Angle - Knowledge (XXG)

1028:. The proposition showed that since both of a pair of vertical angles are supplementary to both of the adjacent angles, the vertical angles are equal in measure. According to a historical note, when Thales visited Egypt, he observed that whenever the Egyptians drew two intersecting lines, they would measure the vertical angles to make sure that they were equal. Thales concluded that one could prove that all vertical angles are equal if one accepted some general notions such as: 559: 1618: 4071: 1707: 1317: 4168:". For example, an orientation represented as −45° is effectively equal to an orientation defined as 360° − 45° or 315°. Although the final position is the same, a physical rotation (movement) of −45° is not the same as a rotation of 315° (for example, the rotation of a person holding a broom resting on a dusty floor would leave visually different traces of swept regions on the floor). 1867:. There are two exterior angles at each vertex of the polygon, each determined by extending one of the two sides of the polygon that meet at the vertex; these two angles are vertical and hence are equal. An exterior angle measures the amount of rotation one must make at a vertex to trace the polygon. If the corresponding interior angle is a reflex angle, the exterior angle should be considered 8140: 975: 1749: 1146: 3906: 3603:. The other option is to introduce a dimensional constant. According to Quincey this approach is "logically rigorous" compared to SI, but requires "the modification of many familiar mathematical and physical equations". A dimensional constant for angle is "rather strange" and the difficulty of modifying equations to add the dimensional constant is likely to preclude widespread use. 2016: 583: 545: 4334: 2258: 45: 6085:, in addition to the issue of "measurement units chosen". A smoother approach is to measure the angle by the length of the corresponding unit circle arc. Here "unit" can be chosen to be dimensionless in the sense that it is the real number 1 associated with the unit segment on the real line. See Radoslav M. Dimitrić, for instance. 3657: 4786: 3548:

but not on the right hand side. Anthony French calls this phenomenon "a perennial problem in the teaching of mechanics". Oberhofer says that the typical advice of ignoring radians during dimensional analysis and adding or removing radians in units according to convention and contextual knowledge is

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radians, then stopping if the angle is acute, otherwise taking the supplementary angle, 180° minus the reduced magnitude. For example, an angle of 30 degrees is already a reference angle, and an angle of 150 degrees also has a reference angle of 30 degrees (180° − 150°). Angles of 210° and 510°

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might refer to any of four angles: the clockwise angle from B to C about A, the anticlockwise angle from B to C about A, the clockwise angle from C to B about A, or the anticlockwise angle from C to B about A, where the direction in which the angle is measured determines its sign (see

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equal to a milliradian. Under these three other definitions, one turn makes up for exactly 6000, 6300, or 6400 mils, spanning the range from 0.05625 to 0.06 degrees (3.375 to 3.6 minutes). In comparison, the milliradian is approximately 0.05729578 degrees (3.43775 minutes). One

1169:, are angles that share a common vertex and edge but do not share any interior points. In other words, they are angles side by side or adjacent, sharing an "arm". Adjacent angles which sum to a right angle, straight angle, or full angle are special and are respectively called 4194:

are measured relative to north. By convention, viewed from above, bearing angles are positive clockwise, so a bearing of 45° corresponds to a north-east orientation. Negative bearings are not used in navigation, so a north-west orientation corresponds to a bearing of 315°.

1883:) to decide the sign of the exterior angle measure. In Euclidean geometry, the sum of the exterior angles of a simple convex polygon, if only one of the two exterior angles is assumed at each vertex, will be one full turn (360°). The exterior angle here could be called a 347:). However, in many geometrical situations, it is evident from the context that the positive angle less than or equal to 180 degrees is meant, and in these cases, no ambiguity arises. Otherwise, to avoid ambiguity, specific conventions may be adopted so that, for instance, 4679: 5535: 2985:

of a turn. Just like with the milliradian, each of the other definitions approximates the milliradian's useful property of subtensions, i.e. that the value of one milliradian approximately equals the angle subtended by a width of 1 meter as seen from 1 km away

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Giacomo Prando writes "the current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations". For example, an object hanging by a string from a pulley will rise or drop by

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as the square of the sine of the angle between the lines. As the sine of an angle and the sine of its supplementary angle are the same, any angle of rotation that maps one of the lines into the other leads to the same value for the spread between the

4692: 3901:{\displaystyle \operatorname {Sin} \theta =\sin \ x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots =\eta \theta -{\frac {(\eta \theta )^{3}}{3!}}+{\frac {(\eta \theta )^{5}}{5!}}-{\frac {(\eta \theta )^{7}}{7!}}+\cdots ,} 4795: 5090: 1431: 1306: 3592:(and dimension) of "plane angle". Quincey's review of proposals outlines two classes of proposal. The first option changes the unit of a radius to meters per radian, but this is incompatible with dimensional analysis for the 1395:

radians). If the two complementary angles are adjacent, their non-shared sides form a right angle. In Euclidean geometry, the two acute angles in a right triangle are complementary because the sum of internal angles of a

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subunit is that many angles common in simple geometry are measured as a whole number of degrees. Fractions of a degree may be written in normal decimal notation (e.g., 3.5° for three and a half degrees), but the

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The angle between a plane and an intersecting straight line is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the

5381: 5309: 285:, . . . ) are also used. In contexts where this is not confusing, an angle may be denoted by the upper case Roman letter denoting its vertex. See the figures in this article for examples. 2231: 1920:

In a triangle, three intersection points, two between an interior angle bisector and the opposite side, and the third between the other exterior angle bisector and the opposite side extended are collinear.

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Metric Committee specified that the radian should explicitly appear in quantities only when different numerical values would be obtained when other angle measures were used, such as in the quantities of

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of the angle; a gradient is often expressed as a percentage. For very small values (less than 5%), the slope of a line is approximately the measure in radians of its angle with the horizontal direction.

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defines a plane angle as the inclination to each other, in a plane, of two lines that meet each other and do not lie straight with respect to each other. According to the Neoplatonic metaphysician

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of these sectors correspond to the angle magnitudes in each case. Unlike the circular angle, the hyperbolic angle is unbounded. When the circular and hyperbolic functions are viewed as

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The size of a geometric angle is usually characterized by the magnitude of the smallest rotation that maps one of the rays into the other. Angles of the same size are said to be

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being omitted. The radian is used in virtually all mathematical work beyond simple, practical geometry due, for example, to the pleasing and "natural" properties that the

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In three-dimensional geometry, "clockwise" and "anticlockwise" have no absolute meaning, so the direction of positive and negative angles must be defined in terms of an

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turn. It is denoted by a single prime ( ′ ). For example, 3° 30′ is equal to 3 × 60 + 30 = 210 minutes or 3 +

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In Euclidean geometry, any sum of two angles in a triangle is supplementary to the third because the sum of the internal angles of a triangle is a straight angle.

2790: turn. As this system is amenable to measuring objects that cycle once per day (such as the relative position of stars), the sexagesimal subunits are called 5329: 2241:

thus defined is independent of the size of the circle: if the length of the radius is changed, then the arc length changes in the same proportion, so the ratio

4781:{\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} 5590: 3059:(albeit to limited precision). Other measures of the angle used in computing may be based on dividing one whole turn into 2 equal parts for other values of 8160: 4244:

and the x-axis (positive or negative). Procedurally, the magnitude of the reference angle for a given angle may determined by taking the angle's magnitude

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When two straight lines intersect at a point, four angles are formed. Pairwise, these angles are named according to their location relative to each other.

2312:, an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. This convention impacts how angles are treated in 7710:

Leonard, B P (1 October 2021). "Proposal for the dimensionally consistent treatment of angle and solid angle by the International System of Units (SI)".

4888:{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} 5180:{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|} 2960:

are calibrated to this definition. In addition, three other related definitions are used for artillery and navigation, often called a 'mil', which are

6757: 1582:{\displaystyle {\begin{aligned}&\sin ^{2}A+\sin ^{2}B=1&&\cos ^{2}A+\cos ^{2}B=1\\&\tan A=\cot B&&\sec A=\csc B\end{aligned}}} 2182: 8054: 3553: 5701:

has an angular diameter of approximately 0.5° when viewed from Earth. One could say, "The Moon's diameter subtends an angle of half a degree." The

2367: = 6.283...). It is the angle subtended by an arc of a circle that has the same length as the circle's radius. The symbol for radian is 8128: 7701: 7663: 7500: 6975: 6555: 2081: 1698:

The sines of supplementary angles are equal. Their cosines and tangents (unless undefined) are equal in magnitude but have opposite signs.

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The angle expressed by another angular unit may then be obtained by multiplying the angle by a suitable conversion constant of the form

8048: 6591: 7435: 7331: 5659:, each intersecting one of the stars. The angle between those lines and the angular separation between the two stars can be measured. 2309: 2012:

in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.

1680:. However, supplementary angles do not have to be on the same line and can be separated in space. For example, adjacent angles of a 4268:

correspond to a reference angle of 30 degrees as well (210° mod 180° = 30°, 510° mod 180° = 150° whose supplementary angle is 30°).

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Defining radian as a base unit may be useful for software, where the disadvantage of longer equations is minimal. For example, the

8155: 333:. Where there is no risk of confusion, the angle may sometimes be referred to by a single vertex alone (in this case, "angle A"). 4296: 2393:, or about 57.2958 degrees. Often, particularly in mathematical texts, one radian is assumed to equal one, resulting in the unit 4674:{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} 6678: 5641: 3120: 223:, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship. 4304: 3957:

is the "complete" function that takes an argument with a dimension of angle and is independent of the units expressed, while

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Aubrecht, Gordon J.; French, Anthony P.; Iona, Mario; Welch, Daniel W. (February 1993). "The radian—That troublesome unit".

5530:{\displaystyle \cos \theta ={\frac {g_{ij}U^{i}V^{j}}{\sqrt {\left|g_{ij}U^{i}U^{j}\right|\left|g_{ij}V^{i}V^{j}\right|}}}.} 288:

The three defining points may also identify angles in geometric figures. For example, the angle with vertex A formed by the

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forms of the hyperbolic functions. This comparison of the two series corresponding to functions of angles was described by

2626: = 1,296,000). It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 8040: 4897:

The latter definition ignores the direction of the vectors. It thus describes the angle between one-dimensional subspaces

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When two adjacent angles form a straight line, they are supplementary. Therefore, if we assume that the measure of angle

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Quincey, Paul (1 April 2016). "The range of options for handling plane angle and solid angle within a system of units".

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This approach requires, however, an additional proof that the measure of the angle does not change with changing radius

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A pair of angles opposite each other, formed by two intersecting straight lines that form an "X"-like shape, are called

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of a right angle = 11.25° = 12.5 grad. Each point is subdivided into four quarter points, so one turn equals 128.

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Quincey, Paul; Brown, Richard J C (1 June 2016). "Implications of adopting plane angle as a base quantity in the SI".

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of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full

1888: 53: 211:, an angle must be either a quality, a quantity, or a relationship. The first concept, angle as quality, was used by 144:, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation. 8182: 5943: 5925: 4564: 4528:{\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} 3026: 1977: 140:. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of a 2719: 1871:. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick an 4214:. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all 6642: 4418: 4278: 3585: 3518:

is the numerical value of the angle through which the pulley turns when expressed in radians. When multiplying

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is only to be used to express angles, not to express ratios of lengths in general. A similar calculation using

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Brinsmade, J. B. (December 1936). "Plane and Solid Angles. Their Pedagogic Value When Introduced Explicitly".

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It is frequently helpful to impose a convention that allows positive and negative angular values to represent

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at the point of intersection. Various names (now rarely, if ever, used) have been given to particular cases:—

4034: 7611: 4361:(mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the 4317: 3112: 2534:= 3.5 degrees. A mixed format with decimal fractions is sometimes used, e.g., 3° 5.72′ = 3 + 2401:

display when their arguments are in radians. The radian is the (derived) unit of angular measurement in the

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sexagesimal subunits of the "degree–minute–second" system (discussed next) are also in use, especially for

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This formula supplies an easy method to find the angle between two planes (or curved surfaces) from their

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is radius. One SI radian corresponds to the (numerical value of the) angle expressed in radians for which

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Angles AOB and COD are conjugate as they form a complete angle. Considering magnitudes, 45° + 315° = 360°.

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An angle larger than a right angle and smaller than a straight angle (between 90° and 180°) is called an

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radian. One "diameter part" is approximately 0.95493°. There are about 376.991 diameter parts per turn.

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subunits of the Babylonian unit. It is straightforward to construct with ruler and compasses. It is the

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I.e., the measure of the angle AOC is the sum of the measure of angle AOB and the measure of angle BOC.

