Angle - Knowledge

563: 1622: 4075: 1032:. 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: 1711: 1321: 8144: 979: 1753: 1150: 2020: 587: 549: 4338: 2262: 45: 4172:". 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). 1871:. 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 3910: 3607:. 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. 6089:, 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. 3661: 4790: 3552:

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

1173:, 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 4198:

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°.

1887:) 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, 4683: 5539: 2989:

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

4696: 3905:{\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 ,} 4799: 5094: 1435: 1310: 3596:(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 1399:

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

5385: 5313: 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. 2235: 1924:

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.

2794: 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 5333: 2245:

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

4785:{\displaystyle \operatorname {Re} \left(\langle \mathbf {u} ,\mathbf {v} \rangle \right)=\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} 3063:(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 4248:

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.

2316:, an angle is defined as a dimensionless quantity, and in particular, the radian unit is dimensionless. This convention impacts how angles are treated in 7714:

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

4892:{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} 5184:{\displaystyle \left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|=\left|\cos(\theta )\right|\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|} 2964:

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

1586:{\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}}} 2186: 5705:

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

2371: = 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 2085: 5594: 1702:

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

8052: 6595: 7335: 5663:, each intersecting one of the stars. The angle between those lines and the angular separation between the two stars can be measured. 6761: 2016:

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

1684:. However, supplementary angles do not have to be on the same line and can be separated in space. For example, adjacent angles of a 8058: 4272:

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

3557: 3288: 8159: 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"). 2397:, or about 57.2958 degrees. Often, particularly in mathematical texts, one radian is assumed to equal one, resulting in the unit 4678:{\displaystyle \langle \mathbf {u} ,\mathbf {v} \rangle =\cos(\theta )\ \left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} 3574: 6682: 3617: 223:, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third: angle as a relationship. 8132: 7705: 7667: 7504: 6979: 6559: 3961:

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".

5534:{\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

2630: = 1,296,000). It is denoted by a double prime ( ″ ). For example, 3° 7′ 30″ is equal to 3 + 4901:

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

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. 4568: 4532:{\displaystyle \mathbf {u} \cdot \mathbf {v} =\cos(\theta )\left\|\mathbf {u} \right\|\left\|\mathbf {v} \right\|.} 4300: 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 5645: 3124: 1875:. Even in a non-simple polygon, it may be possible to define the exterior angle. Still, one will have to pick an 4218:. An angle is defined by its measure and is not dependent upon the lengths of the sides of the angle (e.g., all 6646: 4308: 3467:

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:—

17: 7615: 4365:(mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the 2538:= 3.5 degrees. A mixed format with decimal fractions is sometimes used, e.g., 3° 5.72′ = 3 + 2405:

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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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.

3059:. The binary degree is used in computing so that an angle can be efficiently represented in a single 2802:. These are distinct from, and 15 times larger than, minutes and seconds of arc. 1 hour = 15° = 2723: 5224: 5200: 2363:

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

7375: 5643:(that is, the apparent position of an astronomical object) can be identified using any of several 4994: 4972: 3496:. It is a long-established practice in mathematics and across all areas of science to make use of 7959:

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

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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

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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

2708: 2317: 2067: 1689: 1615: 983: 8: 6737: 6678: 5988: 5978: 5706: 5557: 5348: 4688: 4558: 1950: 1880: 1876: 8099: 7982: 7935: 7893: 7848: 7797: 7784:

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

7768: 7727: 7590: 7559: 7526: 7304: 7185: 7070: 7035: 6936: 5649:, where the references vary according to the particular system. Astronomers measure the 3969:

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

8111: 7994: 7968: 7947: 7879: 7834: 7809: 7739: 7687: 7635: 7602: 7316: 7290: 7197: 7171: 6950: 6741: 6053: 5939: 5651: 5629: 5585: 5318: 4691:, the expression for the cosine above may give non-real values, so it is replaced with 4069: 3485: 2301:(i.e., the angle subtended by the circumference of a circle at its centre) is equal to 2274: 2135: 1776: 1414:, "to fill up". An acute angle is "filled up" by its complement to form a right angle. 397: 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

2309:. Two exceptions are the radian (and its decimal submultiples) and the diameter part. 8128: 8115: 8107: 7998: 7951: 7943: 7813: 7805: 7752: 7743: 7701: 7663: 7639: 7631: 7606: 7598: 7500: 7435: 7320: 7201: 7193: 6975: 6954: 6555: 6518: 5666:

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

5569: 5565: 4418: 4313: 4180: 4113:, an angle is typically defined by its two sides, with its vertex at the origin. The 4059: 1964: 1673: 1025: 449: 220: 212: 120: 79: 7866:

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

3414: 1305:{\displaystyle m\angle \mathrm {AOC} =m\angle \mathrm {AOB} +m\angle \mathrm {BOC} } 8127:, vol. 1B (1 ed.), Hong Kong: Oxford University Press, pp. 161–163, 8123:

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

186: 90: 7616:"The next 50 years of the SI: a review of the opportunities for the e-Science age" 3626:. With this change the formula for the angle subtended at the center of a circle, 1200:

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

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is radius. One SI radian corresponds to the angle expressed in radians for which

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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

1618:" in the names of some trigonometric ratios refers to the word "complementary". 8204: 7990: 7902: 7867: 7735: 7312: 5998: 5671: 5667: 5589: 4426: 2414: 2282: 2174: 1959: 1863: 1804: 1767: 1762: 1103:, either of these angle measures may be used to determine the measure of Angle 908: 289: 171: 129: 95: 68: 49: 6945: 6920: 4244:

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

3172:. In German, the symbol has been used to denote a quadrant. 1 quad = 90° = 2000:

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)".

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is the angle through which the pulley turns in radians. When multiplying

3231: 3227: 3164: 2936: 2712: 2699:. It is a decimal subunit of the quadrant. A right angle is 100 grads. A 2434: 554: 443: 364: 265: 2429:, denoted by a small superscript circle (°), is 1/360 of a turn, so one 1939:

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

4546: 4187: 2896: 2763: 2451: 1954: 580:) angles. The acute and obtuse angles are also known as oblique angles. 363:"Oblique angle" redirects here. For the cinematographic technique, see 8179: 7534: 7078: 7043: 7008: 4429:

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

8168:, vol. 2 (11th ed.), Cambridge University Press, p. 14 7356: 7238: 6526: 5702: 5636: 5621: 5607: 4158: 4091: 4049:'s unit system similarly considers angles to have an angle dimension. 2727: 2700: 2571: 2463: 2447: 1903: 1600: 8059:

University of Texas research department: linguistics research center

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

5568:. The comparison can be visualized as the size of the openings of a 1692:(one whose vertices all fall on a single circle) are supplementary. 8086:

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

7973: 7884: 7295: 7176: 5617: 5610:, the location of any point on the Earth can be identified using a 5308:{\displaystyle \dim({\mathcal {U}}):=k\leq \dim({\mathcal {W}}):=l} 3616:

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

7839: 3283:. It equals 6°, so a whole turn was divided into 60 hexacontades. 978: 8012:

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

7434:(6th ed.). Belmont, CA: Thomson Brooks/Cole. p. 110. 6542: 6472: 6340: 4337: 4249: 4130: 4118: 4079: 3578: 3120: 2663: 2343: 2323:

The following table lists some units used to represent angles.

2286: 2261: 2230:{\displaystyle \theta ={\frac {k}{2\pi }}\cdot {\frac {s}{r}}.} 2079: 2009: 1775:

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

6863:, p. 151: "One radian corresponds to the angle for which 6781:, p. 151: "One radian corresponds to the angle for which 6654:

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.

2297:. Most units of angular measurement are defined such that one 548: 532:

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

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Current SI can be considered relative to this framework as a

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

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

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

5721: 5656: 5577: 4325: 4320:

Although done rarely, one can report the direct results of

3651: 3060: 2970: 1719:

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

1317:

Three special angle pairs involve the summation of angles:

6294: 6139: 3612: 3424: 1867:; that is, an interior angle and an exterior angle form a 1771:

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

8054:

Preliminary Indo-European lexicon — Pokorny PIE data

6267: 5717: 1963:. It may be defined as the acute angle between two lines 1020:

The equality of vertically opposite angles is called the

990:"Vertical angle" redirects here. Not to be confused with 7865: 7686:, The Thirteen Books of Euclid's Elements, vol. 1, 7362: 7256: 7244: 7136: 7131: 6212: 1676:

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

355:

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

44: 7056: 6516: 2406: 2131: 2123:{\displaystyle \theta ={\frac {s}{r}}\,\mathrm {rad} .} 1365:

are angle pairs whose measures sum to one right angle (

1236:

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

270: 7821:

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

7496:

A High School First Course in Euclidean Plane Geometry

7228: 7226: 7213: 7211: 3492:. In SI 2019, the SI radian is defined accordingly as 1803:

turn; the measures of the interior angles of a simple

6369: 6284: 6282: 6175: 6114: 5388: 5321: 5251: 5227: 5203: 5097: 5061: 5027: 4997: 4975: 4941: 4907: 4802: 4699: 4605: 4571: 4468: 4141:-axis. When Cartesian coordinates are represented by 3995: 3975: 3918: 3664: 3489: 2293:(grad), though many others have been used throughout 2189: 2088: 1438: 1432:

are complementary, the following relationships hold:

1244: 302: 7662:(3rd ed.), Pearson Prentice Hall, p. 104, 6443: 5601: 5584:

in their angle argument, the circular ones are just

4391:(Gr. ξυστρίς, a tool for scraping), concavo-convex; 967: 77:

of the angle, sharing a common endpoint, called the

7262: 7223: 7208: 6204:{\displaystyle {\cal {C}}={\frac {2\pi }{\Theta }}} 5082:{\displaystyle \operatorname {span} (\mathbf {v} )} 5048:{\displaystyle \operatorname {span} (\mathbf {u} )} 4962:{\displaystyle \operatorname {span} (\mathbf {v} )} 4928:{\displaystyle \operatorname {span} (\mathbf {u} )} 3950:{\displaystyle x=\eta \theta =\theta /{\text{rad}}} 447:. Two lines that form a right angle are said to be 27:

