Degree (angle)

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Degree
Unit system Non-SI accepted unit
Unit of Angle
Symbol° [1] [2] ordeg [3]
Conversions
1 ° [1] [2] in ...... is equal to ...
    turns    1/360 turn
    radians    π/180 rad ≈ 0.01745.. rad
    milliradians    50·π/9 mrad ≈ 17.45.. mrad
    gons    10/9g
One degree (shown in red) and
eighty nine degrees (shown in blue) Degree diagram.svg
One degree (shown in red) and
eighty nine degrees (shown in blue)

A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), [4] is a measurement of a plane angle in which one full rotation is 360 degrees. [5]

Contents

It is not an SI unit—the SI unit of angular measure is the radian—but it is mentioned in the SI brochure as an accepted unit. [6] Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians.

History

A circle with an equilateral chord (red). One sixtieth of this arc is a degree. Six such chords complete the circle. Equilateral chord with length equal to radius.svg
A circle with an equilateral chord (red). One sixtieth of this arc is a degree. Six such chords complete the circle.

The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. [5] Ancient astronomers noticed that the sun, which follows through the ecliptic path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient calendars, such as the Persian calendar and the Babylonian calendar, used 360 days for a year. The use of a calendar with 360 days may be related to the use of sexagesimal numbers.

Another theory is that the Babylonians subdivided the circle using the angle of an equilateral triangle as the basic unit, and further subdivided the latter into 60 parts following their sexagesimal numeric system. [8] [9] The earliest trigonometry, used by the Babylonian astronomers and their Greek successors, was based on chords of a circle. A chord of length equal to the radius made a natural base quantity. One sixtieth of this, using their standard sexagesimal divisions, was a degree.

Aristarchus of Samos and Hipparchus seem to have been among the first Greek scientists to exploit Babylonian astronomical knowledge and techniques systematically. [10] [11] Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes. [12] Eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts.[ citation needed ]

The division of the circle into 360 parts also occurred in ancient India, as evidenced in the Rigveda: [13]

Twelve spokes, one wheel, navels three.

Who can comprehend this?
On it are placed together
three hundred and sixty like pegs.
They shake not in the least.

Dirghatamas, Rigveda 1.164.48

Another motivation for choosing the number 360 may have been that it is readily divisible: 360 has 24 divisors, [note 1] making it one of only 7 numbers such that no number less than twice as much has more divisors (sequence A072938 in the OEIS ). [14] [15] Furthermore, it is divisible by every number from 1 to 10 except 7. [note 2] This property has many useful applications, such as dividing the world into 24 time zones, each of which is nominally 15° of longitude, to correlate with the established 24-hour day convention.

Finally, it may be the case that more than one of these factors has come into play. According to that theory, the number is approximately 365 because of the apparent movement of the sun against the celestial sphere, and that it was rounded to 360 for some of the mathematical reasons cited above.

Subdivisions

For many practical purposes, a degree is a small enough angle that whole degrees provide sufficient precision. When this is not the case, as in astronomy or for geographic coordinates (latitude and longitude), degree measurements may be written using decimal degrees, with the degree symbol behind the decimals; for example, 40.1875°.

Alternatively, the traditional sexagesimal unit subdivisions can be used. One degree is divided into 60 minutes (of arc), and one minute into 60 seconds (of arc). Use of degrees-minutes-seconds is also called DMS notation. These subdivisions, also called the arcminute and arcsecond , are respectively represented by a single prime (′) and a double prime (″). [4] For example, 40.1875° = 40° 11′ 15″, or, using quotation mark characters, 40° 11' 15". Additional precision can be provided using decimals for the arcseconds component.

