Degree (angle)

Last updated

Degree
Degree diagram.svg
One degree (shown in red) and eighty nine degrees (shown in blue)
General information
Unit system Non-SI accepted unit
Unit of Angle
Symbol° [1] [2] ,deg [3]
Conversions
[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

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

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. [5] 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. 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. [4]

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. [7] [8] 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. [9] [10] Timocharis, Aristarchus, Aristillus, Archimedes, and Hipparchus were the first Greeks known to divide the circle in 360 degrees of 60 arc minutes. [11] Eratosthenes used a simpler sexagesimal system dividing a circle into 60 parts.[ citation needed ]

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 ). [12] 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 (DD notation); 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 represented by a single prime (′) and double prime (″) respectively. For example, 40.1875° = 40° 11′ 15″. Additional precision can be provided using decimal fractions of an arcsecond.

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). [13]

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. [14] Hence, the modern symbols for the minute and second of arc, and the word "second" also refer to this system. [15]

SI prefixes can also be applied as in, e.g., millidegree, microdegree, etc.

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 (corresponding to a cycle or revolution) is used in technology and science.[ citation needed ] 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" in France and nearby countries, [note 3] where the number in a right angle is equal to 100 gon with 400 gon in a full circle (1° = 109 gon). This was called grade (nouveau) or grad . Due to confusion with the existing term grad(e) in some northern European countries (meaning a standard degree, 1/360 of a turn), the new unit was called Neugrad in German (whereas the "old" degree was referred to as Altgrad), likewise nygrad in Danish, Swedish and Norwegian (also gradian), and nýgráða in Icelandic. To end the confusion, the name gon was later adopted for the new unit. Although this idea of metrification 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
0 turn0 rad0g
1/72 turnπ/36 rad5+5/9g
1/24 turnπ/12 rad15°16+2/3g
1/16 turnπ/8 rad22.5°25g
1/12 turnπ/6 rad30°33+1/3g
1/10 turnπ/5 rad36°40g
1/8 turnπ/4 rad45°50g
1/2π turn1 radapprox. 57.3°approx. 63.7g
1/6 turnπ/3 rad60°66+2/3g
1/5 turn2π/5 rad72°80g
1/4 turnπ/2 rad90°100g
1/3 turn2π/3 rad120°133+1/3g
2/5 turn4π/5 rad144°160g
1/2 turnπ rad180°200g
3/4 turn3π/2 rad270°300g
1 turn2π rad360°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

<span class="mw-page-title-main">Angle</span> Figure formed by two rays meeting at a common 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 are also known as plane angles as they 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 may also define an angle, which is the angle of the rays lying tangent to the respective curves at their point of intersection.

<span class="mw-page-title-main">Minute and second of arc</span> 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, or complete rotation, one arcminute is 1/21600 of a turn. The nautical mile (nmi) was originally defined as the arc length of a minute of latitude on a spherical Earth, so the actual Earth circumference is very near 21600 nmi. A minute of arc is π/10800 of a radian.

The duodecimal system is a positional notation numeral system using twelve as its base. The number twelve is instead written as "10" in duodecimal, whereas the digit string "12" means "1 dozen and 2 units". Similarly, in duodecimal, "100" means "1 gross", "1000" means "1 great gross", and "0.1" means "1 twelfth".

<span class="mw-page-title-main">Hipparchus</span> 2nd-century BC Greek astronomer, geographer and mathematician

Hipparchus was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hipparchus was born in Nicaea, Bithynia, and probably died on the island of Rhodes, Greece. He is known to have been a working astronomer between 162 and 127 BC.

<span class="mw-page-title-main">Radian</span> SI derived unit of angle

The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. It is defined such that one radian is the angle subtended at the centre of a circle by an arc that is equal in length to the radius. The unit was formerly an SI supplementary unit and is currently a dimensionless SI derived unit, defined in the SI as 1 rad = 1 and expressed in terms of the SI base unit metre (m) as rad = m/m. Angles without explicitly specified units are generally assumed to be measured in radians, especially in mathematical writing.

<span class="mw-page-title-main">Right angle</span> 90° angle (π/2 radians)

In geometry and trigonometry, a right angle is an angle of exactly 90 degrees or /2 radians corresponding to a quarter turn. If a ray is placed so that its endpoint is on a line and the adjacent angles are equal, then they are right angles. The term is a calque of Latin angulus rectus; here rectus means "upright", referring to the vertical perpendicular to a horizontal base line.

Sexagesimal, also known as base 60, 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.

<span class="mw-page-title-main">Babylonian cuneiform numerals</span> Numeral system

Babylonian cuneiform numerals, also used in Assyria and Chaldea, were written in cuneiform, using a wedge-tipped reed stylus to print a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record.