Figure formed by two rays meeting at a common point

8190:, vol. 2 (9th ed.), 1878, pp. 29–30 6309: 6279: 6203: 6126: 5533: 5327: 5307: 5237: 5213: 5183: 5081: 5047: 5005: 4983: 4961: 4927: 4891: 4794:or, more commonly, using the absolute value, with 4784: 4677: 4589: 4531: 4432: 4402: 4001: 3981: 3957:is the angle in radians. The capitalized function 3949: 3904: 3230:, The degree, minute of arc and second of arc are 2229: 2122: 1585: 1304: 325: 7157: 7155: 7153: 7151: 6887:International Bureau of Weights and Measures 2019 6861:International Bureau of Weights and Measures 2019 6819:, the radian can be defined in terms of the area 6801:International Bureau of Weights and Measures 2019 6779:International Bureau of Weights and Measures 2019 5659:by imagining two lines through the center of the 4281:For an angular unit, it is definitional that the 2558:was historically defined as an arcminute along a 1861:The supplement of an interior angle is called an 8196: 8076: 7657: 7476:The Algebra of Coplanar Vectors and Trigonometry 7376:"UnityDimensions—Wolfram Language Documentation" 6656:. Savage Innovations, LLC. 2007 . Archived from 6478: 6410: 6408: 6363: 6104:Other proposals include the abbreviation "rad" ( 5342: 4425:but could only trisect certain angles. In 1837, 4009:if it is clear that the complete form is meant. 6671: 6535: 3518:is the radius of the pulley in centimeters and 385:An angle equal to 0° or not turned is called a 7650:Elementary geometry: practical and theoretical 7646: 7148: 6971:Minds-on Physics: Advanced topics in mechanics 6754: 6303: 3363:(occasionally used in Islamic mathematics) is 2722: = 400). The grad is used mostly in 2134:, the radian is treated as being equal to the 1406:The adjective complementary is from the Latin 215:, who regarded an angle as a deviation from a 6647:"ooPIC Programmer's Guide - Chapter 15: URCP" 6572: 6511:CRC Standard Mathematical Tables and Formulae 6508: 6405: 4590:{\displaystyle \langle \cdot ,\cdot \rangle } 4345:is defined as the angle between the tangents 2281:, with the most contemporary units being the 1603:of its complement, and its secant equals the 7868:"On the dimension of angles and their units" 7820: 7753:"Dimensional angles and universal constants" 7750: 7658:Henderson, David W.; Taimina, Daina (2005), 7492: 7280: 7161: 7142: 7111: 6816: 6543:International Bureau of Weights and Measures 6273: 6147: 6121: 6115: 5595:Introduction to the Analysis of the Infinite 5119: 5103: 4824: 4808: 4727: 4711: 4622: 4606: 4584: 4572: 1688:are supplementary, and opposite angles of a 263:denoting the size of some angle (the symbol 7333: 5712:Other astronomical approximations include: 5016: 4137:representing rotations toward the negative 4129:representing rotations toward the positive 1779:, the measures of the interior angles of a 161: 7751:Lévy-Leblond, Jean-Marc (September 1998). 7653:(3rd ed.), Cambridge University Press 7541: 7107: 6907:Angular amplitude of swing No dimensions. 6905:. New Haven : Yale University Press. 6730: 6700: 6491: 6489: 6487: 6143: 1868: 1187: 8023: 7972: 7901: 7883: 7856: 7838: 7647:Godfrey, Charles; Siddons, A. W. (1919), 7567: 7512: 7294: 7175: 7091: 7021: 6944: 6760: 6683:"Angles, integers, and modulo arithmetic" 6105: 4561:, we replace the Euclidean dot product ( 4041:units library defines angle units with a 2105: 1747: 1642:Two angles that sum to a straight angle ( 336:In other ways, an angle denoted as, say, 128:conventionally defined as the ratio of a 8154: 8122: 8008:"Angle as a fourth fundamental quantity" 7454: 7429: 7334:Schabel, Matthias C.; Watanabe, Steven. 6898: 6641: 6639: 6578: 6513:, Boca Raton, FL: CRC Press, p. 270 6414: 6327: 6257: 5716:0.5° is the approximate diameter of the 5376:are the components of the metric tensor 5355:is used to define the angle between two 4336: 4332: 4073: 3577:(N⋅m/rad), and not in the quantities of 3558:American Association of Physics Teachers 2260: 2018: 1751: 1709: 1620: 1319: 1227: 1148: 1042:Equals subtracted from equals are equal. 977: 147: 43: 8085: 8081:, American Book Company, pp. 25–27 8032: 7958: 7921: 7713: 7675: 7458: 7268: 7232: 7217: 7127: 7123: 7103: 6967: 6851:), in which case it has the units m⋅m." 6815:, p. 844: "Also, as alluded to in 6812: 6484: 6261: 6235: 6166: 6127:{\displaystyle \langle \theta \rangle } 3407: 2130:Conventionally, in mathematics and the 1943: 1891:. Exterior angles are commonly used in 14: 8197: 8050: 8005: 7695: 7613: 7115: 7095: 6994: 6918: 6677: 6548:The International System of Units (SI) 6461: 6449: 6375: 6246: 6155: 6135: 3257: turn. 1 Babylonian unit = 60° = 326:{\displaystyle {\widehat {\rm {BAC}}}} 7910: 7783: 7119: 6736: 6706: 6636: 6517: 6315: 6288: 4557:To define angles in an abstract real 4276: 4179:, which is typically determined by a 4153:-axis upward, positive rotations are 2269:Throughout history, angles have been 986:are used here to show angle equality. 226: 7614:Foster, Marcus P (1 December 2010). 7576: 7411:from the original on 23 October 2017 7099: 6568:from the original on 18 October 2021 4379:, on both sides, κυρτός, convex) or 4341:The angle between the two curves at 4202: 3982:{\displaystyle \operatorname {Sin} } 3261:/3 rad ≈ 1.047197551 rad. 1668:If the two supplementary angles are 1107:. Using the measure of either angle 370: 185:) meaning "crooked, curved" and the 7700:, W. H. Freeman, pp. 97, 255, 5727:1° is the approximate width of the 5543: 2433:is 360°. One advantage of this old 1975: 1838: − 2)2 right angles, or ( 509:An angle equal to 1 turn (360° or 2 219:; the second, angle as quality, by 24: 7544:"Angles—Let's treat them squarely" 6581:"On Angles and Angle Measurements" 6339: 6196: 6178: 5693:Astronomers also measure objects' 5291: 5263: 5230: 5206: 4023:is assumed to hold, or similarly, 2314:International System of Quantities 2113: 2110: 2107: 1834: − 2)180 degrees, ( 1298: 1295: 1292: 1288: 1278: 1275: 1272: 1268: 1258: 1255: 1252: 1248: 1065:. Similarly, the measure of angle 377: 343: 313: 310: 307: 25: 8216: 8172: 7493:Aboughantous, Charles H. (2010), 6899:Bridgman, Percy Williams (1922). 6462:Willis, Clarence Addison (1922). 6321: 5602:Angles in geography and astronomy 4460:and their lengths by the formula 3548:, radians appear in the units of 2379: radians, and one radian is 1953:(such as two adjacent faces of a 1039:Equals added to equals are equal. 259:, . . . ) as 8142: 8006:Romain, Jacques E. (July 1962). 5315:, this leads to a definition of 5173: 5160: 5115: 5107: 5072: 5038: 4999: 4977: 4952: 4918: 4878: 4865: 4820: 4812: 4771: 4758: 4723: 4715: 4664: 4651: 4618: 4610: 4552: 4518: 4505: 4478: 4470: 4312:between two lines is defined in 4066:Sign (mathematics) § Angles 4053: 3413:This section is an excerpt from 3057:binary angular measurement (BAM) 2138:1, thus being normally omitted. 2012:curve or describing an object's 1931:of a simple polygon to mean the 1095:is supplementary to both angles 585: 561: 547: 48:A green angle formed by two red 7823:"Dimensionless units in the SI" 7542:Brownstein, K. R. (July 1997). 7486: 7464: 7448: 7423: 7393: 7368: 7327: 7274: 7085: 7050: 7015: 6988: 6961: 6919:Prando, Giacomo (August 2020). 