Maritime charts are marked in degrees and decimal minutes to facilitate measurement; 1 minute of latitude is 1 nautical mile. The example above would be given as 40° 11.25′ (commonly written as 11′25 or 11′.25). [16]

The older system of thirds, fourths, etc., which continues the sexagesimal unit subdivision, was used by al-Kashi [ citation needed ] and other ancient astronomers, but is rarely used today. These subdivisions were denoted by writing the Roman numeral for the number of sixtieths in superscript: 1I for a "prime" (minute of arc), 1II for a second, 1III for a third, 1IV for a fourth, etc. [17] Hence, the modern symbols for the minute and second of arc, and the word "second" also refer to this system. [18]

Alternative units

A chart to convert between degrees and radians Degree-Radian Conversion.svg
A chart to convert between degrees and radians

In most mathematical work beyond practical geometry, angles are typically measured in radians rather than degrees. This is for a variety of reasons; for example, the trigonometric functions have simpler and more "natural" properties when their arguments are expressed in radians. These considerations outweigh the convenient divisibility of the number 360. One complete turn (360°) is equal to 2 π radians, so 180° is equal to π radians, or equivalently, the degree is a mathematical constant: 1° = π180.

The turn (or revolution, full circle, full rotation, cycle) is used in technology and science. One turn is equal to 360°.

With the invention of the metric system, based on powers of ten, there was an attempt to replace degrees by decimal "degrees" [note 3] called grad or gon, where the number in a right angle is equal to 100 gon with 400 gon in a full circle (1° = 109 gon). Although that idea was abandoned by Napoleon, grades continued to be used in several fields and many scientific calculators support them. Decigrades (14,000) were used with French artillery sights in World War I.

An angular mil, which is most used in military applications, has at least three specific variants, ranging from 16,400 to 16,000. It is approximately equal to one milliradian (c.16,283). A mil measuring 16,000 of a revolution originated in the imperial Russian army, where an equilateral chord was divided into tenths to give a circle of 600 units. This may be seen on a lining plane (an early device for aiming indirect fire artillery) dating from about 1900 in the St. Petersburg Museum of Artillery.

Conversion of common angles
Turns Radians Degrees Gradians, or gons
000g
1/24π/1215°16+2/3g
1/12π/630°33+1/3g
1/10π/536°40g
1/8π/445°50g
1/2π1c. 57.3°c. 63.7g
1/6π/360°66+2/3g
1/52π/572°80g
1/4π/290°100g
1/32π/3120°133+1/3g
2/54π/5144°160g
1/2π180°200g
3/43π/2270°300g
12π360°400g

See also

Notes

  1. The divisors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, and 360.
  2. Contrast this with the relatively unwieldy 2520, which is the least common multiple for every number from 1 to 10.
  3. These new and decimal "degrees" must not be confused with decimal degrees.

Related Research Articles

Angle Figure formed by two rays meeting at a single point

In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles are also formed by the intersection of two planes. These are called dihedral angles. Two intersecting curves define also an angle, which is the angle of the tangents at the intersection point. For example, the spherical angle formed by two great circles on a sphere equals the dihedral angle between the planes containing the great circles.

Minute and second of arc Units for measuring angles

A minute of arc, arcminute (arcmin), arc minute, or minute arc, denoted by the symbol , is a unit of angular measurement equal to 1/60 of one degree. Since one degree is 1/360 of a turn, one minute of arc is 1/21600 of a turn. The nautical mile was originally defined as a minute of latitude on a hypothetical spherical Earth, so the actual Earth circumference is very near 21 600 nautical miles. A minute of arc is π/10800 of a radian.

Hipparchus Ancient Greek scholar

Hipparchus of Nicaea was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry but is most famous for his incidental discovery of precession of the equinoxes.

Radian SI derived unit of angle

The radian, denoted by the symbol , is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is 180/π degrees or just under 57.3°. The unit was formerly an SI supplementary unit and the radian is now considered an SI derived unit. The radian is defined in the SI as being a dimensionless value, and its symbol is accordingly often omitted, especially in mathematical writing.

Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form—for measuring time, angles, and geographic coordinates.

Babylonian cuneiform numerals Numeral system

Assyro-Chaldean Babylonian cuneiform numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record.

In trigonometry, the gradian, also known as the gon, grad, or grade, is a unit of measurement of an angle, defined as one hundredth of the right angle. It is equivalent to 1/400 of a turn, 9/10 of a degree, or π/200 of a radian. Measuring angles in gradians is said to employ the centesimal system of angular measurement.:22

Chord (geometry) Geometric line segment whose endpoints both lie on the curve

A chord of a circle is a straight line segment whose endpoints both lie on the circle. The infinite line extension of a chord is a secant line, or just secant. More generally, a chord is a line segment joining two points on any curve, for instance, an ellipse. A chord that passes through a circle's center point is the circle's diameter. The word chord is from the Latin chorda meaning bowstring.

