the radian is { displaystyle pi}.
The superior or negative angle is greater than 180 ° and less than 360 °, or the radian is greater than { displaystyle pi} and less than { displaystyle 2 pi}.
The perimeter angle is equal to 360 °, or the radian is { displaystyle 2 pi}.
Combination of angles
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There are three combinations of special angles, whose degrees and are all special values:
Cosine angle: When the sum of the degrees of two angles is equal to 90 °, that is, oneright angleThese two angles are the residual angles.If two adjacent angles are complementary to each other, two non shared edges will form a right angle.stayEuclidean geometryThe two corners that are not right angles are complementary angles to each other.
If angleAandBThey are complementary angles to each other, and the following mathematical formula will hold:
Complementary angle: When the sum of the degrees of two angles is equal to 180 °, that is, oneFlat angleThese two angles are complementary angles.If two adjacent angles are complementary to each other, two non shared 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.Cyclic Quadrilateral The diagonal of is also complementary.
If the point P is a point outside the circle O, and the tangent of the circle is made through the point P, and the tangent points are respectively at point T and point Q, then ∠ TPQ and ∠ TOQ are complementary angles.
The sinuses of two complementary angles are equal, and their cosines and tangents (if defined) are equal, but the signs are different.
In Euclidean geometry, the sum of the two angles of a triangle is the complement of the third triangle.
In Riemannian geometry
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stayRiemannian Geometry Medium, usingMetric tensorTo define twotangentIncluded angle between, whereUandVIs the tangent vector,gijIs a metric tensorGComponent of.
In astronomy
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withGeographyFrom the point of view of, any position on the earth can be represented by a geographical coordinate system, which marks the longitude and latitude of the position, both of which are represented by the angle of the line from this point to the Earth's spherical center. Longitude is based on the Greenwich meridian as the reference datum, while latitude is based on the equator as the reference datum.
In astronomy, a point in the celestial sphere can be represented by any kind of celestial coordinate system, but its datum varies with different coordinate systems.AstrometricAngular distanceWhen, you will imagine that there are two stars respectively connected to the earth in a straight line, and then measure the angle between the two straight lines, which is the angular distance.
Astronomers can also measure the apparent size of an object by its angular diameter.For example, the angular diameter of the full moon is about 0.5 °.The small angle formula can convert the above angle measurement into the ratio of distance and size.