6912: 6892: 6854: 6806: 6772: 6719:from the original on 2019-08-06 6689:from the original on 2019-06-30 6612: 6601:from the original on 2019-01-17 6502: 6455: 6419: 6381: 6098: 6077: 5646:astronomical coordinate systems 4433:Dot product and generalisations 4409:Bisection § Angle bisector 4403:Bisecting and trisecting angles 4361:The angle between a line and a 4164:In many contexts, an angle of − 4090:direction, and negative in the 3965:is the traditional function on 3441:is subtended angle in radians, 3162: turn and also known as a 3125:IEEE 754 recommended operations 2078:of the circle is the number of 1115:, we find the measure of angle 1091:and are congruent. Since angle 271:constant denoted by that symbol 193:". Both are connected with the 7676:Heiberg, Johan Ludvig (1908), 6743:The Growth of Physical Science 6579:Dimitrić, Radoslav M. (2012). 6364:Shute, Shirk & Porter 1960 6333: 6251: 6240: 6229: 5296: 5286: 5268: 5258: 5238:{\displaystyle {\mathcal {W}}} 5214:{\displaystyle {\mathcal {U}}} 5177: 5169: 5164: 5156: 5147: 5141: 5076: 5068: 5042: 5034: 4956: 4948: 4922: 4914: 4882: 4874: 4869: 4861: 4852: 4846: 4775: 4767: 4762: 4754: 4750: 4744: 4668: 4660: 4655: 4647: 4640: 4634: 4522: 4514: 4509: 4501: 4497: 4491: 3873: 3863: 3837: 3827: 3801: 3791: 3553:"pedagogically unsatisfying". 3421:Plane angle may be defined as 3168:. The quadrant is the unit in 2703:was historically defined as a 1895:when drawing regular polygons. 1036:All straight angles are equal. 201:, meaning "to bend" or "bow". 13: 1: 7430:McKeague, Charles P. (2008). 6974:. Kendall Hunt. p. 262. 6468:. Blakiston's Son. p. 8. 6222: 5343:Angles in Riemannian geometry 4549:from their vector equations. 4383:(Gr. κισσός, ivy), biconvex; 3618:introduction of the constant 3469:the area of a circular sector 3415:Radian § Dimensional analysis 3115:) unit is implemented in the 2305:units, for some whole number 1669: 273:). Lower case Roman letters ( 7690:: Cambridge University Press 7401:"Mathwords: Reference Angle" 6968:Leonard, William J. (1999). 6707:Bonin, Walter (2016-01-11). 6479:Henderson & Taimina 2005 5616:. This system specifies the 5613:geographic coordinate system 5006:{\displaystyle \mathbf {v} } 4984:{\displaystyle \mathbf {u} } 4419:ancient Greek mathematicians 4070:Euclidean space § Angle 1889:supplementary exterior angle 1756:Internal and external angles 1188:§ Combining angle pairs 181: 67:is the figure formed by two 7: 8040:Encyclopedia of Mathematics 7757:American Journal of Physics 7548:American Journal of Physics 7515:American Journal of Physics 7351:Angles are treated as units 6588:The Teaching of Mathematics 6509:D. Zwillinger, ed. (1995), 6499:, Dover Publications, 2007. 6497:Advanced Euclidean Geometry 5974:Argument (complex analysis) 5922: 5335:angles called canonical or 4111:Cartesian coordinate system 2265:Definition of 1 radian 1761:An angle that is part of a 1680:. Such angles are called a 1410:, associated with the verb 1222:consecutive interior angles 472:("obtuse" meaning "blunt"). 54:Cartesian coordinate system 10: 8221: 8108:10.1088/0026-1394/22/1/002 7944:10.1088/0026-1394/53/2/840 7914:Modern Elementary Geometry 7806:10.1088/0026-1394/53/3/991 7696:Jacobs, Harold R. (1974), 7632:10.1088/0026-1394/47/6/R01 7599:10.1088/0026-1394/18/1/002 7336:"Boost.Units FAQ – 1.79.0" 7194:10.1088/0026-1394/53/3/998 6762:Murnaghan, Francis Dominic 6304:Godfrey & Siddons 1919 5930:Angle measuring instrument 4406: 4375: 4157:, and negative cycles are 4121:, while the other side or 4063: 4057: 4035:in mathematical formulas. 4016:system where the equation 3412: 3326:unit equal to about 2° or 2915:of a turn. 1 point = 2070:. The ratio of the length 1982:Angle measuring instrument 1979: 1927:Some authors use the name 1146:and are equal in measure. 1014:. They are abbreviated as 1012:vertically opposite angles 989: 630: 362: 175: 36: 29: 8051:Slocum, Jonathan (2007), 7858:10.1088/0026-1394/52/1/40 6946:10.1038/s41567-020-0997-3 4395:(Gr. κοίλη, a hollow) or 4086:count as positive in the 3226:was the unit used by the 2074:of the arc by the radius 1910:(meet at a single point). 1214:alternate interior angles 1210:alternate exterior angles 114:of an angle is called an 8079:Plane and Solid Geometry 8033:Sidorov, L. A. (2001) , 7991:10.1088/1681-7575/ac023f 7911:Moser, James M. (1971), 7903:10.1088/1681-7575/ac7bc2 7736:10.1088/1681-7575/abe0fc 7499:, Universal Publishers, 7313:10.1088/1681-7575/aa7160 7143:Mohr & Phillips 2015 6817:Mohr & Phillips 2015 6070: 5367:are tangent vectors and 5017:Angles between subspaces 4423:compass and straightedge 4283:angle addition postulate 4149:-axis rightward and the 3636:, is modified to become 3590:base unit of measurement 3461:1 SI radian = 1 m/m 2444:geographical coordinates 2256: 1933:explement exterior angle 1234:angle addition postulate 1134:. Therefore, both angle 1028:attributed the proof to 358: 233:mathematical expressions 30:Not to be confused with 8187:Encyclopædia Britannica 8165:Encyclopædia Britannica 8125:New Century Mathematics 6746:. CUP Archive. p. 5639:, a given point on the 4969:spanned by the vectors 4565:) by the inner product 4322:trigonometric functions 4031:allows the omission of 2610:of a minute of arc and 2403:trigonometric functions 1599:of an angle equals the 1169:, often abbreviated as 1142:have measures equal to 1084:have measures equal to 1054:, the measure of angle 7471:Robert Baldwin Hayward 6427:"Supplementary Angles" 6389:"Complementary Angles" 6205: 6128: 6004:Exterior angle theorem 5535: 5329: 5309: 5239: 5215: 5185: 5083: 5049: 5007: 4985: 4963: 4929: 4893: 4786: 4679: 4591: 4533: 4358: 4222:are equal in measure). 4095: 4003: 3983: 3951: 3906: 3206:turn = 100 grad. 3119:scientific calculator 2566: = 21,600). 2266: 2231: 2124: 2053: 1949:The angle between two 1757: 1748:Polygon-related angles 1715: 1638: 1587: 1357: 1306: 1162: 1022:vertical angle theorem 987: 327: 235:, it is common to use 162: 56: 39:Angle (disambiguation) 8025:10.6028/jres.066B.012 7380:reference.wolfram.com 6709:"RE: WP-32S in 2016?" 6624:TheFreeDictionary.com 6206: 6138:), and the constants 6129: 6014:Great circle distance 5745:approximations only. 5724:as viewed from Earth. 5564:is the argument of a 5536: 5330: 5310: 5240: 5216: 5186: 5084: 5050: 5008: 4986: 4964: 4930: 4894: 4787: 4680: 4592: 4534: 4340: 4333:Angles between curves 4109:In a two-dimensional 4077: 4004: 4002:{\displaystyle \sin } 3984: 3952: 3907: 2707:-grad of arc along a 2440:"minute" and "second" 2277:. These are known as 2264: 2232: 2125: 2023:The measure of angle 2022: 1869:linear pair of angles 1755: 1713: 1682:linear pair of angles 1672:(i.e., have a common 1624: 1588: 1352:is the complement of 1323: 1307: 1228:Combining angle pairs 1152: 981: 513:radians) is called a 495:radians) is called a 441:radians) is called a 328: 148:History and etymology 47: 6902:Dimensional analysis 6738:Jeans, James Hopwood 6328:Wong & Wong 2009 6173: 6112: 5959:Angular acceleration 5935:Angles between flats 5678:with respect to the 5386: 5319: 5249: 5225: 5201: 5095: 5059: 5025: 4995: 4973: 4939: 4905: 4800: 4697: 4603: 4569: 4466: 4456:is related to their 3993: 3973: 3916: 3662: 3571:angular acceleration 3534:of a rolling wheel, 3408:Dimensional analysis 3238:equilateral triangle 3045:, also known as the 2318:dimensional analysis 2187: 2086: 2056:To measure an angle 1944:Plane-related angles 1893:Logo Turtle programs 1723:radians) are called 1690:cyclic quadrilateral 1664:supplementary angles 1662:radians) are called 1607:of its complement.) 