Turn (angle) Unit of angle equal to 2𝜋 radians

A turn is a unit of plane angle measurement equal to 2π radians, 360 degrees or 400 gradians. A turn is also referred to as a cycle, revolution, complete rotation or full circle.

Milliradian Angular measurement, thousandth of a radian

A milliradian is an SI derived unit for angular measurement which is defined as a thousandth of a radian (0.001 radian). Milliradians are used in adjustment of firearm sights by adjusting the angle of the sight compared to the barrel. Milliradians are also used for comparing shot groupings, or to compare the difficulty of hitting different sized shooting targets at different distances. When using a scope with both mrad adjustment and a reticle with mrad markings, the shooter can use the reticle as a ‘ruler’ to count the number of mrads a shot was off-target, which directly translates to the sight adjustment needed to hit the target with a follow up shot. Optics with mrad markings in the reticle can also be used to make a range estimation of a known size target, or vice versa, to determine a target size if the distance is known, a practice called “milling”.

Circular sector

A circular sector or circle sector, is the portion of a disk enclosed by two radii and an arc, where the smaller area is known as the minor sector and the larger being the major sector. In the diagram, θ is the central angle, the radius of the circle, and is the arc length of the minor sector.

A square degree (deg2) is a non-SI unit measure of solid angle. Other denotations include sq. deg. and (°)2. Just as degrees are used to measure parts of a circle, square degrees are used to measure parts of a sphere. Analogous to one degree being equal to π/180 radians, a square degree is equal to (π/180)2 steradians (sr), or about 1/3283 sr or 3.0461741978670859934×10−4 sr.

Arc (geometry) Closed segment of a differentiable curve

In Euclidean geometry, an arc is a connected subset of a differentiable curve. Arcs of lines are called segments or rays, depending whether they are bounded or not. A common curved example is an arc of a circle, called a circular arc. In a sphere, an arc of a great circle is called a great arc.

Babylonian mathematics

Babylonian mathematics was any mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Babylonian mathematical texts are plentiful and well edited. In respect of time they fall in two distinct groups: one from the Old Babylonian period, the other mainly Seleucid from the last three or four centuries BC. In respect of content, there is scarcely any difference between the two groups of texts. Babylonian mathematics remained constant, in character and content, for nearly two millennia.

Trigonometry is a branch of mathematics that studies the relationships between the sides and the angles in triangles. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves.

History of trigonometry

Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata, who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics and reaching its modern form with Leonhard Euler (1748).

Āryabhaṭas sine table

The astronomical treatise Āryabhaṭīya was composed during the fifth century by the Indian mathematician and astronomer Āryabhaṭa, for the computation of the half-chords of certain set of arcs of a circle. It is not a table in the modern sense of a mathematical table; that is, it is not a set of numbers arranged into rows and columns.

The table of chords, created by the Greek astronomer, geometer, and geographer Ptolemy in Egypt during the 2nd century AD, is a trigonometric table in Book I, chapter 11 of Ptolemy's Almagest, a treatise on mathematical astronomy. It is essentially equivalent to a table of values of the sine function. It was the earliest trigonometric table extensive enough for many practical purposes, including those of astronomy. Centuries passed before more extensive trigonometric tables were created. One such table is the Canon Sinuum created at the end of the 16th century.

Clock position Relative direction using a dial

A clock position, or clock bearing, is the direction of an object observed from a vehicle, typically a vessel or an aircraft, relative to the orientation of the vehicle to the observer. The vehicle must be considered to have a front, a back, a left side and a right side. These quarters may have specialized names, such as bow and stern for a vessel, or nose and tail for an aircraft. The observer then measures or observes the angle made by the intersection of the line of sight to the longitudinal axis, the dimension of length, of the vessel, using the clock analogy.

Throughout history, angles have been measured in many different units. These are known as angular units, with the most contemporary units being the degree, the radian (rad), and the gradian (grad), though many others have been used throughout history. The purpose of this page is to aggregate other concepts pertaining to the angular unit, where additional explanation can be provided.

References

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