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Two angles that share terminal sides, but differ in size by an integer multiple of a turn, are called

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passing through the angle's vertex and perpendicular to the plane in which the rays of the angle lie.

8091: 7974: 7927: 7885: 7840: 7789: 7760: 7719: 7582: 7551: 7518: 7296: 7177: 7062: 7027: 6928: 6340: 5954: 5930: 4187: 3566: 3233: 3165: 3055:. The binary degree is used in computing so that an angle can be efficiently represented in a single 2798:. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° = 2704: 2313: 2063: 1685: 1611: 5220: 5196: 2359:

is determined by the circumference of a circle that is equal in length to the radius of the circle (

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An angle larger than a straight angle but less than 1 turn (between 180° and 360°) is called a

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Quincey, Paul; Brown, Richard J C (1 August 2017). "A clearer approach for defining unit systems".

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holds. Some quantities related to angles where the angle addition postulate does not hold include:

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that contains the rays. Angles are also formed by the intersection of two planes; these are called

7371: 5639:(that is, the apparent position of an astronomical object) can be identified using any of several 4990: 4968: 3492:. It is a long-established practice in mathematics and across all areas of science to make use of 8107: 7990: 7964: 7955:

Quincey, Paul (1 October 2021). "Angles in the SI: a detailed proposal for solving the problem".

7943: 7875: 7830: 7805: 7735: 7683: 7631: 7598: 7312: 7286: 7193: 7167: 6946: 6459: 6049: 5935: 5647: 5625: 5581: 4061: 3481: 2270: 2131: 1772: 397: 60: 6743: 6422: 6384: 2948:

The true milliradian is defined as a thousandth of a radian, which means that a rotation of one

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At least a dozen scientists between 1936 and 2022 have made proposals to treat the radian as a

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In a triangle, three intersection points, each of an external angle bisector with the opposite

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These measurements depend on the individual subject, and the above should be treated as rough

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radians, 360°, or 1 turn. In general, the measures of the interior angles of a simple convex

8099: 8058: 8015: 7982: 7935: 7893: 7848: 7797: 7768: 7727: 7623: 7590: 7559: 7526: 7304: 7185: 7070: 7035: 7000: 6936: 6576: 6024: 5964: 5959: 5694: 5636: 5545: 5332: 4442: 4408: 4083: 3593: 3578: 3570: 3527: 3284: 2953: 1903: 978:

Angles A and B are a pair of vertical angles; angles C and D are a pair of vertical angles.

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are usually measured in angular units, expressed in terms of time, based on a 24-hour day.

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is the angle subtended by the circumference of a circle at its centre. A turn is equal to 2

6044: 6019: 5989: 5745: 5671: 5577: 5569: 4434: 4065: 3464: 3394: 3390: 3207: 3127: 2949: 2731: 2294: 1868: 1768: 1673: 634: 137: 6674: 8095: 7978: 7931: 7889: 7844: 7793: 7780:

Mills, Ian (1 June 2016). "On the units radian and cycle for the quantity plane angle".

7764: 7723: 7586: 7555: 7522: 7300: 7181: 7066: 7031: 6932: 5645:, where the references vary according to the particular system. Astronomers measure the 3965:

which assumes its argument is a dimensionless number in radians. The capitalised symbol

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is the numerical value of the angle when expressed in radians. The capitalized function

8200: 6737: 5994: 5667: 5663: 5585: 5314: 4687:, the expression for the cosine above may give non-real values, so it is replaced with 4422: 2410: 2297:(i.e., the angle subtended by the circumference of a circle at its centre) is equal to 2278: 2170: 1955: 1859: 1800: 1763: 1758: 1410:, "to fill up". An acute angle is "filled up" by its complement to form a right angle. 904: 375:

There is some common terminology for angles, whose measure is always non-negative (see

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knew how to bisect an angle (divide it into two angles of equal measure) using only a

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would equal exactly 2000π mrad (or approximately 6283.185 mrad). Almost all

2305:. Two exceptions are the radian (and its decimal submultiples) and the diameter part. 8194: 8151: 8146: 8111: 8103: 7994: 7947: 7939: 7809: 7801: 7748: 7739: 7635: 7627: 7602: 7594: 7316: 7197: 7189: 6950: 6029: 5979: 5738: 5724: 5662:

In both geography and astronomy, a sighting direction can be specified in terms of a

5348: 5190: 4538: 4172: 4150: 4109:, an angle is typically defined by its two sides, with its vertex at the origin. The 3643: 3589: 3562: 3558: 3514:

is the numerical value of the radius of the pulley when expressed in centimeters and

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Mohr, Peter J; Shirley, Eric L; Phillips, William D; Trott, Michael (23 June 2022).

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is defined by the measure from the initial side in radians, degrees, or turns, with

1301:{\displaystyle m\angle \mathrm {AOC} =m\angle \mathrm {AOB} +m\angle \mathrm {BOC} } 8123:, vol. 1B (1 ed.), Hong Kong: Oxford University Press, pp. 161–163, 8119:

Wong, Tak-wah; Wong, Ming-sim (2009), "Angles in Intersecting and Parallel Lines",

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of any location in terms of angles subtended at the center of the Earth, using the

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was subdivided into 32 Akhnam, and each akhnam was subdivided into 7 zam so that a

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AB and AC (that is, the half-lines from point A through points B and C) is denoted

236: 7612:"The next 50 years of the SI: a review of the opportunities for the e-Science age" 3622:. With this change the formula for the angle subtended at the center of a circle, 1196:

is a line that intersects a pair of (often parallel) lines and is associated with

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from P touch the circle at points T and Q, then ∠TPQ and ∠TOQ are supplementary.

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the unit radian does not appear in the result. Similarly in the formula for the

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are effectively equivalent. In other contexts, such as identifying a point on a

1614:" in the names of some trigonometric ratios refers to the word "complementary". 550: 439: 364: 265: 7986: 7898: 7863: 7731: 7308: 1706: 1316: 1099:, either of these angle measures may be used to determine the measure of Angle 6941: 6916: 6054: 6034: 6014: 4542: 4240:

in standard position is the positive acute angle between the terminal side of

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gives 1 SI radian as 1 m/m = 1. The key fact is that the SI radian is a

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degrees, or 3.125 degrees. The arcsecond is the angle used to measure a

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is the measure of a complete turn expressed in the chosen unit (for example,

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Angles that have the same measure (i.e., the same magnitude) are said to be

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In particular, Quincey identifies Torrens' proposal to introduce a constant

3168:. In German, the symbol has been used to denote a quadrant. 1 quad = 90° = 2723: 2696: 2567: 2459: 2443: 1996:

In some contexts, such as identifying a point on a circle or describing the

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always refers to the anticlockwise (positive) angle from B to C about A and

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French, Anthony P. (May 1992). "What happens to the 'radians'? (comment)".

2015: 8020: 8003: 4333: 2257: 582: 5613: 2695:. It is a decimal subunit of the quadrant. A right angle is 100 grads. A 1895: 1776: 1600: 1397: 2425:, denoted by a small superscript circle (°), is 1/360 of a turn, so one 1935:

supplement!) of the interior angle. This conflicts with the above usage.

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of two exterior angles and the bisector of the other interior angle are

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The names, intervals, and measuring units are shown in the table below:

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Eder, W E (January 1982). "A Viewpoint on the Quantity "Plane Angle"".

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can be extended to subspaces of finite dimensions. Given two subspaces

4362: 4191: 2957: 2687: 2286: 2174: 1811: 1592: 933: 576:) angles. The acute and obtuse angles are also known as oblique angles. 363:"Oblique angle" redirects here. For the cinematographic technique, see 208: 167: 104: 8175: 7530: 7074: 7039: 7004: 4425:

showed that this construction could not be performed for most angles.

8164:, vol. 2 (11th ed.), Cambridge University Press, p. 14 6538: 6336: 4245: 4126: 4114: 4075: 4045:'s unit system similarly considers angles to have an angle dimension. 3574: 3410: 3116: 2659: 2339: 2282: 2075: 2005: 1607: 1025: 771: 204: 133: 8055:

University of Texas research department: linguistics research center

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can convert such an angular measurement into a distance/size ratio.

5564:. The comparison can be visualized as the size of the openings of a 1688:(one whose vertices all fall on a single circle) are supplementary. 8082:

Torrens, A B (1 January 1986). "On Angles and Angular Quantities".

7969: 7880: 7291: 7172: 5606:, the location of any point on the Earth can be identified using a 5304:{\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l} 3612:

equal to 1 inverse radian (1 rad) in a fashion similar to the

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centered at the vertex of the angle is drawn, e.g., with a pair of

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The difference between an angle and a complete angle is termed the

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is typically not used for this purpose to avoid confusion with the

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This article incorporates text from a publication now in the

7835: 6647: 5683: 5656: 4358: 4332: 4287: 4069: 3279:. It equals 6°, so a whole turn was divided into 60 hexacontades. 2700: 2256: 2054: 2014: 1747: 1705: 1616: 1144: 973: 190: 157: 100: 43: 31: 8008:

Journal of Research of the National Bureau of Standards Section B

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Oberhofer, E. S. (March 1992). "What happens to the 'radians'?".

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An angle smaller than a right angle (less than 90°) is called an

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Shute, William G.; Shirk, William W.; Porter, George F. (1960),

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Experiencing Geometry / Euclidean and Non-Euclidean with History

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The difference between an angle and a right angle is termed the

44: 7430:(6th ed.). Belmont, CA: Thomson Brooks/Cole. p. 110. 2319:

The following table lists some units used to represent angles.

2226:{\displaystyle \theta ={\frac {k}{2\pi }}\cdot {\frac {s}{r}}.} 1771:

has at least one interior angle, that is, a reflex angle. In

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If a point P is exterior to a circle with center O, and if the

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may also define an angle, which is the angle of the rays lying

6859:, p. 151: "One radian corresponds to the angle for which 6777:, p. 151: "One radian corresponds to the angle for which 6650:

Manual & Technical Specifications - ooPIC Compiler Ver 6.0

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10° is the approximate width of a closed fist at arm's length.

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The definition of the angle between one-dimensional subspaces

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is effectively equivalent to an angle of "one full turn minus

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in opposite directions or "sense" relative to some reference.

2293:. Most units of angular measurement are defined such that one 528:

An angle that is not a multiple of a right angle is called an

37:

This article is about angles in geometry. For other uses, see

6295: 6293: 4370: 4008:

Current SI can be considered relative to this framework as a

1400:

is 180 degrees, and the right angle accounts for 90 degrees.

180: 174: 6174: 5287: 5259: 5226: 5202: 3437:

is the (numerical value of the) subtended angle in radians,

2707:

of the Earth, so the kilometer is the decimal analog to the

5733:

20° is the approximate width of a handspan at arm's length.

4316:

Although done rarely, one can report the direct results of

2402: 2127: 1715:

Two angles that sum to a complete angle (1 turn, 360°, or 2

1313:

Three special angle pairs involve the summation of angles:

270: 1863:; that is, an interior angle and an exterior angle form a 1767:

if it lies on the inside of that simple polygon. A simple

107:

to the respective curves at their point of intersection.

83:

of the angle. Angles formed by two rays are also known as

8050:

Preliminary Indo-European lexicon — Pokorny PIE data

3485: 1959:. It may be defined as the acute angle between two lines 1016:

The equality of vertically opposite angles is called the

986:"Vertical angle" redirects here. Not to be confused with 7682:, The Thirteen Books of Euclid's Elements, vol. 1, 7358: 7252: 7240: 7127: 6208: 1672:

and share just one side), their non-shared sides form a

355:

the anticlockwise (positive) angle from C to B about A.

2119:{\displaystyle \theta ={\frac {s}{r}}\,\mathrm {rad} .} 1361:

are angle pairs whose measures sum to one right angle (

1232:

states that if B is in the interior of angle AOC, then

7817:

Mohr, Peter J; Phillips, William D (1 February 2015).