1436: 1363:Complementary angles 1242: 1218:corresponding angles 491: turn (180° or 378:§ Signed angles 344:§ Signed angles 300: 166:, meaning "corner". 8100:1986Metro..22....1T 7983:2021Metro..58e3002Q 7936:2016Metro..53..840Q 7894:2022Metro..59e3001M 7849:2015Metro..52...40M 7798:2016Metro..53..991M 7769:1998AmJPh..66..814L 7728:2021Metro..58e2001L 7591:1982Metro..18....1E 7560:1997AmJPh..65..605B 7527:1936AmJPh...4..175B 7305:2017Metro..54..454Q 7186:2016Metro..53..998Q 7071:1993PhTea..31...84A 7059:The Physics Teacher 7036:1992PhTea..30..170O 7024:The Physics Teacher 6997:The Physics Teacher 6937:2020NatPh..16..888P 5989:Clock angle problem 5979:Astrological aspect 5707:small-angle formula 5701:. For example, the 5558:hyperbolic function 5349:Riemannian geometry 5339:between subspaces. 4689:inner product space 4559:inner product space 4117:is on the positive 4078:Measuring from the 3575:torsional stiffness 3514:centimeters, where 3445:is arc length, and 2973:mil" is defined as 2014:cumulative rotation 1830: radians, or ( 1725:explementary angles 195:Proto-Indo-European 118:or simply "angle". 99:. Two intersecting 89:as they lie in the 6685:. blogs.msdn.com. 6519:Weisstein, Eric W. 6495:Johnson, Roger A. 6431:www.mathsisfun.com 6393:www.mathsisfun.com 6330:, pp. 161–163 6264:, pp. 177–178 6201: 6124: 6054:Transcendent angle 5948:standard deviation 5940:Angular statistics 5652:angular separation 5630:Greenwich meridian 5628:and (usually) the 5586:alternating series 5531: 5325: 5305: 5235: 5211: 5181: 5079: 5045: 5003: 4981: 4959: 4925: 4889: 4782: 4675: 4587: 4529: 4359: 4277:Related quantities 4236:(sometimes called 4096: 3999: 3979: 3947: 3902: 3486:dimensionless unit 3279:is a unit used by 2458: = 360) 2332:Number in one turn 2267: 2227: 2136:dimensionless unit 2120: 2054: 1787:radians, 180°, or 1777:Euclidean geometry 1758: 1716: 1639: 1583: 1581: 1358: 1302: 1163: 1125:) = 180° − 180° + 988: 475:An angle equal to 403:An angle equal to 396:("acute" meaning " 323: 227:Identifying angles 170:words include the 61:Euclidean geometry 57: 8134:978-0-19-800177-5 7707:978-0-7167-0456-0 7669:978-0-13-143748-7 7535:10.1119/1.1999110 7506:978-1-59942-822-2 7405:www.mathwords.com 7112:Lévy-Leblond 1998 7079:10.1119/1.2343667 7044:10.1119/1.2343500 7009:10.1119/1.2343535 6981:978-0-7872-5412-4 6921:"A spectral unit" 6766:Analytic Geometry 6679:Hargreaves, Shawn 6561:978-92-822-2272-0 6366:, pp. 25–27. 6351:Proposition I:13. 6274:Aboughantous 2010 6199: 6148:Lévy-Leblond 1998 5920: 5919: 5570:hyperbolic sector 5566:circular function 5526: 5525: 5328:{\displaystyle k} 5013:correspondingly. 4645: 4447:Euclidean vectors 4397:angulus lunularis 4314:rational geometry 4227:coterminal angles 4203:Equivalent angles 4145:, defined by the 4143:standard position 4060:Angle of rotation 4029:radian convention 3945: 3891: 3855: 3819: 3768: 3743: 3718: 3685: 3405: 3404: 3393:In old Arabia, a 3170:Euclid's Elements 3009:= 0.0009817... ≈ 2774:The astronomical 2222: 2209: 2103: 1822:sides add up to ( 1030:Thales of Miletus 1026:Eudemus of Rhodes 965: 964: 929:(180, 360)° 602:Name 576:), and straight ( 371:Individual angles 320: 221:Carpus of Antioch 213:Eudemus of Rhodes 121:Angle of rotation 16:(Redirected from 8212: 8191: 8183: 8169: 8148: 8146: 8145: 8137: 8119: 8082: 8073: 8072: 8070: 8061:, archived from 8047: 8029: 8027: 8002: 7976: 7955: 7918: 7907: 7905: 7887: 7862: 7860: 7842: 7817: 7780: 7747: 7710: 7691: 7672: 7654: 7643: 7610: 7573: 7571: 7538: 7509: 7480: 7468: 7462: 7452: 7446: 7445: 7427: 7421: 7420: 7418: 7416: 7397: 7391: 7390: 7388: 7386: 7372: 7366: 7363:Mohr et al. 2022 7360: 7354: 7353: 7348: 7346: 7331: 7325: 7324: 7298: 7278: 7272: 7266: 7260: 7257:Mohr et al. 2022 7254: 7248: 7245:Mohr et al. 2022 7242: 7236: 7230: 7221: 7215: 7206: 7205: 7179: 7159: 7146: 7140: 7134: 7132:Mohr et al. 2022 7089: 7083: 7082: 7054: 7048: 7047: 7019: 7013: 7012: 6992: 6986: 6985: 6965: 6959: 6958: 6948: 6916: 6910: 6909: 6896: 6890: 6884: 6878: 6876: 6872: 6858: 6852: 6850: 6843: 6841: 6840: 6837: 6834: 6810: 6804: 6798: 6792: 6790: 6776: 6770: 6769: 6758: 6752: 6751: 6734: 6728: 6727: 6725: 6724: 6704: 6698: 6697: 6695: 6694: 6675: 6669: 6668: 6666: 6665: 6643: 6634: 6633: 6631: 6630: 6616: 6610: 6609: 6607: 6606: 6600: 6585: 6576: 6570: 6569: 6554:(9th ed.), 6553: 6539: 6533: 6532: 6531: 6522:"Exterior Angle" 6514: 6506: 6500: 6493: 6482: 6476: 6470: 6469: 6459: 6453: 6447: 6441: 6440: 6438: 6437: 6423: 6417: 6412: 6403: 6402: 6400: 6399: 6385: 6379: 6373: 6367: 6361: 6352: 6350: 6337: 6331: 6325: 6319: 6313: 6307: 6301: 6292: 6286: 6277: 6271: 6265: 6255: 6249: 6244: 6238: 6233: 6216: 6213:Mohr et al. 2022 6210: 6208: 6207: 6202: 6200: 6195: 6187: 6182: 6181: 6133: 6131: 6130: 6125: 6108:), the notation 6102: 6090: 6088: 6081: 6029:Irrational angle 5969:Angular velocity 5964:Angular diameter 5915: 5914: 5910: 5903: 5902: 5898: 5891: 5890: 5886: 5885: 5866: 5865: 5861: 5854: 5853: 5849: 5842: 5841: 5837: 5836: 5815: 5814: 5810: 5803: 5802: 5798: 5797: 5759: 5758: 5731:at arm's length. 5699:angular diameter 5686:with respect to 5641:celestial sphere 5550:hyperbolic angle 5544:Hyperbolic angle 5540: 5538: 5537: 5532: 5527: 5524: 5520: 5519: 5518: 5509: 5508: 5499: 5498: 5481: 5477: 5476: 5475: 5466: 5465: 5456: 5455: 5438: 5437: 5436: 5435: 5426: 5425: 5416: 5415: 5402: 5337:principal angles 5334: 5332: 5331: 5326: 5314: 5312: 5311: 5306: 5295: 5294: 5267: 5266: 5244: 5242: 5241: 5236: 5234: 5233: 5220: 5218: 5217: 5212: 5210: 5209: 5190: 5188: 5187: 5182: 5180: 5176: 5167: 5163: 5154: 5150: 5126: 5122: 5118: 5110: 5088: 5086: 5085: 5080: 5075: 5054: 5052: 5051: 5046: 5041: 5012: 5010: 5009: 5004: 5002: 4990: 4988: 4987: 4982: 4980: 4968: 4966: 4965: 4960: 4955: 4934: 4932: 4931: 4926: 4921: 4898: 4896: 4895: 4890: 4885: 4881: 4872: 4868: 4859: 4855: 4831: 4827: 4823: 4815: 4791: 4789: 4788: 4783: 4778: 4774: 4765: 4761: 4734: 4730: 4726: 4718: 4684: 4682: 4681: 4676: 4671: 4667: 4658: 4654: 4643: 4621: 4613: 4596: 4594: 4593: 4588: 4538: 4536: 4535: 4530: 4525: 4521: 4512: 4508: 4481: 4473: 4413:Angle trisection 4378: 4377: 4299:is equal to the 4270: 4266: 4264: 4263: 4260: 4257: 4240:) for any angle 4088:counterclockwise 4082:, angles on the 4044: 4034: 4026: 4022: 4008: 4006: 4005: 4000: 3988: 3986: 3985: 3980: 3964: 3960: 3956: 3954: 3953: 3948: 3946: 3943: 3941: 3911: 3909: 3908: 3903: 3892: 3890: 3882: 3881: 3880: 3861: 3856: 3854: 3846: 3845: 3844: 3825: 3820: 3818: 3810: 3809: 3808: 3789: 3769: 3767: 3759: 3758: 3749: 3744: 3742: 3734: 3733: 3724: 3719: 3717: 3709: 3708: 3699: 3683: 3657: 3645: 3635: 3615: 3606: 3598:area of a circle 3583:angular momentum 3551: 3547: 3532:angular velocity 3529: 3525: 3521: 3517: 3513: 3499: 3495: 3483: 3466: 3462: 3458: 3448: 3444: 3440: 3436: 3378: 3376: 3375: 3372: 3369: 3344: 3342: 3341: 3338: 3335: 3331: 3314: 3312: 3311: 3308: 3305: 3301: 3260: 3256: 3254: 3253: 3250: 3247: 3205: 3203: 3202: 3199: 3196: 3189: 3187: 3186: 3183: 3180: 3179: 3161: 3159: 3158: 3155: 3152: 3114: 3108: 3093: 3084: 3082: 3081: 3078: 