<span class="mw-page-title-main">Gradian</span> Unit of measurement of an angle, equal to 1/400th of a circle

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; in other words, 100 gradians is equal to 90 degrees. 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, initiated as part of metrication and decimalisation efforts.

<span class="mw-page-title-main">Chord (geometry)</span> 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 a circular arc. If a chord were to be extended infinitely on both directions into a line, the object is a secant line. The perpendicular line passing through the chord's midpoint is called sagitta.

60 (sixty) is the natural number following 59 and preceding 61. Being three times 20, it is called threescore in older literature.

360 is the natural number following 359 and preceding 361.

<span class="mw-page-title-main">Turn (angle)</span> Unit of plane angle where a full circle equals 1

One turn is a unit of plane angle measurement equal to  radians, 360 degrees or 400 gradians. Thus it is the angular measure subtended by a complete circle at its center.

<span class="mw-page-title-main">Milliradian</span> 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".

In astrology, the Equatorial Ascendant, or the East Point, is the sign and degree rising over the Eastern Horizon at the Earth's equator at any given time. In the celestial sphere it corresponds to the intersection of the ecliptic with a great circle containing the celestial poles and the East point of the horizon.

<span class="mw-page-title-main">Babylonian astronomy</span> Study of celestial objects during the early history of Mesopotamia

Babylonian astronomy was the study or recording of celestial objects during the early history of Mesopotamia.

<span class="mw-page-title-main">Outline of trigonometry</span> Overview of and topical guide to trigonometry

The following outline is provided as an overview of and topical guide to trigonometry:

<span class="mw-page-title-main">History of trigonometry</span>

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).

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. Since the 8th and 9th centuries, the sine and other trigonometric functions have been used in Islamic mathematics and astronomy, reforming the production of sine tables. Khwarizmi and Habash al-Hasib later produced a set of trigonometric tables.

<span class="mw-page-title-main">Clock position</span> 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.

References

  1. 1 2 HP 48G Series – User's Guide (UG) (8 ed.). Hewlett-Packard. December 1994 [1993]. HP 00048-90126, (00048-90104). Retrieved 6 September 2015.
  2. 1 2 HP 50g graphing calculator user's guide (UG) (1 ed.). Hewlett-Packard. 1 April 2006. HP F2229AA-90006. Retrieved 10 October 2015.
  3. HP Prime Graphing Calculator User Guide (UG) (PDF) (1 ed.). Hewlett-Packard Development Company, L.P. October 2014. HP 788996-001. Archived from the original (PDF) on 3 September 2014. Retrieved 13 October 2015.
  4. 1 2 Weisstein, Eric W. "Degree". mathworld.wolfram.com. Retrieved 31 August 2020.
  5. Bureau international des poids et mesures, Le Système international d'unités (SI) / The International System of Units (SI), 9th ed. [ permanent dead link ] (Sèvres: 2019), ISBN   978-92-822-2272-0, c. 4, pp. 145–146.
  6. Euclid (2008). "Book 4". Euclid's Elements of Geometry [Euclidis Elementa, editit et Latine interpretatus est I. L. Heiberg, in aedibus B. G. Teubneri 1883–1885]. Translated by Heiberg, Johan Ludvig; Fitzpatrick, Richard (2 ed.). Princeton University Press. ISBN   978-0-6151-7984-1.
  7. Jeans, James Hopwood (1947). The Growth of Physical Science. Cambridge University Press (CUP). p.  7.
  8. Murnaghan, Francis Dominic (1946). Analytic Geometry. p. 2.
  9. Rawlins, Dennis. "On Aristarchus". DIO - the International Journal of Scientific History.
  10. Toomer, Gerald James. Hipparchus and Babylonian astronomy.
  11. "2 (Footnote 24)" (PDF). Aristarchos Unbound: Ancient Vision / The Hellenistic Heliocentrists' Colossal Universe-Scale / Historians' Colossal Inversion of Great & Phony Ancients / History-of-Astronomy and the Moon in Retrograde!. March 2008. p. 19. ISSN   1041-5440 . Retrieved 16 October 2015.{{cite book}}: |journal= ignored (help)
  12. Brefeld, Werner. "Teilbarkeit hochzusammengesetzter Zahlen" [Divisibility highly composite numbers] (in German).
  13. Hopkinson, Sara (2012). RYA day skipper handbook - sail. Hamble: The Royal Yachting Association. p. 76. ISBN   9781-9051-04949.
  14. Al-Biruni (1879) [1000]. The Chronology of Ancient Nations. Translated by Sachau, C. Edward. pp. 147–149.
  15. Flegg, Graham H. (1989). Numbers Through the Ages. Macmillan International Higher Education. pp. 156–157. ISBN   1-34920177-4.