7492:

A High School First Course in Euclidean Plane Geometry

7224: 7222: 7209: 7207: 3488:. In SI 2019, the SI radian is defined accordingly as 1799:

turn; the measures of the interior angles of a simple

6280: 6278: 6171: 6110: 5384: 5317: 5247: 5223: 5199: 5093: 5057: 5023: 4993: 4971: 4937: 4903: 4798: 4695: 4601: 4567: 4464: 4137:-axis. When Cartesian coordinates are represented by 3991: 3971: 3914: 3660: 2289:(grad), though many others have been used throughout 2185: 2084: 1434: 1428:

are complementary, the following relationships hold:

1240: 302: 7658:(3rd ed.), Pearson Prentice Hall, p. 104, 5580:

in their angle argument, the circular ones are just

4387:(Gr. ξυστρίς, a tool for scraping), concavo-convex; 77:

of the angle, sharing a common endpoint, called the

6200:{\displaystyle {\cal {C}}={\frac {2\pi }{\Theta }}} 5078:{\displaystyle \operatorname {span} (\mathbf {v} )} 5044:{\displaystyle \operatorname {span} (\mathbf {u} )} 4958:{\displaystyle \operatorname {span} (\mathbf {v} )} 4924:{\displaystyle \operatorname {span} (\mathbf {u} )} 3946:{\displaystyle x=\eta \theta =\theta /{\text{rad}}} 443:. Two lines that form a right angle are said to be 27:

Figure formed by two rays meeting at a common point

8186:, vol. 2 (9th ed.), 1878, pp. 29–30 6199: 6122: 5529: 5323: 5303: 5233: 5209: 5179: 5077: 5043: 5001: 4979: 4957: 4923: 4887: 4790:or, more commonly, using the absolute value, with 4780: 4673: 4585: 4527: 3997: 3977: 3945: 3900: 3226:, The degree, minute of arc and second of arc are 2225: 2118: 1581: 1300: 325: 7153: 7151: 7149: 7147: 6883:International Bureau of Weights and Measures 2019 6857:International Bureau of Weights and Measures 2019 6815:, the radian can be defined in terms of the area 6797:International Bureau of Weights and Measures 2019 6775:International Bureau of Weights and Measures 2019 5655:by imagining two lines through the center of the 4277:For an angular unit, it is definitional that the 2554:was historically defined as an arcminute along a 1857:The supplement of an interior angle is called an 7472:The Algebra of Coplanar Vectors and Trigonometry 7372:"UnityDimensions—Wolfram Language Documentation" 6652:. Savage Innovations, LLC. 2007 . Archived from 6474: 6406: 6404: 6359: 6100:Other proposals include the abbreviation "rad" ( 4421:but could only trisect certain angles. In 1837, 4005:if it is clear that the complete form is meant. 385:An angle equal to 0° or not turned is called a 7646:Elementary geometry: practical and theoretical 6967:Minds-on Physics: Advanced topics in mechanics 6299: 3359:(occasionally used in Islamic mathematics) is 2718: = 400). The grad is used mostly in 2130:, the radian is treated as being equal to the 1402:The adjective complementary is from the Latin 215:, who regarded an angle as a deviation from a 6643:"ooPIC Programmer's Guide - Chapter 15: URCP" 6507:CRC Standard Mathematical Tables and Formulae 4586:{\displaystyle \langle \cdot ,\cdot \rangle } 4341:is defined as the angle between the tangents 2277:, with the most contemporary units being the 1599:of its complement, and its secant equals the 8: 7864:"On the dimension of angles and their units" 7749:"Dimensional angles and universal constants" 7654:Henderson, David W.; Taimina, Daina (2005), 7138: 7107: 6812: 6539:International Bureau of Weights and Measures 6269: 6143: 6117: 6111: 5591:Introduction to the Analysis of the Infinite 5115: 5099: 4820: 4804: 4723: 4707: 4618: 4602: 4580: 4568: 1684:are supplementary, and opposite angles of a 263:denoting the size of some angle (the symbol 5708:Other astronomical approximations include: 4133:representing rotations toward the negative 4125:representing rotations toward the positive 1775:, the measures of the interior angles of a 161: 7747:Lévy-Leblond, Jean-Marc (September 1998). 7649:(3rd ed.), Cambridge University Press 7103: 6903:Angular amplitude of swing No dimensions. 6901:. New Haven : Yale University Press. 6487: 6485: 6483: 6139: 1864: 1183: 8019: 7968: 7897: 7879: 7852: 7834: 7643:Godfrey, Charles; Siddons, A. W. (1919), 7563: 7290: 7171: 7087: 6940: 6679:"Angles, integers, and modulo arithmetic" 6182: 6173: 6172: 6170: 6109: 6101: 5510: 5500: 5487: 5467: 5457: 5444: 5427: 5417: 5404: 5397: 5383: 5316: 5286: 5285: 5258: 5257: 5246: 5225: 5224: 5222: 5201: 5200: 5198: 5168: 5155: 5110: 5102: 5092: 5067: 5056: 5033: 5022: 4994: 4992: 4972: 4970: 4947: 4936: 4913: 4902: 4873: 4860: 4815: 4807: 4797: 4766: 4753: 4718: 4710: 4694: 4659: 4646: 4613: 4605: 4600: 4566: 4557:, we replace the Euclidean dot product ( 4513: 4500: 4473: 4465: 4463: 4037:units library defines angle units with a 3990: 3970: 3938: 3933: 3913: 3872: 3856: 3836: 3820: 3800: 3784: 3750: 3744: 3725: 3719: 3700: 3694: 3659: 2210: 2192: 2184: 2102: 2101: 2091: 2083: 1638:Two angles that sum to a straight angle ( 1507: 1488: 1463: 1444: 1435: 1433: 1287: 1267: 1247: 1239: 336:In other ways, an angle denoted as, say, 306: 304: 303: 301: 128:conventionally defined as the ratio of a 8004:"Angle as a fourth fundamental quantity" 7450: 7330:Schabel, Matthias C.; Watanabe, Steven. 6637: 6635: 6509:, Boca Raton, FL: CRC Press, p. 270 6410: 6323: 6253: 5754: 5712:0.5° is the approximate diameter of the 5372:are the components of the metric tensor 5351:is used to define the angle between two 3573:(N⋅m/rad), and not in the quantities of 3554:American Association of Physics Teachers 2321: 1315: 1038:Equals subtracted from equals are equal. 594: 8077:, American Book Company, pp. 25–27 7454: 7264: 7228: 7213: 7123: 7119: 7099: 6847:), in which case it has the units m⋅m." 6811:, p. 844: "Also, as alluded to in 6808: 6257: 6231: 6224: 6162: 6123:{\displaystyle \langle \theta \rangle } 6093: 6072: 2126:Conventionally, in mathematics and the 1887:. Exterior angles are commonly used in 7111: 7091: 6544:The International System of Units (SI) 6445: 6371: 6242: 6151: 6131: 3253: turn. 1 Babylonian unit = 60° = 326:{\displaystyle {\widehat {\rm {BAC}}}} 7115: 6311: 6284: 4553:To define angles in an abstract real 4175:, which is typically determined by a 4149:-axis upward, positive rotations are 2265:Throughout history, angles have been 982:are used here to show angle equality. 7: 7610:Foster, Marcus P (1 December 2010). 7407:from the original on 23 October 2017 7095: 6564:from the original on 18 October 2021 4375:, on both sides, κυρτός, convex) or 4337:The angle between the two curves at 3978:{\displaystyle \operatorname {Sin} } 3257:/3 rad ≈ 1.047197551 rad. 1664:If the two supplementary angles are 1103:. Using the measure of either angle 185:) meaning "crooked, curved" and the 7696:, W. H. Freeman, pp. 97, 255, 5723:1° is the approximate width of the 2429:is 360°. One advantage of this old 1834: − 2)2 right angles, or ( 505:An angle equal to 1 turn (360° or 2 219:; the second, angle as quality, by 7540:"Angles—Let's treat them squarely" 6577:"On Angles and Angle Measurements" 6192: 5689:Astronomers also measure objects' 4019:is assumed to hold, or similarly, 2310:International System of Quantities 2109: 2106: 2103: 1830: − 2)180 degrees, ( 1294: 1291: 1288: 1284: 1274: 1271: 1268: 1264: 1254: 1251: 1248: 1244: 1061:. Similarly, the measure of angle 377: 343: 313: 310: 307: 25: 7489:Aboughantous, Charles H. (2010), 6895:Bridgman, Percy Williams (1922). 6458:Willis, Clarence Addison (1922). 5598:Angles in geography and astronomy 4456:and their lengths by the formula 3544:, radians appear in the units of 2375: radians, and one radian is 1949:(such as two adjacent faces of a 1035:Equals added to equals are equal. 259:, . . . ) as 8138: 8002:Romain, Jacques E. (July 1962). 5311:, this leads to a definition of 5169: 5156: 5111: 5103: 5068: 5034: 4995: 4973: 4948: 4914: 4874: 4861: 4816: 4808: 4767: 4754: 4719: 4711: 4660: 4647: 4614: 4606: 4514: 4501: 4474: 4466: 4308:between two lines is defined in 4062:Sign (mathematics) § Angles 3409:This section is an excerpt from 3053:binary angular measurement (BAM) 2134:1, thus being normally omitted. 2008:curve or describing an object's 1927:of a simple polygon to mean the 1091:is supplementary to both angles 581: 557: 543: 48:A green angle formed by two red 7819:"Dimensionless units in the SI" 7538:Brownstein, K. R. (July 1997). 6915:Prando, Giacomo (August 2020). 6715:from the original on 2019-08-06 6685:from the original on 2019-06-30 6597:from the original on 2019-01-17 5642:astronomical coordinate systems 4429:Dot product and generalisations 4405:Bisection § Angle bisector 4399:Bisecting and trisecting angles 4357:The angle between a line and a 4160:In many contexts, an angle of − 4086:direction, and negative in the 3961:is the traditional function on 3158: turn and also known as a 3121:IEEE 754 recommended operations 2074:of the circle is the number of 1111:, we find the measure of angle 1087:and are congruent. Since angle 271:constant denoted by that symbol 193:". Both are connected with the 7672:Heiberg, Johan Ludvig (1908), 6739:The Growth of Physical Science 6575:Dimitrić, Radoslav M. (2012). 6360:Shute, Shirk & Porter 1960 5292: 5282: 5264: 5254: 5234:{\displaystyle {\mathcal {W}}} 5210:{\displaystyle {\mathcal {U}}} 5173: 5165: 5160: 5152: 5143: 5137: 5072: 5064: 5038: 5030: 4952: 4944: 4918: 4910: 4878: 4870: 4865: 4857: 4848: 4842: 4771: 4763: 4758: 4750: 4746: 4740: 4664: 4656: 4651: 4643: 4636: 4630: 4518: 4510: 4505: 4497: 4493: 4487: 3869: 3859: 3833: 3823: 3797: 3787: 3549:"pedagogically unsatisfying". 3417:Plane angle may be defined as 3164:. The quadrant is the unit in 2699:was historically defined as a 1891:when drawing regular polygons. 1032:All straight angles are equal. 201:, meaning "to bend" or "bow". 1: 7426:McKeague, Charles P. (2008). 6970:. Kendall Hunt. p. 262. 6464:. Blakiston's Son. p. 8. 5339:Angles in Riemannian geometry 4545:from their vector equations. 4379:(Gr. κισσός, ivy), biconvex; 3614:introduction of the constant 3465:the area of a circular sector 3411:Radian § Dimensional analysis 3111:) unit is implemented in the 2301:units, for some whole number 1665: 273:). Lower case Roman letters ( 7686:: Cambridge University Press 7397:"Mathwords: Reference Angle" 6964:Leonard, William J. (1999). 6703:Bonin, Walter (2016-01-11). 6475:Henderson & Taimina 2005 5612:. This system specifies the 5609:geographic coordinate system 5002:{\displaystyle \mathbf {v} } 4980:{\displaystyle \mathbf {u} } 4415:ancient Greek mathematicians 4066:Euclidean space § Angle 1885:supplementary exterior angle 1752:Internal and external angles 1184:§ Combining angle pairs 181: 67:is the figure formed by two 8036:Encyclopedia of Mathematics 7753:American Journal of Physics 7544:American Journal of Physics 7511:American Journal of Physics 7347:Angles are treated as units 6584:The Teaching of Mathematics 6505:D. Zwillinger, ed. (1995), 6495:, Dover Publications, 2007. 6493:Advanced Euclidean Geometry 5970:Argument (complex analysis) 5331:angles called canonical or 4107:Cartesian coordinate system 2261:Definition of 1 radian 1757:An angle that is part of a 1676:. Such angles are called a 1406:, associated with the verb 1218:consecutive interior angles 468:("obtuse" meaning "blunt"). 54:Cartesian coordinate system 8217: 8104:10.1088/0026-1394/22/1/002 7940:10.1088/0026-1394/53/2/840 7910:Modern Elementary Geometry 7802:10.1088/0026-1394/53/3/991 7692:Jacobs, Harold R. (1974), 7628:10.1088/0026-1394/47/6/R01 7595:10.1088/0026-1394/18/1/002 7332:"Boost.Units FAQ – 1.79.0" 7190:10.1088/0026-1394/53/3/998 6758:Murnaghan, Francis Dominic 6300:Godfrey & Siddons 1919 5926:Angle measuring instrument 4402: 4371: 4153:, and negative cycles are 4117:, while the other side or 4059: 4053: 4031:in mathematical formulas. 