3075: 3024: 3022: 3021: 3018: 3015: 3008: 3006: 3005: 3002: 2999: 2998: 2988: 2986: 2985: 2982: 2979: 2946: 2930: 2928: 2927: 2924: 2921: 2914: 2912: 2911: 2908: 2905: 2870: 2868: 2867: 2864: 2861: 2857: 2851: 2849: 2848: 2845: 2842: 2835: 2833: 2832: 2829: 2826: 2819: 2817: 2816: 2813: 2810: 2809: 2793: 2791: 2790: 2787: 2784: 2753: 2726:and continental 2661: 2659: 2658: 2655: 2652: 2645: 2643: 2642: 2639: 2636: 2625: 2623: 2622: 2619: 2616: 2609: 2607: 2606: 2603: 2600: 2553: 2551: 2550: 2547: 2544: 2537: 2535: 2534: 2531: 2528: 2521: 2519: 2518: 2515: 2512: 2505: 2503: 2502: 2499: 2496: 2396: 2394: 2393: 2392: 2388: 2385: 2378: 2370: 2353: 2326: 2325: 2244: 2236: 2234: 2233: 2228: 2223: 2215: 2210: 2208: 2197: 2177:or 400 grad for 2172: 2161: 2159: 2158: 2157: 2152: 2149: 2129: 2127: 2126: 2121: 2116: 2104: 2096: 2051: 2049: 2047: 2046: 2041: 2038: 2028: 1995:equal in measure 1976:Measuring angles 1857: 1855: 1854: 1851: 1848: 1829: 1813: 1802: 1800: 1799: 1796: 1793: 1786: 1737:of the angle or 1729:conjugate angles 1722: 1661: 1657: 1655: 1654: 1651: 1648: 1592: 1590: 1589: 1584: 1582: 1556: 1532: 1516: 1515: 1497: 1496: 1486: 1472: 1471: 1453: 1452: 1442: 1398: 1396: 1395: 1392: 1389: 1388: 1380: 1378: 1377: 1374: 1371: 1311: 1309: 1308: 1303: 1301: 1281: 1261: 1133: 1090: 1075: 1064: 973: 972: 958:(200, 400) 952:(100, 200) 923:(90, 180)° 903: 900: 892: 889: 883: 875: 872: 865: 862: 856: 852: 850: 849: 846: 843: 833: 830: 826: 824: 823: 820: 817: 808: 805: 801: 799: 798: 795: 792: 782: 770: 765: 763: 761: 760: 757: 754: 744: 742: 740: 739: 736: 733: 724: 722: 720: 719: 716: 713: 706: 704: 703: 700: 697: 687: 685: 683: 682: 679: 676: 667: 665: 663: 662: 659: 656: 646: 599: 598: 589: 565: 551: 512: 494: 490: 488: 487: 484: 481: 440: 438: 437: 434: 431: 430: 418: 416: 415: 412: 409: 354: 350: 339: 332: 330: 329: 324: 322: 321: 316: 305: 295: 268: 184: 178: 177: 165: 21: 8220: 8219: 8215: 8214: 8213: 8211: 8210: 8209: 8195: 8194: 8178: 8175: 8158:, ed. (1911), " 8143: 8141: 8135: 8068: 8066: 8065:on 27 June 2010 7917:, Prentice-Hall 7777:10.1119/1.18964 7708: 7670: 7569:10.1119/1.18616 7507: 7489: 7484: 7483: 7469: 7465: 7453: 7449: 7442: 7428: 7424: 7414: 7412: 7399: 7398: 7394: 7384: 7382: 7374: 7373: 7369: 7361: 7357: 7344: 7342: 7332: 7328: 7279: 7275: 7267: 7263: 7259:, pp. 8–9. 7255: 7251: 7243: 7239: 7231: 7224: 7216: 7209: 7170:(3): 998–1002. 7160: 7149: 7141: 7137: 7108:Brownstein 1997 7090: 7086: 7055: 7051: 7020: 7016: 6993: 6989: 6982: 6966: 6962: 6917: 6913: 6897: 6893: 6885: 6881: 6874: 6864: 6859: 6855: 6838: 6835: 6832: 6831: 6829: 6824: 6811: 6807: 6799: 6795: 6782: 6777: 6773: 6759: 6755: 6735: 6731: 6722: 6720: 6705: 6701: 6692: 6690: 6676: 6672: 6663: 6661: 6645: 6644: 6637: 6628: 6626: 6618: 6617: 6613: 6604: 6602: 6598: 6583: 6577: 6573: 6562: 6551: 6545:(20 May 2019), 6540: 6536: 6507: 6503: 6494: 6485: 6477: 6473: 6460: 6456: 6448: 6444: 6435: 6433: 6425: 6424: 6420: 6413: 6406: 6397: 6395: 6387: 6386: 6382: 6374: 6370: 6362: 6355: 6338: 6334: 6326: 6322: 6314: 6310: 6302: 6295: 6287: 6280: 6272: 6268: 6256: 6252: 6245: 6241: 6234: 6230: 6225: 6220: 6219: 6188: 6186: 6177: 6176: 6174: 6171: 6170: 6164: 6144:Brownstein 1997 6113: 6110: 6109: 6103: 6099: 6094: 6093: 6084: 6082: 6078: 6073: 6068: 6049:Spherical angle 6024:Inscribed angle 5994:Decimal degrees 5925: 5912: 5908: 5907: 5900: 5896: 5895: 5888: 5883: 5882: 5881: 5863: 5859: 5858: 5851: 5847: 5846: 5839: 5834: 5833: 5832: 5812: 5808: 5807: 5800: 5795: 5794: 5793: 5750:right ascension 5682:as well as the 5632:as references. 5604: 5582:infinite series 5574:circular sector 5546: 5514: 5510: 5504: 5500: 5491: 5487: 5486: 5482: 5471: 5467: 5461: 5457: 5448: 5444: 5443: 5439: 5431: 5427: 5421: 5417: 5408: 5404: 5403: 5401: 5387: 5384: 5383: 5375: 5345: 5320: 5317: 5316: 5290: 5289: 5262: 5261: 5250: 5247: 5246: 5229: 5228: 5226: 5223: 5222: 5205: 5204: 5202: 5199: 5198: 5172: 5168: 5159: 5155: 5134: 5130: 5114: 5106: 5102: 5098: 5096: 5093: 5092: 5071: 5060: 5057: 5056: 5037: 5026: 5023: 5022: 5019: 4998: 4996: 4993: 4992: 4976: 4974: 4971: 4970: 4951: 4940: 4937: 4936: 4917: 4906: 4903: 4902: 4877: 4873: 4864: 4860: 4839: 4835: 4819: 4811: 4807: 4803: 4801: 4798: 4797: 4770: 4766: 4757: 4753: 4722: 4714: 4710: 4706: 4698: 4695: 4694: 4663: 4659: 4650: 4646: 4617: 4609: 4604: 4601: 4600: 4570: 4567: 4566: 4555: 4517: 4513: 4504: 4500: 4477: 4469: 4467: 4464: 4463: 4439:Euclidean space 4435: 4415: 4407:Main articles: 4405: 4356: 4352: 4348: 4335: 4279: 4268: 4267:turn, 180°, or 4261: 4258: 4255: 4254: 4252: 4234:reference angle 4205: 4135:negative angles 4127:positive angles 4072: 4062: 4056: 4051: 4050: 4045:dimension, and 4042: 4032: 4024: 4017: 3994: 3991: 3990: 3989:can be denoted 3974: 3971: 3970: 3962: 3958: 3942: 3937: 3917: 3914: 3913: 3883: 3876: 3872: 3862: 3860: 3847: 3840: 3836: 3826: 3824: 3811: 3804: 3800: 3790: 3788: 3760: 3754: 3750: 3748: 3735: 3729: 3725: 3723: 3710: 3704: 3700: 3698: 3663: 3660: 3659: 3655: 3637: 3627: 3624: 3611: 3601: 3549: 3535: 3527: 3523: 3519: 3515: 3505: 3497: 3493: 3471: 3464: 3460: 3450: 3446: 3442: 3438: 3422: 3418: 3410: 3373: 3370: 3367: 3366: 3364: 3339: 3336: 3333: 3332: 3329: 3327: 3309: 3306: 3303: 3302: 3299: 3297: 3258: 3251: 3248: 3245: 3244: 3242: 3200: 3197: 3194: 3193: 3191: 3184: 3181: 3177: 3176: 3175: 3173: 3156: 3153: 3150: 3149: 3147: 3112: 3106: 3091: 3079: 3076: 3073: 3072: 3070: 3019: 3016: 3013: 3012: 3010: 3003: 3000: 2996: 2994: 2993: 2991: 2983: 2980: 2977: 2976: 2974: 2941: 2925: 2922: 2919: 2918: 2916: 2909: 2906: 2903: 2902: 2900: 2877:(compass) point 2865: 2862: 2859: 2858: 2855: 2853: 2846: 2843: 2840: 2839: 2837: 2830: 2827: 2824: 2823: 2821: 2814: 2811: 2807: 2806: 2805: 2803: 2788: 2785: 2782: 2781: 2779: 2751: 2656: 2653: 2650: 2649: 2647: 2640: 2637: 2634: 2633: 2631: 2620: 2617: 2614: 2613: 2611: 2604: 2601: 2598: 2597: 2595: 2562:of the Earth. ( 2548: 2545: 2542: 2541: 2539: 2532: 2529: 2526: 2525: 2523: 2516: 2513: 2510: 2509: 2507: 2500: 2497: 2494: 2493: 2491: 2390: 2389: 2386: 2383: 2382: 2380: 2376: 2375:. One turn is 2 2368: 2348: 2289:(rad), and the 2259: 2240: 2214: 2201: 2196: 2188: 2185: 2184: 2167: 2155: 2153: 2150: 2145: 2144: 2142: 2106: 2095: 2087: 2084: 2083: 2077: 2073: 2061: 2042: 2039: 2034: 2033: 2031: 2030: 2024: 1984: 1978: 1946: 1852: 1849: 1846: 1845: 1843: 1842: − 2) 1827: 1826: − 2) 1811: 1797: 1794: 1791: 1790: 1788: 1784: 1773:concave polygon 1750: 1745: 1720: 1659: 1658:turn, 180°, or 1652: 1649: 1646: 1645: 1643: 1632: 1628: 1580: 1579: 1555: 1530: 1529: 1511: 1507: 1492: 1488: 1485: 1467: 1463: 1448: 1444: 1439: 1437: 1434: 1433: 1393: 1390: 1386: 1385: 1384: 1382: 1375: 1372: 1369: 1368: 1366: 1355: 1351: 1347: 1339: 1335: 1331: 1291: 1271: 1251: 1243: 1240: 1239: 1230: 1206:interior angles 1202:exterior angles 1194: 1167:Adjacent angles 1121:180° − (180° − 1120: 1085: 1070: 1059: 1008:opposite angles 1004:vertical angles 995: 976: 970: 969: 898: 895: 887: 881: 878: 870: 868: 860: 854: 847: 844: 841: 840: 838: 836: 828: 821: 818: 815: 814: 812: 811: 803: 796: 793: 790: 789: 787: 785: 780: 768: 758: 755: 752: 751: 749: 747: 737: 734: 731: 730: 728: 727: 717: 714: 711: 710: 708: 701: 698: 695: 694: 692: 690: 680: 677: 674: 673: 671: 670: 660: 657: 654: 653: 651: 649: 644: 617:straight angle 597: 596: 595: 594: 593: 590: 582: 581: 579: 575: 571: 566: 558: 557: 552: 510: 492: 485: 482: 479: 478: 476: 435: 432: 428: 427: 426: 424: 413: 410: 407: 406: 404: 373: 368: 361: 352: 348: 337: 306: 304: 303: 301: 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: 8218: 8208: 8207: 8193: 8192: 8174: 8173:External links 8171: 8156:Chisholm, Hugh 8139: 8138: 8133: 8120: 8083: 8074: 8048: 8030: 8003: 7956: 7930:(2): 840–845. 7919: 7908: 7863: 7818: 7792:(3): 991–997. 7781: 7763:(9): 814–815. 7748: 7711: 7706: 7693: 7673: 7668: 7655: 7644: 7626:(6): R41–R51. 7611: 7574: 7554:(7): 605–614. 7539: 7521:(4): 175–179. 7510: 7505: 7488: 7485: 7482: 7481: 7463: 7447: 7441:978-0495382607 7440: 7422: 7392: 7367: 7355: 7326: 7289:(4): 454–460. 7273: 7261: 7249: 7237: 7222: 7207: 7147: 7135: 7092:Brinsmade 1936 7084: 7049: 7030:(3): 170–171. 7014: 7003:(5): 260–261. 6987: 6980: 6960: 6925:Nature Physics 6911: 6891: 6889:, p. 137. 6879: 6853: 6805: 6803:, p. 151. 6793: 6771: 6753: 6729: 6699: 6670: 6635: 6620:"angular unit" 6611: 6594:(2): 133–140. 6571: 6560: 6534: 6501: 6483: 6481:, p. 104. 6471: 6465:Plane Geometry 6454: 6442: 6418: 6404: 6380: 6378:, p. 255. 6368: 6353: 6332: 6320: 6308: 6293: 6278: 6266: 6250: 6239: 6227: 6226: 6224: 6221: 6218: 6217: 6198: 6194: 6191: 6185: 6180: 6162: 6123: 6120: 6117: 6106:Brinsmade 1936 6096: 6095: 6092: 6091: 6075: 6074: 6072: 6069: 6067: 6066: 6061: 6056: 6051: 6046: 6041: 6036: 6031: 6026: 6021: 6016: 6011: 6006: 6001: 5999:Dihedral angle 5996: 5991: 5986: 5981: 5976: 5971: 5966: 5961: 5956: 5954:Angle bisector 5951: 5937: 5932: 5926: 5924: 5921: 5918: 5917: 5905: 5893: 5879: 5876: 5873: 5869: 5868: 5856: 5844: 5830: 5827: 5824: 5820: 5819: 5817: 5805: 5791: 5788: 5785: 5781: 5780: 5777: 5774: 5771: 5768: 5763: 5748:In astronomy, 5739: 5738: 5735: 5732: 5725: 5668:vertical angle 5603: 5600: 5590:Leonhard Euler 5562:circular angle 5545: 5542: 5530: 5523: 5517: 5513: 5507: 5503: 5497: 5494: 5490: 5485: 5480: 5474: 5470: 5464: 5460: 5454: 5451: 5447: 5442: 5434: 5430: 5424: 5420: 5414: 5411: 5407: 5400: 5397: 5394: 5391: 5371: 5344: 5341: 5324: 5304: 5301: 5298: 5293: 5288: 5285: 5282: 5279: 5276: 5273: 5270: 5265: 5260: 5257: 5254: 5232: 5208: 5179: 5175: 5171: 5166: 5162: 5158: 5153: 5149: 5146: 5143: 5140: 5137: 5133: 5129: 5125: 5121: 5117: 5113: 5109: 5105: 5101: 5078: 5074: 5070: 5067: 5064: 5044: 5040: 5036: 5033: 5030: 5018: 5015: 5001: 4979: 4958: 4954: 4950: 4947: 4944: 4924: 4920: 4916: 4913: 4910: 4888: 4884: 4880: 4876: 4871: 4867: 4863: 4858: 4854: 4851: 4848: 4845: 4842: 4838: 4834: 4830: 4826: 4822: 4818: 4814: 4810: 4806: 4781: 4777: 4773: 4769: 4764: 4760: 4756: 4752: 4749: 4746: 4743: 4740: 4737: 4733: 4729: 4725: 4721: 4717: 4713: 4709: 4705: 4702: 4674: 4670: 4666: 4662: 4657: 4653: 4649: 4642: 4639: 4636: 4633: 4630: 4627: 4624: 4620: 4616: 4612: 4608: 4586: 4583: 4580: 4577: 4574: 4554: 4551: 4543:normal vectors 4528: 4524: 4520: 4516: 4511: 4507: 4503: 4499: 4496: 4493: 4490: 4487: 4484: 4480: 4476: 4472: 4434: 4431: 4427:Pierre Wantzel 4404: 4401: 4354: 4350: 4346: 4334: 4331: 4330: 4329: 4324:, such as the 4318: 4304: 4278: 4275: 4274: 4273: 4230: 4223: 4204: 4201: 4058:Main article: 4055: 4054:Signed angles 4052: 3998: 3978: 3940: 3936: 3933: 3930: 3927: 3924: 3921: 3901: 3898: 3895: 3889: 3886: 3879: 3875: 3871: 3868: 3865: 3859: 3853: 3850: 3843: 3839: 3835: 3832: 3829: 3823: 3817: 3814: 3807: 3803: 3799: 3796: 3793: 3787: 3784: 3781: 3778: 3775: 3772: 3766: 3763: 3757: 3753: 3747: 3741: 3738: 3732: 3728: 3722: 3716: 3713: 3707: 3703: 3697: 3694: 3691: 3688: 3682: 3679: 3676: 3673: 3670: 3667: 3622: 3463:= 1. However, 3419: 3411: 3409: 3406: 3403: 3402: 3391: 3388: 3385: 3381: 3380: 3357: 3354: 3351: 3347: 3346: 3316: 3294: 3291: 3285: 3284: 3273: 3270: 3267: 3263: 3262: 3220: 3217: 3214: 3208: 3207: 3140: 3137: 3134: 3128: 3127: 3101: 3098: 3095: 3088: 3087: 3039: 3036: 3033: 3027: 3026: 2950: 2947: 2939: 2933: 2932: 2885: 2882: 2879: 2873: 2872: 2800:second of time 2796:minute of time 2772: 2769: 2766: 2760: 2759: 2744: 2741: 2738: 2732: 2731: 2685:, also called 2679: 2676: 2673: 2667: 2666: 2580: 2577: 2574: 2568: 2567: 2506:of a degree = 2472: 2469: 2466: 2460: 2459: 2423: 2420: 2417: 2411: 2410: 2367: = 2 2357: 2354: 2346: 2340: 2339: 2336: 2333: 2330: 2258: 2255: 2253:is unaltered. 2226: 2221: 2218: 2213: 2207: 2204: 2200: 2195: 2192: 2119: 2115: 2112: 2109: 2102: 2099: 2094: 2091: 2082:in the angle: 2075: 2071: 2057: 1977: 1974: 1973: 1972: 1968: 1967:to the planes. 1960:dihedral angle 1957:) is called a 1945: 1942: 1941: 1940: 1929:exterior angle 1925: 1922: 1911: 1896: 1864:exterior angle 1859: 1768:interior angle 1763:simple polygon 1749: 1746: 1744: 1743: 1717: 1640: 1630: 1626: 1578: 1575: 1572: 1569: 1566: 1563: 1560: 1557: 1554: 1551: 1548: 1545: 1542: 1539: 1536: 1533: 1531: 1528: 1525: 1522: 1519: 1514: 1510: 1506: 1503: 1500: 1495: 1491: 1487: 1484: 1481: 1478: 1475: 1470: 1466: 1462: 1459: 1456: 1451: 1447: 1443: 1441: 1421:of the angle. 1381:turn, 90°, or 1359: 1353: 1349: 1345: 1337: 1333: 1329: 1300: 1297: 1294: 1290: 1287: 1284: 1280: 1277: 1274: 1270: 1267: 1264: 1260: 1257: 1254: 1250: 1247: 1229: 1226: 1193: 1192: 1164: 1044: 1043: 1040: 1037: 999: 975: 966: 963: 962: 959: 956: 953: 950: 947: 946:(0, 100) 944: 941: 934: 933: 930: 927: 924: 921: 918: 917:(0, 90)° 915: 912: 905: 904: 893: 876: 866: 834: 809: 783: 778: 772: 771: 766: 745: 725: 688: 668: 647: 642: 635: 634: 629: 625: 624: 621: 618: 615: 612: 609: 606: 603: 591: 584: 583: 577: 573: 569: 567: 560: 559: 553: 546: 545: 544: 543: 542: 538: 537: 530: 519:complete angle 507: 500: 497:straight angle 473: 466: 401: 390: 372: 369: 360: 357: 319: 315: 312: 309: 256: 252: 248: 244: 240: 228: 225: 149: 146: 132:length to its 26: 9: 6: 4: 3: 2: 8217: 8206: 8203: 8202: 8200: 8189: 8188: 8182: 8181:"Angle" 8177: 8176: 8170: 8167: 8166: 8161: 8157: 8152: 8151:public domain 8136: 8130: 8126: 8121: 8117: 8113: 8109: 8105: 8101: 8097: 8093: 8089: 8084: 8080: 8075: 8064: 8060: 8056: 8055: 8049: 8046: 8042: 8041: 8036: 8031: 8026: 8021: 8017: 8013: 8009: 8004: 8000: 7996: 7992: 7988: 7984: 7980: 7975: 7970: 7967:(5): 053002. 7966: 7962: 7957: 7953: 7949: 7945: 7941: 7937: 7933: 7929: 7925: 7920: 7916: 7915: 7909: 7904: 7899: 7895: 7891: 7886: 7881: 7878:(5): 053001. 7877: 7873: 7869: 7864: 7859: 7854: 7850: 7846: 7841: 7836: 7832: 7828: 7824: 7819: 7815: 7811: 7807: 7803: 7799: 7795: 7791: 7787: 7782: 7778: 7774: 7770: 7766: 7762: 7758: 7754: 7749: 7745: 7741: 7737: 7733: 7729: 7725: 7722:(5): 052001. 