4012:system where the equation 3408: 3322:unit equal to about 2° or 2911:of a turn. 1 point = 2066:. The ratio of the length 1978:Angle measuring instrument 1975: 1923:Some authors use the name 1142:and are equal in measure. 1010:. They are abbreviated as 1008:vertically opposite angles 985: 626: 362: 175: 36: 29: 8047:Slocum, Jonathan (2007), 7854:10.1088/0026-1394/52/1/40 6942:10.1038/s41567-020-0997-3 4391:(Gr. κοίλη, a hollow) or 4082:count as positive in the 3222:was the unit used by the 2070:of the arc by the radius 1906:(meet at a single point). 1210:alternate interior angles 1206:alternate exterior angles 114:of an angle is called an 8075:Plane and Solid Geometry 8029:Sidorov, L. A. (2001) , 7987:10.1088/1681-7575/ac023f 7907:Moser, James M. (1971), 7899:10.1088/1681-7575/ac7bc2 7732:10.1088/1681-7575/abe0fc 7495:, Universal Publishers, 7309:10.1088/1681-7575/aa7160 7139:Mohr & Phillips 2015 6813:Mohr & Phillips 2015 5363:are tangent vectors and 5013:Angles between subspaces 4419:compass and straightedge 4279:angle addition postulate 4145:-axis rightward and the 3632:, is modified to become 3586:base unit of measurement 3457:1 SI radian = 1 m/m 2440:geographical coordinates 1929:explement exterior angle 1230:angle addition postulate 1130:. Therefore, both angle 1024:attributed the proof to 233:mathematical expressions 30:Not to be confused with 8183:Encyclopædia Britannica 8161:Encyclopædia Britannica 8121:New Century Mathematics 6742:. CUP Archive. p. 5635:, a given point on the 4965:spanned by the vectors 4561:) by the inner product 4318:trigonometric functions 4027:allows the omission of 2606:of a minute of arc and 2399:trigonometric functions 1595:of an angle equals the 1165:, often abbreviated as 1138:have measures equal to 1080:have measures equal to 1050:, the measure of angle 7467:Robert Baldwin Hayward 6423:"Supplementary Angles" 6385:"Complementary Angles" 6201: 6124: 6000:Exterior angle theorem 5531: 5325: 5305: 5235: 5211: 5181: 5079: 5045: 5003: 4981: 4959: 4925: 4889: 4782: 4675: 4587: 4529: 4354: 4218:are equal in measure). 4091: 3999: 3979: 3947: 3902: 3202:turn = 100 grad. 3115:scientific calculator 2562: = 21,600). 2262: 2227: 2120: 2049: 1945:The angle between two 1753: 1744:Polygon-related angles 1711: 1634: 1583: 1353: 1302: 1158: 1018:vertical angle theorem 983: 327: 235:, it is common to use 162: 56: 39:Angle (disambiguation) 8021:10.6028/jres.066B.012 7376:reference.wolfram.com 6705:"RE: WP-32S in 2016?" 6620:TheFreeDictionary.com 6202: 6134:), and the constants 6125: 6010:Great circle distance 5741:approximations only. 5720:as viewed from Earth. 5560:is the argument of a 5532: 5326: 5306: 5236: 5212: 5182: 5080: 5046: 5004: 4982: 4960: 4926: 4890: 4783: 4676: 4588: 4530: 4336: 4329:Angles between curves 4105:In a two-dimensional 4073: 4000: 3998:{\displaystyle \sin } 3980: 3948: 3903: 2703:-grad of arc along a 2436:"minute" and "second" 2273:. These are known as 2260: 2228: 2121: 2019:The measure of angle 2018: 1865:linear pair of angles 1751: 1709: 1678:linear pair of angles 1668:(i.e., have a common 1620: 1584: 1348:is the complement of 1319: 1303: 1224:Combining angle pairs 1148: 977: 509:radians) is called a 491:radians) is called a 437:radians) is called a 328: 148:History and etymology 47: 6898:Dimensional analysis 6734:Jeans, James Hopwood 6324:Wong & Wong 2009 6169: 6108: 5955:Angular acceleration 5931:Angles between flats 5674:with respect to the 5382: 5315: 5245: 5221: 5197: 5091: 5055: 5021: 4991: 4969: 4935: 4901: 4796: 4693: 4599: 4565: 4462: 4452:is related to their 3989: 3969: 3912: 3658: 3567:angular acceleration 3530:of a rolling wheel, 3404:Dimensional analysis 3234:equilateral triangle 3041:, also known as the 2314:dimensional analysis 2183: 2082: 2052:To measure an angle 1940:Plane-related angles 1889:Logo Turtle programs 1719:radians) are called 1686:cyclic quadrilateral 1660:supplementary angles 1658:radians) are called 1603:of its complement.) 1432: 1359:Complementary angles 1238: 1214:corresponding angles 487: turn (180° or 378:§ Signed angles 344:§ Signed angles 300: 166:, meaning "corner". 8096:1986Metro..22....1T 7979:2021Metro..58e3002Q 7932:2016Metro..53..840Q 7890:2022Metro..59e3001M 7845:2015Metro..52...40M 7794:2016Metro..53..991M 7765:1998AmJPh..66..814L 7724:2021Metro..58e2001L 7587:1982Metro..18....1E 7556:1997AmJPh..65..605B 7523:1936AmJPh...4..175B 7301:2017Metro..54..454Q 7182:2016Metro..53..998Q 7067:1993PhTea..31...84A 7055:The Physics Teacher 7032:1992PhTea..30..170O 7020:The Physics Teacher 6993:The Physics Teacher 6933:2020NatPh..16..888P 5985:Clock angle problem 5975:Astrological aspect 5703:small-angle formula 5697:. For example, the 5554:hyperbolic function 5345:Riemannian geometry 5335:between subspaces. 4685:inner product space 4555:inner product space 4113:is on the positive 4074:Measuring from the 3571:torsional stiffness 3510:centimeters, where 3441:is arc length, and 2969:mil" is defined as 2010:cumulative rotation 1826: radians, or ( 1721:explementary angles 419: turn (90° or 195:Proto-Indo-European 118:or simply "angle". 99:. Two intersecting 89:as they lie in the 6681:. blogs.msdn.com. 6515:Weisstein, Eric W. 6491:Johnson, Roger A. 6427:www.mathsisfun.com 6389:www.mathsisfun.com 6326:, pp. 161–163 6260:, pp. 177–178 6197: 6120: 6050:Transcendent angle 5944:standard deviation 5936:Angular statistics 5648:angular separation 5626:Greenwich meridian 5624:and (usually) the 5582:alternating series 5527: 5321: 5301: 5231: 5207: 5177: 5075: 5041: 4999: 4977: 4955: 4921: 4885: 4778: 4671: 4583: 4525: 4355: 4273:Related quantities 4232:(sometimes called 4092: 3995: 3975: 3943: 3898: 3482:dimensionless unit 3275:is a unit used by 2454: = 360) 2328:Number in one turn 2263: 2223: 2132:dimensionless unit 2116: 2050: 1783:radians, 180°, or 1773:Euclidean geometry 1754: 1712: 1635: 1579: 1577: 1354: 1298: 1159: 1121:) = 180° − 180° + 984: 471:An angle equal to 403:An angle equal to 396:("acute" meaning " 323: 227:Identifying angles 170:words include the 61:Euclidean geometry 57: 8130:978-0-19-800177-5 7703:978-0-7167-0456-0 7665:978-0-13-143748-7 7531:10.1119/1.1999110 7502:978-1-59942-822-2 7401:www.mathwords.com 7108:Lévy-Leblond 1998 7075:10.1119/1.2343667 7040:10.1119/1.2343500 7005:10.1119/1.2343535 6977:978-0-7872-5412-4 6917:"A spectral unit" 6762:Analytic Geometry 6675:Hargreaves, Shawn 6557:978-92-822-2272-0 6362:, pp. 25–27. 6347:Proposition I:13. 6270:Aboughantous 2010 6195: 6144:Lévy-Leblond 1998 5916: 5915: 5566:hyperbolic sector 5562:circular function 5522: 5521: 5324:{\displaystyle k} 5009:correspondingly. 4641: 4443:Euclidean vectors 4393:angulus lunularis 4310:rational geometry 4223:coterminal angles 4199:Equivalent angles 4141:, defined by the 4139:standard position 4056:Angle of rotation 4025:radian convention 3941: 3887: 3851: 3815: 3764: 3739: 3714: 3681: 3401: 3400: 3389:In old Arabia, a 3166:Euclid's Elements 3005:= 0.0009817... ≈ 2770:The astronomical 2218: 2205: 2099: 1818:sides add up to ( 1026:Thales of Miletus 1022:Eudemus of Rhodes 961: 960: 925:(180, 360)° 598:Name 572:), and straight ( 371:Individual angles 320: 221:Carpus of Antioch 213:Eudemus of Rhodes 121:Angle of rotation 16:(Redirected from 8208: 8187: 8179: 8165: 8144: 8142: 8141: 8133: 8115: 8078: 8069: 8068: 8066: 8057:, archived from 8043: 8025: 8023: 7998: 7972: 7951: 7914: 7903: 7901: 7883: 7858: 7856: 7838: 7813: 7776: 7743: 7706: 7687: 7668: 7650: 7639: 7606: 7569: 7567: 7534: 7505: 7476: 7464: 7458: 7448: 7442: 7441: 7423: 7417: 7416: 7414: 7412: 7393: 7387: 7386: 7384: 7382: 7368: 7362: 7359:Mohr et al. 2022 7356: 7350: 7349: 7344: 7342: 7327: 7321: 7320: 7294: 7274: 7268: 7262: 7256: 7253:Mohr et al. 2022 7250: 7244: 7241:Mohr et al. 2022 7238: 7232: 7226: 7217: 7211: 7202: 7201: 7175: 7155: 7142: 7136: 7130: 7128:Mohr et al. 2022 7085: 7079: 7078: 7050: 7044: 7043: 7015: 7009: 7008: 6988: 6982: 6981: 6961: 6955: 6954: 6944: 6912: 6906: 6905: 6892: 6886: 6880: 6874: 6872: 6868: 6854: 6848: 6846: 6839: 6837: 6836: 6833: 6830: 6806: 6800: 6794: 6788: 6786: 6772: 6766: 6765: 6754: 6748: 6747: 6730: 6724: 6723: 6721: 6720: 6700: 6694: 6693: 6691: 6690: 6671: 6665: 6664: 6662: 6661: 6639: 6630: 6629: 6627: 6626: 6612: 6606: 6605: 6603: 6602: 6596: 6581: 6572: 6566: 6565: 6550:(9th ed.), 6549: 6535: 6529: 6528: 6527: 6518:"Exterior Angle" 6510: 6502: 6496: 6489: 6478: 6472: 6466: 6465: 6455: 6449: 6443: 6437: 6436: 6434: 6433: 6419: 6413: 6408: 6399: 6398: 6396: 6395: 6381: 6375: 6369: 6363: 6357: 6348: 6346: 6333: 6327: 6321: 6315: 6309: 6303: 6297: 6288: 6282: 6273: 6267: 6261: 6251: 6245: 6240: 6234: 6229: 6212: 6209:Mohr et al. 2022 6206: 6204: 6203: 6198: 6196: 6191: 6183: 6178: 6177: 6129: 6127: 6126: 6121: 6104:), the notation 6098: 6086: 6084: 6077: 6025:Irrational angle 5965:Angular velocity 5960:Angular diameter 5911: 5910: 5906: 5899: 5898: 5894: 5887: 5886: 5882: 5881: 5862: 5861: 5857: 5850: 5849: 5845: 5838: 5837: 5833: 5832: 5811: 5810: 5806: 5799: 5798: 5794: 5793: 5755: 5727:at arm's length. 