7721: 7717: 7712: 7709: 7703: 7699: 7694: 7689: 7685: 7684: 7679: 7674: 7671: 7665: 7661: 7656: 7652: 7651: 7645: 7641: 7637: 7633: 7629: 7625: 7621: 7617: 7612: 7608: 7604: 7600: 7596: 7592: 7588: 7584: 7580: 7575: 7570: 7565: 7561: 7557: 7553: 7549: 7545: 7540: 7536: 7532: 7528: 7524: 7520: 7516: 7511: 7508: 7502: 7498: 7497: 7491: 7490: 7479:, chapter six 7478: 7477: 7472: 7467: 7461:, p. 178 7460: 7456: 7455:Chisholm 1911 7451: 7443: 7437: 7433: 7426: 7410: 7406: 7402: 7396: 7381: 7377: 7371: 7364: 7359: 7352: 7341: 7340:www.boost.org 7337: 7330: 7322: 7318: 7314: 7310: 7306: 7302: 7297: 7292: 7288: 7284: 7277: 7270: 7265: 7258: 7253: 7246: 7241: 7234: 7229: 7227: 7219: 7214: 7212: 7203: 7199: 7195: 7191: 7187: 7183: 7178: 7173: 7169: 7165: 7158: 7156: 7154: 7152: 7144: 7139: 7133: 7129: 7125: 7121: 7117: 7113: 7109: 7105: 7101: 7097: 7093: 7088: 7080: 7076: 7072: 7068: 7064: 7060: 7053: 7045: 7041: 7037: 7033: 7029: 7025: 7018: 7010: 7006: 7002: 6998: 6991: 6983: 6977: 6973: 6972: 6964: 6956: 6952: 6947: 6942: 6938: 6934: 6930: 6926: 6922: 6915: 6908: 6904: 6903: 6895: 6888: 6883: 6871: 6867: 6862: 6857: 6849: 6846: 6827: 6823:of a sector ( 6822: 6818: 6814: 6809: 6802: 6797: 6789: 6785: 6780: 6775: 6767: 6763: 6757: 6749: 6745: 6744: 6739: 6733: 6718: 6714: 6710: 6703: 6688: 6684: 6680: 6674: 6660:on 2008-06-28 6659: 6655: 6653: 6648: 6642: 6640: 6625: 6621: 6615: 6597: 6593: 6589: 6582: 6575: 6567: 6563: 6557: 6550: 6549: 6544: 6538: 6529: 6528: 6523: 6520: 6512: 6505: 6498: 6492: 6490: 6488: 6480: 6475: 6467: 6466: 6458: 6452:, p. 97. 6451: 6446: 6432: 6428: 6422: 6416: 6415:Chisholm 1911 6411: 6409: 6394: 6390: 6384: 6377: 6372: 6365: 6360: 6358: 6348: 6347: 6342: 6336: 6329: 6324: 6318:, p. 71. 6317: 6312: 6305: 6300: 6298: 6291:, p. 41. 6290: 6285: 6283: 6276:, p. 18. 6275: 6270: 6263: 6259: 6258:Chisholm 1911 6254: 6248: 6243: 6237: 6232: 6228: 6214: 6192: 6189: 6183: 6168: 6161: 6157: 6153: 6149: 6145: 6141: 6137: 6118: 6107: 6101: 6097: 6087: 6080: 6076: 6065: 6062: 6060: 6057: 6055: 6052: 6050: 6047: 6045: 6042: 6040: 6037: 6035: 6034:Phase (waves) 6032: 6030: 6027: 6025: 6022: 6020: 6017: 6015: 6012: 6010: 6007: 6005: 6002: 6000: 5997: 5995: 5992: 5990: 5987: 5985: 5984:Central angle 5982: 5980: 5977: 5975: 5972: 5970: 5967: 5965: 5962: 5960: 5957: 5955: 5952: 5949: 5945: 5941: 5938: 5936: 5933: 5931: 5928: 5927: 5906: 5894: 5880: 5877: 5874: 5871: 5870: 5857: 5845: 5831: 5828: 5825: 5822: 5821: 5818: 5806: 5792: 5789: 5786: 5783: 5782: 5778: 5775: 5772: 5769: 5767: 5764: 5761: 5760: 5757: 5755: 5751: 5746: 5744: 5743:rule of thumb 5736: 5733: 5730: 5729:little finger 5726: 5723: 5719: 5715: 5714: 5713: 5710: 5708: 5704: 5700: 5696: 5695:apparent size 5691: 5689: 5685: 5681: 5677: 5673: 5669: 5664: 5662: 5658: 5654: 5653: 5648: 5647: 5642: 5638: 5633: 5631: 5627: 5623: 5619: 5615: 5614: 5609: 5599: 5597: 5596: 5591: 5587: 5583: 5579: 5575: 5571: 5567: 5563: 5559: 5555: 5551: 5541: 5528: 5521: 5515: 5511: 5505: 5501: 5495: 5492: 5488: 5483: 5478: 5472: 5468: 5462: 5458: 5452: 5449: 5445: 5440: 5432: 5428: 5422: 5418: 5412: 5409: 5405: 5398: 5395: 5392: 5389: 5381: 5379: 5374: 5370: 5366: 5362: 5358: 5354: 5353:metric tensor 5350: 5340: 5338: 5322: 5302: 5299: 5283: 5280: 5277: 5274: 5271: 5255: 5252: 5196: 5195:Hilbert space 5191: 5151: 5144: 5138: 5135: 5131: 5127: 5123: 5111: 5099: 5090: 5065: 5062: 5031: 5028: 5014: 4945: 4942: 4911: 4908: 4899: 4886: 4856: 4849: 4843: 4840: 4836: 4832: 4828: 4816: 4804: 4795: 4792: 4779: 4747: 4741: 4738: 4735: 4731: 4719: 4707: 4703: 4700: 4692: 4690: 4687:In a complex 4685: 4672: 4637: 4631: 4628: 4625: 4614: 4598: 4581: 4578: 4575: 4564: 4560: 4553:Inner product 4550: 4548: 4544: 4539: 4526: 4494: 4488: 4485: 4482: 4474: 4461: 4459: 4455: 4451: 4448: 4444: 4440: 4430: 4428: 4424: 4420: 4414: 4410: 4400: 4399:, biconcave. 4398: 4394: 4390: 4386: 4382: 4372: 4368: 4364: 4344: 4339: 4328:of the angle. 4327: 4323: 4319: 4315: 4311: 4310: 4305: 4302: 4298: 4294: 4293: 4288: 4287: 4286: 4284: 4251: 4247: 4243: 4239: 4238:related angle 4235: 4231: 4228: 4224: 4221: 4217: 4216: 4211: 4207: 4206: 4200: 4197: 4193: 4189: 4184: 4182: 4181:normal vector 4178: 4173: 4171: 4167: 4162: 4160: 4156: 4155:anticlockwise 4152: 4148: 4144: 4140: 4136: 4132: 4128: 4124: 4123:terminal side 4120: 4116: 4112: 4107: 4105: 4101: 4093: 4089: 4085: 4081: 4076: 4071: 4067: 4061: 4048: 4040: 4036: 4030: 4020: 4015: 4010: 3996: 3976: 3968: 3938: 3934: 3931: 3928: 3925: 3922: 3919: 3899: 3896: 3893: 3887: 3884: 3877: 3869: 3866: 3857: 3851: 3848: 3841: 3833: 3830: 3821: 3815: 3812: 3805: 3797: 3794: 3785: 3782: 3779: 3776: 3773: 3770: 3764: 3761: 3755: 3751: 3745: 3739: 3736: 3730: 3726: 3720: 3714: 3711: 3705: 3701: 3695: 3692: 3689: 3686: 3680: 3677: 3674: 3671: 3668: 3665: 3653: 3649: 3648:Taylor series 3644: 3640: 3634: 3630: 3625: 3621: 3614: 3608: 3605: 3599: 3595: 3594:base quantity 3591: 3586: 3584: 3580: 3576: 3573:(rad/s), and 3572: 3568: 3567:angular speed 3564: 3563:angle measure 3559: 3554: 3546: 3542: 3538: 3533: 3512: 3508: 3501: 3491: 3487: 3482: 3478: 3474: 3470: 3457: 3453: 3435: 3431: 3427: 3426: 3416: 3400: 3396: 3392: 3389: 3386: 3383: 3382: 3362: 3361:diameter part 3358: 3355: 3352: 3350:diameter part 3349: 3348: 3325: 3321: 3317: 3295: 3292: 3290: 3287: 3286: 3282: 3278: 3274: 3271: 3268: 3265: 3264: 3240: 3239: 3236:angle of the 3233: 3229: 3225: 3221: 3218: 3215: 3213: 3210: 3209: 3171: 3167: 3166: 3145: 3141: 3138: 3135: 3133: 3130: 3129: 3126: 3122: 3118: 3110: 3105:multiples of 3102: 3099: 3096: 3090: 3089: 3086: 3066: 3062: 3058: 3054: 3050: 3049: 3048:binary radian 3044: 3043:binary degree 3040: 3037: 3034: 3032: 3031:binary degree 3029: 3028: 2972: 2967: 2966:approximately 2963: 2959: 2955: 2951: 2948: 2945: 2940: 2938: 2935: 2934: 2898: 2894: 2890: 2886: 2883: 2880: 2878: 2875: 2874: 2852: turn = 2836: quad = 2801: 2797: 2777: 2773: 2770: 2767: 2765: 2762: 2761: 2757: 2749: 2745: 2742: 2739: 2737: 2734: 2733: 2729: 2725: 2724:triangulation 2721: 2717: 2716:nautical mile 2714: 2710: 2706: 2702: 2698: 2694: 2693: 2688: 2684: 2680: 2677: 2674: 2672: 2669: 2668: 2665: 2629: 2626:of a degree ( 2593: 2589: 2585: 2584:second of arc 2581: 2578: 2575: 2573: 2570: 2569: 2565: 2561: 2557: 2556:nautical mile 2489: 2485: 2481: 2477: 2476:minute of arc 2473: 2470: 2467: 2465: 2462: 2461: 2457: 2453: 2449: 2445: 2441: 2436: 2432: 2428: 2424: 2421: 2418: 2416: 2413: 2412: 2408: 2404: 2400: 2374: 2366: 2362: 2358: 2355: 2352: 2347: 2345: 2342: 2341: 2337: 2334: 2331: 2328: 2327: 2324: 2321: 2319: 2315: 2310: 2308: 2304: 2300: 2296: 2292: 2288: 2284: 2280: 2279:angular units 2276: 2272: 2263: 2254: 2252: 2248: 2243: 2239:The value of 2237: 2224: 2219: 2216: 2211: 2205: 2202: 2198: 2193: 2190: 2182: 2180: 2176: 2170: 2165: 2148: 2139: 2137: 2133: 2117: 2100: 2097: 2092: 2089: 2081: 2069: 2065: 2060: 2045: 2037: 2027: 2021: 2017: 2015: 2011: 2007: 2003: 1998: 1996: 1992: 1989: 1983: 1969: 1966: 1962: 1961: 1956: 1952: 1948: 1947: 1938: 1934: 1930: 1926: 1923: 1920: 1916: 1915:extended side 1912: 1909: 1905: 1901: 1897: 1894: 1890: 1886: 1882: 1878: 1874: 1870: 1866: 1865: 1860: 1841: 1837: 1833: 1825: 1821: 1817: 1809: 1808:quadrilateral 1806: 1782: 1778: 1774: 1770: 1769: 1765:is called an 1764: 1760: 1759: 1754: 1742: 1741:of an angle. 1740: 1736: 1730: 1726: 1718: 1712: 1708: 1706: 1703: 1700: 1698: 1697:tangent lines 1693: 1691: 1687: 1686:parallelogram 1683: 1679: 1678:straight line 1675: 1671: 1665: 1641: 1636: 1635:supplementary 1623: 1619: 1617: 1613: 1608: 1606: 1602: 1598: 1593: 1576: 1573: 1570: 1567: 1564: 1561: 1558: 1552: 1549: 1546: 1543: 1540: 1537: 1534: 1526: 1523: 1520: 1517: 1512: 1508: 1504: 1501: 1498: 1493: 1489: 1482: 1479: 1476: 1473: 1468: 1464: 1460: 1457: 1454: 1449: 1445: 1431: 1427: 1422: 1420: 1415: 1413: 1409: 1403: 1364: 1361: 1360: 1343: 1327: 1326:complementary 1322: 1318: 1315: 1312: 1285: 1282: 1265: 1262: 1245: 1237: 1235: 1225: 1223: 1219: 1215: 1211: 1207: 1203: 1199: 1190: 1189: 1184: 1180: 1179:supplementary 1176: 1175:complementary 1172: 1168: 1165: 1161:are adjacent. 