5695:angular diameter 5682:with respect to 5637:celestial sphere 5546:hyperbolic angle 5540:Hyperbolic angle 5536: 5534: 5533: 5528: 5523: 5520: 5516: 5515: 5514: 5505: 5504: 5495: 5494: 5477: 5473: 5472: 5471: 5462: 5461: 5452: 5451: 5434: 5433: 5432: 5431: 5422: 5421: 5412: 5411: 5398: 5333:principal angles 5330: 5328: 5327: 5322: 5310: 5308: 5307: 5302: 5291: 5290: 5263: 5262: 5240: 5238: 5237: 5232: 5230: 5229: 5216: 5214: 5213: 5208: 5206: 5205: 5186: 5184: 5183: 5178: 5176: 5172: 5163: 5159: 5150: 5146: 5122: 5118: 5114: 5106: 5084: 5082: 5081: 5076: 5071: 5050: 5048: 5047: 5042: 5037: 5008: 5006: 5005: 5000: 4998: 4986: 4984: 4983: 4978: 4976: 4964: 4962: 4961: 4956: 4951: 4930: 4928: 4927: 4922: 4917: 4894: 4892: 4891: 4886: 4881: 4877: 4868: 4864: 4855: 4851: 4827: 4823: 4819: 4811: 4787: 4785: 4784: 4779: 4774: 4770: 4761: 4757: 4730: 4726: 4722: 4714: 4680: 4678: 4677: 4672: 4667: 4663: 4654: 4650: 4639: 4617: 4609: 4592: 4590: 4589: 4584: 4534: 4532: 4531: 4526: 4521: 4517: 4508: 4504: 4477: 4469: 4409:Angle trisection 4374: 4373: 4295:is equal to the 4266: 4262: 4260: 4259: 4256: 4253: 4236:) for any angle 4084:counterclockwise 4078:, angles on the 4040: 4030: 4022: 4018: 4004: 4002: 4001: 3996: 3984: 3982: 3981: 3976: 3960: 3956: 3952: 3950: 3949: 3944: 3942: 3939: 3937: 3907: 3905: 3904: 3899: 3888: 3886: 3878: 3877: 3876: 3857: 3852: 3850: 3842: 3841: 3840: 3821: 3816: 3814: 3806: 3805: 3804: 3785: 3765: 3763: 3755: 3754: 3745: 3740: 3738: 3730: 3729: 3720: 3715: 3713: 3705: 3704: 3695: 3679: 3653: 3641: 3631: 3611: 3602: 3594:area of a circle 3579:angular momentum 3547: 3543: 3528:angular velocity 3525: 3521: 3517: 3513: 3509: 3495: 3491: 3479: 3462: 3458: 3454: 3444: 3440: 3436: 3432: 3374: 3372: 3371: 3368: 3365: 3340: 3338: 3337: 3334: 3331: 3327: 3310: 3308: 3307: 3304: 3301: 3297: 3256: 3252: 3250: 3249: 3246: 3243: 3201: 3199: 3198: 3195: 3192: 3185: 3183: 3182: 3179: 3176: 3175: 3157: 3155: 3154: 3151: 3148: 3110: 3104: 3089: 3080: 3078: 3077: 3074: 3071: 3020: 3018: 3017: 3014: 3011: 3004: 3002: 3001: 2998: 2995: 2994: 2984: 2982: 2981: 2978: 2975: 2942: 2926: 2924: 2923: 2920: 2917: 2910: 2908: 2907: 2904: 2901: 2866: 2864: 2863: 2860: 2857: 2853: 2847: 2845: 2844: 2841: 2838: 2831: 2829: 2828: 2825: 2822: 2815: 2813: 2812: 2809: 2806: 2805: 2789: 2787: 2786: 2783: 2780: 2749: 2722:and continental 2657: 2655: 2654: 2651: 2648: 2641: 2639: 2638: 2635: 2632: 2621: 2619: 2618: 2615: 2612: 2605: 2603: 2602: 2599: 2596: 2549: 2547: 2546: 2543: 2540: 2533: 2531: 2530: 2527: 2524: 2517: 2515: 2514: 2511: 2508: 2501: 2499: 2498: 2495: 2492: 2392: 2390: 2389: 2388: 2384: 2381: 2374: 2366: 2349: 2322: 2240: 2232: 2230: 2229: 2224: 2219: 2211: 2206: 2204: 2193: 2173:or 400 grad for 2168: 2157: 2155: 2154: 2153: 2148: 2145: 2125: 2123: 2122: 2117: 2112: 2100: 2092: 2047: 2045: 2043: 2042: 2037: 2034: 2024: 1991:equal in measure 1972:Measuring angles 1853: 1851: 1850: 1847: 1844: 1825: 1809: 1798: 1796: 1795: 1792: 1789: 1782: 1733:of the angle or 1725:conjugate angles 1718: 1657: 1653: 1651: 1650: 1647: 1644: 1588: 1586: 1585: 1580: 1578: 1552: 1528: 1512: 1511: 1493: 1492: 1482: 1468: 1467: 1449: 1448: 1438: 1394: 1392: 1391: 1388: 1385: 1384: 1376: 1374: 1373: 1370: 1367: 1307: 1305: 1304: 1299: 1297: 1277: 1257: 1129: 1086: 1071: 1060: 969: 968: 954:(200, 400) 948:(100, 200) 919:(90, 180)° 899: 896: 888: 885: 879: 871: 868: 861: 858: 852: 848: 846: 845: 842: 839: 829: 826: 822: 820: 819: 816: 813: 804: 801: 797: 795: 794: 791: 788: 778: 766: 761: 759: 757: 756: 753: 750: 740: 738: 736: 735: 732: 729: 720: 718: 716: 715: 712: 709: 702: 700: 699: 696: 693: 683: 681: 679: 678: 675: 672: 663: 661: 659: 658: 655: 652: 642: 595: 585: 561: 547: 508: 490: 486: 484: 483: 480: 477: 436: 434: 433: 430: 427: 426: 418: 416: 415: 412: 409: 354: 350: 339: 332: 330: 329: 324: 322: 321: 316: 305: 295: 268: 184: 178: 177: 165: 21: 8216: 8215: 8211: 8210: 8209: 8207: 8206: 8205: 8191: 8190: 8174: 8171: 8154:, ed. (1911), " 8150: 8139: 8137: 8131: 8118: 8081: 8072: 8064: 8062: 8061:on 27 June 2010 8046: 8028: 8001: 7954: 7917: 7913:, Prentice-Hall 7906: 7861: 7816: 7779: 7773:10.1119/1.18964 7746: 7709: 7704: 7691: 7671: 7666: 7653: 7642: 7609: 7572: 7565:10.1119/1.18616 7537: 7508: 7503: 7488: 7485: 7480: 7479: 7465: 7461: 7449: 7445: 7438: 7425: 7424: 7420: 7410: 7408: 7395: 7394: 7390: 7380: 7378: 7370: 7369: 7365: 7357: 7353: 7340: 7338: 7329: 7328: 7324: 7276: 7275: 7271: 7263: 7259: 7255:, pp. 8–9. 7251: 7247: 7239: 7235: 7227: 7220: 7212: 7205: 7166:(3): 998–1002. 7157: 7156: 7145: 7137: 7133: 7104:Brownstein 1997 7086: 7082: 7052: 7051: 7047: 7017: 7016: 7012: 6990: 6989: 6985: 6978: 6963: 6962: 6958: 6914: 6913: 6909: 6894: 6893: 6889: 6881: 6877: 6870: 6860: 6855: 6851: 6834: 6831: 6828: 6827: 6825: 6820: 6807: 6803: 6795: 6791: 6778: 6773: 6769: 6756: 6755: 6751: 6732: 6731: 6727: 6718: 6716: 6702: 6701: 6697: 6688: 6686: 6673: 6672: 6668: 6659: 6657: 6641: 6640: 6633: 6624: 6622: 6614: 6613: 6609: 6600: 6598: 6594: 6579: 6574: 6573: 6569: 6558: 6547: 6541:(20 May 2019), 6537: 6536: 6532: 6513: 6512: 6504: 6503: 6499: 6490: 6481: 6473: 6469: 6457: 6456: 6452: 6444: 6440: 6431: 6429: 6421: 6420: 6416: 6409: 6402: 6393: 6391: 6383: 6382: 6378: 6370: 6366: 6358: 6351: 6335: 6334: 6330: 6322: 6318: 6310: 6306: 6298: 6291: 6283: 6276: 6268: 6264: 6252: 6248: 6241: 6237: 6230: 6226: 6221: 6216: 6215: 6184: 6167: 6166: 6160: 6140:Brownstein 1997 6106: 6105: 6099: 6095: 6090: 6089: 6080: 6078: 6074: 6069: 6064: 6045:Spherical angle 6020:Inscribed angle 5990:Decimal degrees 5921: 5908: 5904: 5903: 5896: 5892: 5891: 5884: 5879: 5878: 5877: 5859: 5855: 5854: 5847: 5843: 5842: 5835: 5830: 5829: 5828: 5808: 5804: 5803: 5796: 5791: 5790: 5789: 5746:right ascension 5678:as well as the 5628:as references. 5600: 5578:infinite series 5570:circular sector 5542: 5506: 5496: 5483: 5482: 5478: 5463: 5453: 5440: 5439: 5435: 5423: 5413: 5400: 5399: 5380: 5379: 5371: 5341: 5313: 5312: 5243: 5242: 5219: 5218: 5195: 5194: 5164: 5151: 5130: 5126: 5098: 5094: 5089: 5088: 5053: 5052: 5019: 5018: 5015: 4989: 4988: 4967: 4966: 4933: 4932: 4899: 4898: 4869: 4856: 4835: 4831: 4803: 4799: 4794: 4793: 4762: 4749: 4706: 4702: 4691: 4690: 4655: 4642: 4597: 4596: 4563: 4562: 4551: 4509: 4496: 4460: 4459: 4435:Euclidean space 4431: 4411: 4403:Main articles: 4401: 4352: 4348: 4344: 4331: 4275: 4264: 4263:turn, 180°, or 4257: 4254: 4251: 4250: 4248: 4230:reference angle 4201: 4131:negative angles 4123:positive angles 4068: 4058: 4052: 4047: 4046: 4041:dimension, and 4038: 4028: 4020: 4013: 3987: 3986: 3985:can be denoted 3967: 3966: 3958: 3954: 3910: 3909: 3879: 3868: 3858: 3843: 3832: 3822: 3807: 3796: 3786: 3756: 3746: 3731: 3721: 3706: 3696: 3656: 3655: 3651: 3633: 3623: 3620: 3607: 3597: 3545: 3531: 3523: 3519: 3515: 3511: 3501: 3493: 3489: 3467: 3460: 3456: 3446: 3442: 3438: 3434: 3418: 3414: 3406: 3369: 3366: 3363: 3362: 3360: 3335: 3332: 3329: 3328: 3325: 3323: 3305: 3302: 3299: 3298: 3295: 3293: 3254: 3247: 3244: 3241: 3240: 3238: 3196: 3193: 3190: 3189: 3187: 3180: 3177: 3173: 3172: 3171: 3169: 3152: 3149: 3146: 3145: 3143: 3108: 3102: 3087: 3075: 3072: 3069: 3068: 3066: 3015: 3012: 3009: 3008: 3006: 2999: 2996: 2992: 2990: 2989: 2987: 2979: 2976: 2973: 2972: 2970: 2937: 2921: 2918: 2915: 2914: 2912: 2905: 2902: 2899: 2898: 2896: 2873:(compass) point 2861: 2858: 2855: 2854: 2851: 2849: 2842: 2839: 2836: 2835: 2833: 2826: 2823: 2820: 2819: 2817: 2810: 2807: 2803: 2802: 2801: 2799: 2784: 2781: 2778: 2777: 2775: 2747: 2652: 2649: 2646: 2645: 2643: 2636: 2633: 2630: 2629: 2627: 2616: 2613: 2610: 2609: 2607: 2600: 2597: 2594: 2593: 2591: 2558:of the Earth. ( 2544: 2541: 2538: 2537: 2535: 2528: 2525: 2522: 2521: 2519: 2512: 2509: 2506: 2505: 2503: 2496: 2493: 2490: 2489: 2487: 2386: 2385: 2382: 2379: 2378: 2376: 2372: 2371:. One turn is 2 2364: 2344: 2285:(rad), and the 2255: 2236: 2197: 2181: 2180: 2163: 2151: 2149: 2146: 2141: 2140: 2138: 2080: 2079: 2073: 2069: 2057: 2038: 2035: 2030: 2029: 2027: 2026: 2020: 1980: 1974: 1942: 1848: 1845: 1842: 1841: 1839: 1838: − 2) 1823: 1822: − 2) 1807: 1793: 1790: 1787: 1786: 1784: 1780: 1769:concave polygon 1746: 1741: 1716: 1655: 1654:turn, 180°, or 1648: 1645: 1642: 1641: 1639: 1628: 1624: 1576: 1575: 1551: 1526: 1525: 1503: 1484: 1481: 1459: 1440: 1430: 1429: 1389: 1386: 1382: 1381: 1380: 1378: 1371: 1368: 1365: 1364: 1362: 1351: 1347: 1343: 1335: 1331: 1327: 1236: 1235: 1226: 1202:interior angles 1198:exterior angles 1190: 1163:Adjacent angles 1117:180° − (180° − 1116: 1081: 1066: 1055: 1004:opposite angles 1000:vertical angles 991: 972: 966: 965: 894: 891: 883: 877: 874: 866: 864: 856: 850: 843: 840: 837: 836: 834: 832: 824: 817: 814: 811: 810: 808: 807: 799: 792: 789: 786: 785: 783: 781: 776: 764: 754: 751: 748: 747: 745: 743: 733: 730: 727: 726: 724: 723: 713: 710: 707: 706: 704: 697: 694: 691: 690: 688: 686: 676: 673: 670: 669: 667: 666: 656: 653: 650: 649: 647: 645: 640: 613:straight angle 593: 592: 591: 590: 589: 586: 578: 577: 575: 571: 567: 562: 554: 553: 548: 506: 488: 481: 478: 475: 474: 472: 431: 428: 424: 423: 422: 420: 413: 410: 407: 406: 404: 373: 368: 361: 352: 348: 337: 298: 297: 293: 264: 258: 254: 250: 246: 242: 229: 156:comes from the 150: 138:negative number 136:, and may be a 116:angular measure 96:dihedral angles 42: 35: 28: 23: 22: 15: 12: 11: 5: 8214: 8212: 8204: 8203: 8193: 8192: 8189: 8188: 8170: 8169:External links 8167: 8152:Chisholm, Hugh 8135: 8134: 8129: 8116: 8079: 8070: 8044: 8026: 7999: 7952: 7926:(2): 840–845. 7915: 7904: 7859: 7814: 7788:(3): 991–997. 7777: 7759:(9): 814–815. 7744: 7707: 7702: 7689: 7669: 7664: 7651: 7640: 7622:(6): R41–R51. 7607: 7570: 7550:(7): 605–614. 7535: 7517:(4): 175–179. 7506: 7501: 7484: 7481: 7478: 7477: 7459: 7443: 7437:978-0495382607 7436: 7418: 7388: 7363: 7351: 7322: 7285:(4): 454–460. 7269: 7257: 7245: 7233: 7218: 7203: 7143: 7131: 7088:Brinsmade 1936 7080: 7045: 7026:(3): 170–171. 7010: 6999:(5): 260–261. 6983: 6976: 6956: 6921:Nature Physics 6907: 6887: 6885:, p. 137. 6875: 6849: 6801: 6799:, p. 151. 6789: 6767: 6749: 6725: 6695: 6666: 6631: 6616:"angular unit" 6607: 6590:(2): 133–140. 6567: 6556: 6530: 6497: 6479: 6477:, p. 104. 6467: 6461:Plane Geometry 6450: 6438: 6414: 6400: 6376: 6374:, p. 255. 6364: 6349: 6328: 6316: 6304: 6289: 6274: 6262: 6246: 6235: 6223: 6222: 6220: 6217: 6214: 6213: 6194: 6190: 6187: 6181: 6176: 6158: 6119: 6116: 6113: 6102:Brinsmade 1936 6092: 6091: 6088: 6087: 6071: 6070: 6068: 6065: 6063: 6062: 6057: 6052: 6047: 6042: 6037: 6032: 6027: 6022: 6017: 6012: 6007: 6002: 5997: 5995:Dihedral angle 5992: 5987: 5982: 5977: 5972: 5967: 5962: 5957: 5952: 5950:Angle bisector 5947: 5933: 5928: 5922: 5920: 5917: 5914: 5913: 5901: 5889: 5875: 5872: 5869: 5865: 5864: 5852: 5840: 5826: 5823: 5820: 5816: 5815: 5813: 5801: 5787: 5784: 5781: 5777: 5776: 5773: 5770: 5767: 5764: 5759: 5744:In astronomy, 5735: 5734: 5731: 5728: 5721: 5664:vertical angle 5599: 5596: 5586:Leonhard Euler 5558:circular angle 5541: 5538: 5526: 5519: 5513: 5509: 5503: 5499: 5493: 5490: 5486: 5481: 5476: 5470: 5466: 5460: 5456: 5450: 5447: 5443: 5438: 5430: 5426: 5420: 5416: 5410: 5407: 5403: 5396: 5393: 5390: 5387: 5367: 5340: 5337: 5320: 5300: 5297: 5294: 5289: 5284: 5281: 5278: 5275: 5272: 5269: 5266: 5261: 5256: 5253: 5250: 5228: 5204: 5175: 5171: 5167: 5162: 5158: 5154: 5149: 5145: 5142: 5139: 5136: 5133: 5129: 5125: 5121: 5117: 5113: 5109: 5105: 5101: 5097: 5074: 5070: 5066: 5063: 5060: 5040: 5036: 5032: 5029: 5026: 5014: 5011: 4997: 4975: 4954: 4950: 4946: 4943: 4940: 4920: 4916: 4912: 4909: 4906: 4884: 4880: 4876: 4872: 4867: 4863: 4859: 4854: 4850: 4847: 4844: 4841: 4838: 4834: 4830: 4826: 4822: 4818: 4814: 4810: 4806: 4802: 4777: 4773: 4769: 4765: 4760: 4756: 4752: 4748: 4745: 4742: 4739: 4736: 4733: 4729: 4725: 4721: 4717: 4713: 4709: 4705: 4701: 4698: 4670: 4666: 4662: 4658: 4653: 4649: 4645: 4638: 4635: 4632: 4629: 4626: 4623: 4620: 4616: 4612: 4608: 4604: 4582: 4579: 4576: 4573: 4570: 4550: 4547: 4539:normal vectors 4524: 4520: 4516: 4512: 4507: 4503: 4499: 4495: 4492: 4489: 4486: 4483: 4480: 4476: 4472: 4468: 4430: 4427: 4423:Pierre Wantzel 4400: 4397: 4350: 4346: 4342: 4330: 4327: 4326: 4325: 4320:, such as the 4314: 4300: 4274: 4271: 4270: 4269: 4226: 4219: 4200: 4197: 4054:Main article: 4051: 4050:Signed angles 4048: 3994: 3974: 3936: 3932: 3929: 3926: 3923: 3920: 3917: 3897: 3894: 3891: 3885: 3882: 3875: 3871: 3867: 3864: 3861: 3855: 3849: 3846: 3839: 3835: 3831: 3828: 3825: 3819: 3813: 3810: 3803: 3799: 3795: 3792: 3789: 3783: 3780: 3777: 3774: 3771: 3768: 3762: 3759: 3753: 3749: 3743: 3737: 3734: 3728: 3724: 3718: 3712: 3709: 3703: 3699: 3693: 3690: 3687: 3684: 3678: 3675: 3672: 3669: 3666: 3663: 3618: 3459:= 1. However, 3415: 3407: 3405: 3402: 3399: 3398: 3387: 3384: 3381: 3377: 3376: 3353: 3350: 3347: 3343: 3342: 3312: 3290: 3287: 3281: 3280: 3269: 3266: 3263: 3259: 3258: 3216: 3213: 3210: 3204: 3203: 3136: 3133: 3130: 3124: 3123: 3097: 3094: 3091: 3084: 3083: 3035: 3032: 3029: 3023: 3022: 2946: 2943: 2935: 2929: 2928: 2881: 2878: 2875: 2869: 2868: 2796:second of time 2792:minute of time 2768: 2765: 2762: 2756: 2755: 2740: 2737: 2734: 2728: 2727: 2681:, also called 2675: 2672: 2669: 2663: 2662: 2576: 2573: 2570: 2564: 2563: 2502:of a degree = 2468: 2465: 2462: 2456: 2455: 2419: 2416: 2413: 2407: 2406: 2363: = 2 2353: 2350: 2342: 2336: 2335: 2332: 2329: 2326: 2254: 2251: 2249:is unaltered. 2222: 2217: 2214: 2209: 2203: 2200: 2196: 2191: 2188: 2115: 2111: 2108: 2105: 2098: 2095: 2090: 2087: 2078:in the angle: 2071: 2067: 2053: 1973: 1970: 1969: 1968: 1964: 1963:to the planes. 1956:dihedral angle 1953:) is called a 1941: 1938: 1937: 1936: 1925:exterior angle 1921: 1918: 1907: 1892: 1860:exterior angle 1855: 1764:interior angle 1759:simple polygon 1745: 1742: 1740: 1739: 1713: 1636: 1626: 1622: 1574: 1571: 1568: 1565: 1562: 1559: 1556: 1553: 1550: 1547: 1544: 1541: 1538: 1535: 1532: 1529: 1527: 1524: 1521: 1518: 1515: 1510: 1506: 1502: 1499: 1496: 1491: 1487: 1483: 1480: 1477: 1474: 1471: 1466: 1462: 1458: 1455: 1452: 1447: 1443: 1439: 1437: 1417:of the angle. 1377:turn, 90°, or 1355: 1349: 1345: 1341: 1333: 1329: 1325: 1296: 1293: 1290: 1286: 1283: 1280: 1276: 1273: 1270: 1266: 1263: 1260: 1256: 1253: 1250: 1246: 1243: 1225: 1222: 1189: 1188: 1160: 1040: 1039: 1036: 1033: 995: 971: 962: 959: 958: 955: 952: 949: 946: 943: 942:(0, 100) 940: 937: 930: 929: 926: 923: 920: 917: 914: 913:(0, 90)° 911: 908: 901: 900: 889: 872: 862: 830: 805: 779: 774: 768: 767: 762: 741: 721: 684: 664: 643: 638: 631: 630: 625: 621: 620: 617: 614: 611: 608: 605: 602: 599: 587: 580: 579: 573: 569: 565: 563: 556: 555: 549: 542: 541: 540: 539: 538: 534: 533: 526: 515:complete angle 503: 496: 493:straight angle 469: 462: 401: 390: 372: 369: 360: 357: 319: 315: 312: 309: 256: 252: 248: 244: 240: 228: 225: 149: 146: 132:length to its 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 8213: 8202: 8199: 8198: 8196: 8185: 8184: 8178: 8177:"Angle" 8173: 8172: 8168: 8166: 8163: 8162: 8157: 8153: 8148: 8147:public domain 8132: 8126: 8122: 8117: 8113: 8109: 8105: 8101: 8097: 8093: 8089: 8085: 8080: 8076: 8071: 8060: 8056: 8052: 8051: 8045: 8042: 8038: 8037: 8032: 8027: 8022: 8017: 8013: 8009: 8005: 8000: 7996: 7992: 7988: 7984: 7980: 7976: 7971: 7966: 7963:(5): 053002. 7962: 7958: 7953: 7949: 7945: 7941: 7937: 7933: 7929: 7925: 7921: 7916: 7912: 7911: 7905: 7900: 7895: 7891: 7887: 7882: 7877: 7874:(5): 053001. 7873: 7869: 7865: 7860: 7855: 7850: 7846: 7842: 7837: 7832: 7828: 7824: 7820: 7815: 7811: 7807: 7803: 7799: 7795: 7791: 7787: 7783: 7778: 7774: 7770: 7766: 7762: 7758: 7754: 7750: 7745: 7741: 7737: 7733: 7729: 7725: 7721: 7718:(5): 052001. 7717: 7713: 7708: 7705: 7699: 7695: 7690: 7685: 7681: 7680: 7675: 7670: 7667: 7661: 7657: 7652: 7648: 7647: 7641: 7637: 7633: 7629: 7625: 7621: 7617: 7613: 7608: 7604: 7600: 7596: 7592: 7588: 7584: 7580: 7576: 7571: 7566: 7561: 7557: 7553: 7549: 7545: 7541: 7536: 7532: 7528: 7524: 7520: 7516: 7512: 7507: 7504: 7498: 7494: 7493: 7487: 7486: 7482: 7475:, chapter six 7474: 7473: 7468: 7463: 7460: 7457:, p. 178 7456: 7452: 7451:Chisholm 1911 7447: 7444: 7439: 7433: 7429: 7422: 7419: 7406: 7402: 7398: 7392: 7389: 7377: 7373: 7367: 7364: 7360: 7355: 7352: 7348: 7337: 7336:www.boost.org 7333: 7326: 7323: 7318: 7314: 7310: 7306: 7302: 7298: 7293: 7288: 7284: 7280: 7273: 7270: 7266: 7261: 7258: 7254: 7249: 7246: 7242: 7237: 7234: 7230: 7225: 7223: 7219: 7215: 7210: 7208: 7204: 7199: 7195: 7191: 7187: 7183: 7179: 7174: 7169: 7165: 7161: 7154: 7152: 7150: 7148: 7144: 7140: 7135: 7132: 7129: 7125: 7121: 7117: 7113: 7109: 7105: 7101: 7097: 7093: 7089: 7084: 7081: 7076: 7072: 7068: 7064: 7060: 7056: 7049: 7046: 7041: 7037: 7033: 7029: 7025: 7021: 7014: 7011: 7006: 7002: 6998: 6994: 6987: 6984: 6979: 6973: 6969: 6968: 6960: 6957: 6952: 6948: 6943: 6938: 6934: 6930: 6926: 6922: 6918: 6911: 6908: 6904: 6900: 6899: 6891: 6888: 6884: 6879: 6876: 6867: 6863: 6858: 6853: 6850: 6845: 6842: 6823: 6819:of a sector ( 6818: 6814: 6810: 6805: 6802: 6798: 6793: 6790: 6785: 6781: 6776: 6771: 6768: 6763: 6759: 6753: 6750: 6745: 6741: 6740: 6735: 6729: 6726: 6714: 6710: 6706: 6699: 6696: 6684: 6680: 6676: 6670: 6667: 6656:on 2008-06-28 6655: 6651: 6649: 6644: 6638: 6636: 6632: 6621: 6617: 6611: 6608: 6593: 6589: 6585: 6578: 6571: 6568: 6563: 6559: 6553: 6546: 6545: 6540: 6534: 6531: 6525: 6524: 6519: 6516: 6508: 6501: 6498: 6494: 6488: 6486: 6484: 6480: 6476: 6471: 6468: 6463: 6462: 6454: 6451: 6448:, p. 97. 6447: 6442: 6439: 6428: 6424: 6418: 6415: 6412: 6411:Chisholm 1911 6407: 6405: 6401: 6390: 6386: 6380: 6377: 6373: 6368: 6365: 6361: 6356: 6354: 6350: 6344: 6343: 6338: 6332: 6329: 6325: 6320: 6317: 6314:, p. 71. 6313: 6308: 6305: 6301: 6296: 6294: 6290: 6287:, p. 41. 6286: 6281: 6279: 6275: 6272:, p. 18. 6271: 6266: 6263: 6259: 6255: 6254:Chisholm 1911 6250: 6247: 6244: 6239: 6236: 6233: 6228: 6225: 6218: 6210: 6188: 6185: 6179: 6164: 6157: 6153: 6149: 6145: 6141: 6137: 6133: 6114: 6103: 6097: 6094: 6083: 6076: 6073: 6066: 6061: 6058: 6056: 6053: 6051: 6048: 6046: 6043: 6041: 6038: 6036: 6033: 6031: 6030:Phase (waves) 6028: 6026: 6023: 6021: 6018: 6016: 6013: 6011: 6008: 6006: 6003: 6001: 5998: 5996: 5993: 5991: 5988: 5986: 5983: 5981: 5980:Central angle 5978: 5976: 5973: 5971: 5968: 5966: 5963: 5961: 5958: 5956: 5953: 5951: 5948: 5945: 5941: 5937: 5934: 5932: 5929: 5927: 5924: 5923: 5918: 5902: 5890: 5876: 5873: 5870: 5867: 5866: 5853: 5841: 5827: 5824: 5821: 5818: 5817: 5814: 5802: 5788: 5785: 5782: 5779: 5778: 5774: 5771: 5768: 5765: 5763: 5760: 5757: 5756: 5753: 5751: 5747: 5742: 5740: 5739:rule of thumb 5732: 5729: 5726: 5725:little finger 5722: 5719: 5715: 5711: 5710: 5709: 5706: 5704: 5700: 5696: 5692: 5691:apparent size 5687: 5685: 5681: 5677: 5673: 5669: 5665: 5660: 5658: 5654: 5650: 5649: 5644: 5643: 5638: 5634: 5629: 5627: 5623: 5619: 5615: 5611: 5610: 5605: 5597: 5595: 5593: 5592: 5587: 5583: 5579: 5575: 5571: 5567: 5563: 5559: 5555: 5551: 5547: 5539: 5537: 5524: 5517: 5511: 5507: 5501: 5497: 5491: 5488: 5484: 5479: 5474: 5468: 5464: 5458: 5454: 5448: 5445: 5441: 5436: 5428: 5424: 5418: 5414: 5408: 5405: 5401: 5394: 5391: 5388: 5385: 5377: 5375: 5370: 5366: 5362: 5358: 5354: 5350: 5349:metric tensor 5346: 5338: 5336: 5334: 5318: 5298: 5295: 5279: 5276: 5273: 5270: 5267: 5251: 5248: 5192: 5191:Hilbert space 5187: 5147: 5140: 5134: 5131: 5127: 5123: 5119: 5107: 5095: 5086: 5061: 5058: 5027: 5024: 5012: 5010: 4941: 4938: 4907: 4904: 4895: 4882: 4852: 4845: 4839: 4836: 4832: 4828: 4824: 4812: 4800: 4791: 4788: 4775: 4743: 4737: 4734: 4731: 4727: 4715: 4703: 4699: 4696: 4688: 4686: 4683:In a complex 4681: 4668: 4633: 4627: 4624: 4621: 4610: 4594: 4577: 4574: 4571: 4560: 4556: 4549:Inner product 4548: 4546: 4544: 4540: 4535: 4522: 4490: 4484: 4481: 4478: 4470: 4457: 4455: 4451: 4447: 4444: 4440: 4436: 4428: 4426: 4424: 4420: 4416: 4410: 4406: 4398: 4396: 4395:, biconcave. 4394: 4390: 4386: 4382: 4378: 4368: 4364: 4360: 4340: 4335: 4328: 4324:of the angle. 4323: 4319: 4315: 4311: 4307: 4306: 4301: 4298: 4294: 4290: 4289: 4284: 4283: 4282: 4280: 4272: 4247: 4243: 4239: 4235: 4234:related angle 4231: 4227: 4224: 4220: 4217: 4213: 4212: 4207: 4203: 4202: 4198: 4196: 4193: 4189: 4185: 4180: 4178: 4177:normal vector 4174: 4169: 4167: 4163: 4158: 4156: 4152: 4151:anticlockwise 4148: 4144: 4140: 4136: 4132: 4128: 4124: 4120: 4119:terminal side 4116: 4112: 4108: 4103: 4101: 4097: 4089: 4085: 4081: 4077: 4072: 4067: 4063: 4057: 4049: 4044: 4036: 4032: 4026: 4016: 4011: 4006: 3992: 3972: 3964: 3934: 3930: 3927: 3924: 3921: 3918: 3915: 3895: 3892: 3889: 3883: 3880: 3873: 3865: 3862: 3853: 3847: 3844: 3837: 3829: 3826: 3817: 3811: 3808: 3801: 3793: 3790: 3781: 3778: 3775: 3772: 3769: 3766: 3760: 3757: 3751: 3747: 3741: 3735: 3732: 3726: 3722: 3716: 3710: 3707: 3701: 3697: 3691: 3688: 3685: 3682: 3676: 3673: 3670: 3667: 3664: 3661: 3649: 3645: 3644:Taylor series 3640: 3636: 3630: 3626: 3621: 3617: 3610: 3604: 3601: 3595: 3591: 3590:base quantity 3587: 3582: 3580: 3576: 3572: 3569:(rad/s), and 3568: 3564: 3563:angular speed 3560: 3559:angle measure 3555: 3550: 3542: 3538: 3534: 3529: 3508: 3504: 3497: 3487: 3483: 3478: 3474: 3470: 3466: 3453: 3449: 3431: 3427: 3423: 3422: 3412: 3403: 3396: 3392: 3388: 3385: 3382: 3379: 3378: 3358: 3357:diameter part 3354: 3351: 3348: 3346:diameter part 3345: 3344: 3321: 3317: 3313: 3291: 3288: 3286: 3283: 3282: 3278: 3274: 3270: 3267: 3264: 3261: 3260: 3236: 3235: 3232:angle of the 3229: 3225: 3221: 3217: 3214: 3211: 3209: 3206: 3205: 3167: 3163: 3162: 3141: 3137: 3134: 3131: 3129: 3126: 3125: 3122: 3118: 3114: 3106: 3101:multiples of 3098: 3095: 3092: 3086: 3085: 3082: 3062: 3058: 3054: 3050: 3046: 3045: 3044:binary radian 3040: 3039:binary degree 3036: 3033: 3030: 3028: 3027:binary degree 3025: 3024: 2968: 2963: 2962:approximately 2959: 2955: 2951: 2947: 2944: 2941: 2936: 2934: 2931: 2930: 2894: 2890: 2886: 2882: 2879: 2876: 2874: 2871: 2870: 2848: turn = 2832: quad = 2797: 2793: 2773: 2769: 2766: 2763: 2761: 2758: 2757: 2753: 2745: 2741: 2738: 2735: 2733: 2730: 2729: 2725: 2721: 2720:triangulation 2717: 2713: 2712:nautical mile 2710: 2706: 2702: 2698: 2694: 2690: 2689: 2684: 2680: 2676: 2673: 2670: 2668: 2665: 2664: 2661: 2625: 2622:of a degree ( 2589: 2585: 2581: 2580:second of arc 2577: 2574: 2571: 2569: 2566: 2565: 2561: 2557: 2553: 2552:nautical mile 2485: 2481: 2477: 2473: 2472:minute of arc 2469: 2466: 2463: 2461: 2458: 2457: 2453: 2449: 2445: 2441: 2437: 2432: 2428: 2424: 2420: 2417: 2414: 2412: 2409: 2408: 2404: 2400: 2396: 2370: 2362: 2358: 2354: 2351: 2348: 2343: 2341: 2338: 2337: 2333: 2330: 2327: 2324: 2323: 2320: 2317: 2315: 2311: 2306: 2304: 2300: 2296: 2292: 2288: 2284: 2280: 2276: 2275:angular units 2272: 2268: 2259: 2252: 2250: 2248: 2244: 2239: 2235:The value of 2233: 2220: 2215: 2212: 2207: 2201: 2198: 2194: 2189: 2186: 2178: 2176: 2172: 2166: 2161: 2144: 2135: 2133: 2129: 2113: 2096: 2093: 2088: 2085: 2077: 2065: 2061: 2056: 2041: 2033: 2023: 2017: 2013: 2011: 2007: 2003: 1999: 1994: 1992: 1988: 1985: 1979: 1971: 1965: 1962: 1958: 1957: 1952: 1948: 1944: 1943: 1939: 1934: 1930: 1926: 1922: 1919: 1916: 1912: 1911:extended side 1908: 1905: 1901: 1897: 1893: 1890: 1886: 1882: 1878: 1874: 1870: 1866: 1862: 1861: 1856: 1837: 1833: 1829: 1821: 1817: 1813: 1805: 1804:quadrilateral 1802: 1778: 1774: 1770: 1766: 1765: 1761:is called an 1760: 1756: 1755: 1750: 1743: 1738: 1737:of an angle. 1736: 1732: 1726: 1722: 1714: 1708: 1704: 1702: 1699: 1696: 1694: 1693:tangent lines 1689: 1687: 1683: 1682:parallelogram 1679: 1675: 1674:straight line 1671: 1667: 1661: 1637: 1632: 1631:supplementary 1619: 1615: 1613: 1609: 1604: 1602: 1598: 1594: 1589: 1572: 1569: 1566: 1563: 1560: 1557: 1554: 1548: 1545: 1542: 1539: 1536: 1533: 1530: 1522: 1519: 1516: 1513: 1508: 1504: 1500: 1497: 1494: 1489: 1485: 1478: 1475: 1472: 1469: 1464: 1460: 1456: 1453: 1450: 1445: 1441: 1427: 1423: 1418: 1416: 1411: 1409: 1405: 1399: 1360: 1357: 1356: 1339: 1323: 1322:complementary 1318: 1314: 1311: 1308: 1281: 1278: 1261: 1258: 1241: 1233: 1231: 1223: 1221: 1219: 1215: 1211: 1207: 1203: 1199: 1195: 1186: 1185: 1180: 1176: 1175:supplementary 1172: 1171:complementary 1168: 1164: 1161: 1157:are adjacent. 1156: 1152: 1147: 1143: 1141: 1137: 1133: 1128: 1124: 1120: 1114: 1110: 1106: 1102: 1098: 1094: 1090: 1085: 1079: 1075: 1072:. Both angle 1070: 1064: 1059: 1053: 1049: 1045: 1037: 1034: 1031: 1030: 1029: 1027: 1023: 1019: 1013: 1012:vert. opp. ∠s 1009: 1005: 1001: 997: 996: 994: 989: 981: 976: 964:Vertical and 963: 956: 953: 950: 947: 944: 941: 938: 936: 935: 932: 931: 927: 924: 921: 918: 915: 912: 909: 907: 906: 903: 902: 897: 890: 886: 880: 873: 869: 863: 859: 853: 831: 827: 806: 802: 780: 775: 773: 770: 769: 763: 742: 722: 685: 665: 644: 639: 637: 636: 633: 632: 629: 623: 622: 618: 616:reflex angle 615: 612: 610:obtuse angle 609: 606: 603: 600: 597: 596: 584: 560: 552: 546: 537: 531: 530:oblique angle 527: 524: 520: 516: 512: 504: 501: 497: 494: 470: 467: 463: 460: 459: 458:perpendicular 454: 453: 448: 447: 442: 441: 402: 399: 395: 391: 388: 384: 383: 382: 380: 379: 370: 366: 358: 356: 346: 345: 334: 317: 291: 286: 284: 280: 276: 272: 267: 262: 238: 237:Greek letters 234: 226: 224: 222: 218: 217:straight line 214: 210: 206: 202: 200: 196: 192: 188: 183: 173: 169: 164: 159: 155: 147: 145: 143: 139: 135: 131: 127: 123: 122: 117: 113: 108: 106: 102: 98: 97: 92: 88: 87: 82: 81: 76: 75: 71:, called the 70: 66: 62: 55: 51: 46: 40: 33: 19: 8181: 8159: 8136: 8120: 8087: 8083: 8074: 8063:, retrieved 8059:the original 8049: 8034: 8011: 8007: 7960: 7956: 7923: 7919: 7909: 7871: 7867: 7829:(1): 40–47. 7826: 7822: 7785: 7781: 7756: 7752: 7715: 7711: 7693: 7678: 7674:Heath, T. L. 7655: 7645: 7619: 7615: 7578: 7574: 7547: 7543: 7514: 7510: 7491: 7483:Bibliography 7471: 7462: 7455:Heiberg 1908 7446: 7428:Trigonometry 7427: 7421: 7409:. Retrieved 7400: 7391: 7379:. Retrieved 7375: 7366: 7361:, p. 3. 7354: 7346: 7339:. Retrieved 7335: 7325: 7282: 7278: 7272: 7265:Quincey 2021 7260: 7248: 7243:, p. 6. 7236: 7229:Torrens 1986 7214:Quincey 2016 7163: 7159: 7134: 7124:Leonard 2021 7120:Quincey 2021 7100:Torrens 1986 7083: 7061:(2): 84–87. 7058: 7054: 7048: 7023: 7019: 7013: 6996: 6992: 6986: 6966: 6959: 6924: 6920: 6910: 6902: 6897: 6890: 6878: 6865: 6861: 6852: 6843: 6840: 6821: 6816: 6809:Quincey 2016 6804: 6792: 6783: 6779: 6770: 6764:. p. 2. 6761: 6752: 6738: 6728: 6717:. Retrieved 6708: 6698: 6687:. Retrieved 6669: 6658:. Retrieved 6654:the original 6646: 6623:. Retrieved 6619: 6610: 6599:. Retrieved 6587: 6583: 6570: 6543: 6533: 6521: 6511:as cited in 6506: 6500: 6492: 6470: 6460: 6453: 6441: 6430:. Retrieved 6426: 6417: 6392:. Retrieved 6388: 6379: 6367: 6342:The Elements 6341: 6331: 6319: 6307: 6302:, p. 9. 6265: 6258:Heiberg 1908 6249: 6238: 6232:Sidorov 2001 6227: 6163:Quincey 2021 6155: 6147: 6096: 6081: 6075: 6060:Zenith angle 6005:Golden angle 5743: 5736: 5707: 5690: 5688: 5661: 5646: 5640: 5630: 5607: 5601: 5589: 5557: 5556:just as the 5543: 5378: 5373: 5368: 5364: 5360: 5356: 5342: 5188: 5087: 5016: 4896: 4792: 4789: 4689: 4682: 4595: 4558: 4552: 4541:and between 4536: 4458: 4449: 4445: 4441:between two 4438: 4437:, the angle 4432: 4412: 4392: 4388: 4384: 4380: 4376: 4366: 4356: 4338: 4303: 4292: 4286: 4276: 4241: 4237: 4233: 4229: 4222: 4216:right angles 4215: 4209: 4205: 4181: 4170: 4165: 4161: 4159: 4146: 4142: 4138: 4134: 4130: 4122: 4118: 4111:initial side 4110: 4104: 4096:orientations 4093: 4024: 4014: 4010:natural unit 4007: 3963:pure numbers 3650:of an angle 3638: 3634: 3628: 3624: 3615: 3605: 3599: 3583: 3552:In 1993 the 3551: 3540: 3536: 3532: 3506: 3502: 3498: 3476: 3472: 3468: 3451: 3447: 3429: 3425: 3419: 3416: 3397:is 224 zam. 3356: 3315: 3277:Eratosthenes 3272: 3231: 3219: 3186: rad = 3159: 3139: 3119:. See also: 3100: 3064: 3060: 3052: 3048: 3042: 3038: 2961: 2954:scope sights 2939: 2888: 2884: 2867: grad. 2816: rad = 2795: 2791: 2771: 2743: 2715: 2692: 2686: 2682: 2678: 2623: 2587: 2583: 2579: 2559: 2556:great circle 2483: 2479: 2475: 2471: 2451: 2426: 2422: 2394: 2368: 2360: 2356: 2346: 2334:Description 2318: 2307: 2302: 2298: 2274: 2264: 2246: 2242: 2237: 2234: 2179: 2164: 2159: 2142: 2136: 2060:circular arc 2051: 2039: 2031: 2021: 2009: 1997: 1995: 1990: 1986: 1983: 1981: 1954: 1932: 1928: 1924: 1884: 1858: 1835: 1831: 1827: 1819: 1815: 1762: 1734: 1730: 1728: 1724: 1720: 1703: 1700: 1697: 1690: 1677: 1663: 1659: 1630: 1605: 1590: 1425: 1421: 1419: 1414: 1412: 1407: 1404:complementum 1403: 1401: 1358: 1337: 1321: 1312: 1309: 1234: 1229: 1227: 1217: 1213: 1209: 1205: 1201: 1197: 1191: 1182: 1181:angles (see 1179:explementary 1178: 1174: 1170: 1166: 1162: 1154: 1150: 1139: 1135: 1131: 1126: 1122: 1118: 1112: 1108: 1104: 1100: 1096: 1092: 1088: 1083: 1077: 1073: 1068: 1062: 1057: 1051: 1047: 1043: 1041: 1017: 1015: 1011: 1007: 1003: 999: 992: 988:Zenith angle 893: 882: 876: 865: 855: 849: 823: 798: 607:right angle 604:acute angle 588:Reflex angle 535: 529: 522: 518: 514: 510: 500:reflex angle 499: 492: 466:obtuse angle 465: 456: 450: 444: 438: 393: 386: 376: 374: 342: 335: 287: 282: 278: 274: 230: 203: 198: 153: 151: 130:circular arc 119: 115: 109: 94: 86:plane angles 85: 84: 78: 72: 64: 58: 18:Obtuse angle 7581:(1): 1–12. 7112:Foster 2010 7092:Romain 1962 6446:Jacobs 1974 6372:Jacobs 1974 6243:Slocum 2007 6152:Foster 2010 6132:Romain 1962 6040:Solid angle 5750:declination 5716:and of the 4454:dot product 4389:amphicoelic 4367:amphicyrtic 4173:orientation 4080:unit circle 4043:Mathematica 4039:plane_angle 3273:hexacontade 3262:hexacontade 3228:sexagesimal 3224:Babylonians 3161:right angle 3117:WP 43S 3081:of a turn. 2933:milliradian 2709:sexagesimal 2550:degrees. 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4238:θ 4225:. 4166:θ 4162:θ 4147:y 4143:x 4135:y 4029:η 4015:η 3935:/ 3928:= 3919:= 3916:x 3896:, 3890:+ 3884:! 3881:7 3874:7 3870:) 3860:( 3848:! 3845:5 3838:5 3834:) 3824:( 3818:+ 3812:! 3809:3 3802:3 3798:) 3788:( 3773:= 3767:+ 3761:! 3758:7 3752:7 3748:x 3736:! 3733:5 3727:5 3723:x 3717:+ 3711:! 3708:3 3702:3 3698:x 3689:x 3686:= 3683:x 3671:= 3652:θ 3635:s 3625:s 3619:0 3616:ε 3609:η 3600:r 3598:π 3546:ω 3541:r 3539:/ 3537:v 3533:ω 3524:θ 3520:r 3516:θ 3512:r 3503:y 3486:1 3477:r 3475:/ 3473:A 3469:θ 3452:r 3448:s 3443:r 3439:s 3435:θ 3430:r 3428:/ 3426:s 3421:θ 3413:. 3367:/ 3364:1 3336:2 3333:/ 3330:1 3326:+ 3311:° 3306:2 3303:/ 3300:1 3296:+ 3255:π 3248:6 3245:/ 3242:1 3212:6 3197:4 3194:/ 3191:1 3181:2 3178:/ 3174:π 3153:4 3150:/ 3147:1 3132:4 3109:π 3103:π 3093:2 3088:π 3073:/ 3070:1 3061:n 3013:/ 3010:1 2997:/ 2993:π 2991:2 2986:( 2977:/ 2974:1 2965:" 2940:π 2922:8 2919:/ 2916:1 2903:/ 2900:1 2862:3 2859:/ 2856:2 2852:+ 2840:/ 2837:1 2827:6 2824:/ 2821:1 2808:/ 2804:π 2782:/ 2779:1 2748:π 2736:1 2716:n 2714:( 2650:/ 2634:/ 2631:7 2624:n 2614:/ 2611:1 2598:/ 2595:1 2560:n 2542:/ 2526:/ 2510:/ 2507:1 2494:/ 2491:1 2452:n 2450:( 2387:π 2383:/ 2373:π 2365:π 2361:n 2347:π 2345:2 2303:n 2299:n 2247:r 2245:/ 2243:s 2238:θ 2221:. 2216:r 2213:s 2199:2 2195:k 2190:= 2165:k 2160:k 2152:π 2150:2 2147:/ 2143:k 2114:. 2110:d 2107:a 2104:r 2097:r 2094:s 2089:= 2072:r 2068:s 2055:θ 2048:. 2040:r 2036:/ 2032:s 2022:θ 1931:( 1917:. 1849:2 1846:/ 1843:1 1836:n 1832:n 1828:n 1824:π 1820:n 1816:n 1808:π 1794:2 1791:/ 1788:1 1781:π 1717:π 1656:π 1649:2 1646:/ 1643:1 1627:b 1623:a 1610:" 1573:B 1564:= 1561:A 1549:B 1540:= 1537:A 1523:1 1520:= 1517:B 1509:2 1501:+ 1498:A 1490:2 1479:1 1476:= 1473:B 1465:2 1457:+ 1454:A 1446:2 1426:B 1422:A 1390:2 1387:/ 1383:π 1372:4 1369:/ 1366:1 1350:b 1346:a 1342:a 1334:b 1332:( 1330:b 1326:a 1295:C 1292:O 1289:B 1282:m 1279:+ 1275:B 1272:O 1269:A 1262:m 1259:= 1255:C 1252:O 1249:A 1242:m 1155:B 1151:A 1140:x 1136:B 1132:A 1127:x 1123:x 1119:x 1113:B 1109:D 1105:C 1101:B 1097:D 1093:C 1089:B 1084:x 1078:D 1074:C 1069:x 1063:D 1058:x 1052:C 1048:x 1044:A 990:. 895:π 892:2 884:π 878:π 875:( 867:π 857:π 851:π 844:2 841:/ 838:1 833:( 825:π 818:2 815:/ 812:1 800:π 793:2 790:/ 787:1 755:2 752:/ 749:1 744:( 734:2 731:/ 728:1 714:2 711:/ 708:1 698:4 695:/ 692:1 687:( 677:4 674:/ 671:1 657:4 654:/ 651:1 574:c 570:b 566:a 532:. 525:. 507:π 502:. 495:. 489:π 482:2 479:/ 476:1 461:. 432:2 429:/ 425:π 414:4 411:/ 408:1 389:. 367:. 314:C 311:A 308:B 283:c 279:b 275:a 266:π 257:φ 253:θ 249:γ 245:β 241:α 239:( 179:( 41:. 34:. 20:)

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