1160: 1156: 1151: 1147: 1145: 1141: 1137: 1132: 1128: 1124: 1118: 1114: 1110: 1106: 1102: 1098: 1094: 1089: 1083: 1079: 1076:. Both angle 1074: 1068: 1063: 1057: 1053: 1049: 1041: 1038: 1035: 1034: 1033: 1031: 1027: 1023: 1017: 1016:vert. opp. ∠s 1013: 1009: 1005: 1001: 1000: 998: 993: 985: 980: 968:Vertical and 960: 957: 954: 951: 948: 945: 942: 940: 939: 936: 935: 931: 928: 925: 922: 919: 916: 913: 911: 910: 907: 906: 901: 894: 890: 884: 877: 873: 867: 863: 857: 835: 831: 810: 806: 784: 779: 777: 774: 773: 767: 746: 726: 689: 669: 648: 643: 641: 640: 637: 636: 633: 627: 626: 622: 620:reflex angle 619: 616: 614:obtuse angle 613: 610: 607: 604: 601: 600: 588: 564: 556: 550: 541: 535: 534:oblique angle 531: 528: 524: 520: 516: 508: 505: 501: 498: 474: 471: 467: 464: 463: 462:perpendicular 458: 457: 452: 451: 446: 445: 422: 402: 399: 395: 391: 388: 384: 383: 382: 380: 379: 366: 356: 346: 345: 334: 317: 291: 286: 284: 280: 276: 272: 267: 262: 238: 237:Greek letters 234: 224: 222: 218: 217:straight line 214: 210: 206: 202: 200: 196: 192: 188: 183: 173: 169: 164: 159: 155: 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: 18:Oblique angle 8185: 8163: 8140: 8124: 8091: 8087: 8078: 8067:, retrieved 8063:the original 8053: 8038: 8015: 8011: 7964: 7960: 7927: 7923: 7913: 7875: 7871: 7833:(1): 40–47. 7830: 7826: 7789: 7785: 7760: 7756: 7719: 7715: 7697: 7682: 7678:Heath, T. L. 7659: 7649: 7623: 7619: 7582: 7578: 7551: 7547: 7518: 7514: 7495: 7487:Bibliography 7475: 7466: 7459:Heiberg 1908 7450: 7432:Trigonometry 7431: 7425: 7413:. Retrieved 7404: 7395: 7383:. Retrieved 7379: 7370: 7365:, p. 3. 7358: 7350: 7343:. Retrieved 7339: 7329: 7286: 7282: 7276: 7269:Quincey 2021 7264: 7252: 7247:, p. 6. 7240: 7233:Torrens 1986 7218:Quincey 2016 7167: 7163: 7138: 7128:Leonard 2021 7124:Quincey 2021 7104:Torrens 1986 7087: 7065:(2): 84–87. 7062: 7058: 7052: 7027: 7023: 7017: 7000: 6996: 6990: 6970: 6963: 6928: 6924: 6914: 6906: 6901: 6894: 6882: 6869: 6865: 6856: 6847: 6844: 6825: 6820: 6813:Quincey 2016 6808: 6796: 6787: 6783: 6774: 6768:. p. 2. 6765: 6756: 6742: 6732: 6721:. Retrieved 6712: 6702: 6691:. Retrieved 6673: 6662:. Retrieved 6658:the original 6650: 6627:. Retrieved 6623: 6614: 6603:. Retrieved 6591: 6587: 6574: 6547: 6537: 6525: 6515:as cited in 6510: 6504: 6496: 6474: 6464: 6457: 6445: 6434:. Retrieved 6430: 6421: 6396:. Retrieved 6392: 6383: 6371: 6346:The Elements 6345: 6335: 6323: 6311: 6306:, p. 9. 6269: 6262:Heiberg 1908 6253: 6242: 6236:Sidorov 2001 6231: 6167:Quincey 2021 6159: 6151: 6100: 6085: 6079: 6064:Zenith angle 6009:Golden angle 5747: 5740: 5711: 5694: 5692: 5665: 5650: 5644: 5634: 5611: 5605: 5593: 5561: 5560:just as the 5547: 5382: 5377: 5372: 5368: 5364: 5360: 5346: 5192: 5091: 5020: 4900: 4796: 4793: 4693: 4686: 4599: 4562: 4556: 4545:and between 4540: 4462: 4453: 4449: 4445:between two 4442: 4441:, the angle 4436: 4416: 4396: 4392: 4388: 4384: 4380: 4370: 4360: 4342: 4307: 4296: 4290: 4280: 4245: 4241: 4237: 4233: 4226: 4220:right angles 4219: 4213: 4209: 4185: 4174: 4169: 4165: 4163: 4150: 4146: 4142: 4138: 4134: 4126: 4122: 4115:initial side 4114: 4108: 4100:orientations 4097: 4028: 4018: 4014:natural unit 4011: 3967:pure numbers 3654:of an angle 3642: 3638: 3632: 3628: 3619: 3609: 3603: 3587: 3556:In 1993 the 3555: 3544: 3540: 3536: 3510: 3506: 3502: 3480: 3476: 3472: 3455: 3451: 3433: 3429: 3423: 3420: 3401:is 224 zam. 3360: 3319: 3281:Eratosthenes 3276: 3235: 3223: 3190: rad = 3163: 3143: 3123:. See also: 3104: 3068: 3064: 3056: 3052: 3046: 3042: 2965: 2958:scope sights 2943: 2892: 2888: 2871: grad. 2820: rad = 2799: 2795: 2775: 2747: 2719: 2696: 2690: 2686: 2682: 2627: 2591: 2587: 2583: 2563: 2560:great circle 2487: 2483: 2479: 2475: 2455: 2430: 2426: 2398: 2372: 2364: 2360: 2350: 2338:Description 2322: 2311: 2306: 2302: 2278: 2268: 2250: 2246: 2241: 2238: 2183: 2168: 2163: 2146: 2140: 2064:circular arc 2055: 2043: 2035: 2025: 2013: 2001: 1999: 1994: 1990: 1987: 1985: 1958: 1936: 1932: 1928: 1888: 1862: 1839: 1835: 1831: 1823: 1819: 1766: 1738: 1734: 1732: 1728: 1724: 1707: 1704: 1701: 1694: 1681: 1667: 1663: 1634: 1609: 1594: 1429: 1425: 1423: 1418: 1416: 1411: 1408:complementum 1407: 1405: 1362: 1341: 1325: 1316: 1313: 1238: 1233: 1231: 1221: 1217: 1213: 1209: 1205: 1201: 1195: 1186: 1185:angles (see 1183:explementary 1182: 1178: 1174: 1170: 1166: 1158: 1154: 1143: 1139: 1135: 1130: 1126: 1122: 1116: 1112: 1108: 1104: 1100: 1096: 1092: 1087: 1081: 1077: 1072: 1066: 1061: 1055: 1051: 1047: 1045: 1021: 1019: 1015: 1011: 1007: 1003: 996: 992:Zenith angle 897: 886: 880: 869: 859: 853: 827: 802: 611:right angle 608:acute angle 592:Reflex angle 539: 533: 526: 522: 518: 514: 504:reflex angle 503: 496: 470:obtuse angle 469: 460: 454: 448: 442: 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: 7585:(1): 1–12. 7116:Foster 2010 7096:Romain 1962 6450:Jacobs 1974 6376:Jacobs 1974 6247:Slocum 2007 6156:Foster 2010 6136:Romain 1962 6044:Solid angle 5754:declination 5720:and of the 4458:dot product 4393:amphicoelic 4371:amphicyrtic 4177:orientation 4084:unit circle 4047:Mathematica 4043:plane_angle 3277:hexacontade 3266:hexacontade 3232:sexagesimal 3228:Babylonians 3165:right angle 3121:WP 43S 3085:of a turn. 2937:milliradian 2713:sexagesimal 2554:degrees. A 2435:sexagesimal 2285:( ° ), the 2273:in various 2002:orientation 1877:orientation 1858: turn. 1810:add up to 2 1625:The angles 1198:transversal 984:Hatch marks 974:angle pairs 605:zero angle 572:), obtuse ( 555:Right angle 523:round angle 444:right angle 394:acute angle 365:Dutch angle 8094:(1): 1–7. 8088:Metrologia 7974:2108.05704 7961:Metrologia 7924:Metrologia 7885:2203.12392 7872:Metrologia 7827:Metrologia 7786:Metrologia 7716:Metrologia 7620:Metrologia 7579:Metrologia 7296:1705.03765 7283:Metrologia 7177:1604.02373 7164:Metrologia 7120:Mills 2016 6931:(8): 888. 6723:2019-08-05 6693:2019-08-05 6664:2019-08-05 6629:2020-08-31 6605:2019-08-06 6436:2020-08-17 6398:2020-08-17 6316:Moser 1971 6289:Moser 1971 6223:References 6059:Trisection 6039:Protractor 6019:Horn angle 5576:since the 4547:skew lines 4389:sistroidal 4385:xystroidal 4188:navigation 4094:direction. 4064:See also: 3646:, and the 3585:(kg⋅m/s). 3581:(N⋅m) and 3324:Babylonian 3293:144 to 180 2897:navigation 2895:, used in 2776:hour angle 2764:hour angle 2590:, or just 2486:, or just 2452:ballistics 2335:In degrees 1980:See also: 1955:polyhedron 1908:concurrent 1783:add up to 1424:If angles 1419:complement 1342:complement 1138:and angle 1080:and angle 515:full angle 456:orthogonal 387:zero angle 8116:250801509 8045:EMS Press 8018:(3): 97. 7999:236547235 7952:125438811 7840:1409.2794 7814:126032642 7744:234036217 7688:Cambridge 7640:117711734 7607:250750831 7321:119418270 7202:119294905 7100:Eder 1982 6955:225445454 6875:1 rad = 1 6713:HP Museum 6527:MathWorld 6197:Θ 6193:π 6122:⟩ 6119:θ 6116:⟨ 5703:full moon 5676:elevation 5637:astronomy 5622:longitude 5608:geography 5396:θ 5393:⁡ 5284:⁡ 5278:≤ 5256:⁡ 5145:θ 5139:⁡ 5120:⟩ 5104:⟨ 5089:given by 5066:⁡ 5032:⁡ 4946:⁡ 4912:⁡ 4850:θ 4844:⁡ 4825:⟩ 4809:⟨ 4748:θ 4742:⁡ 4728:⟩ 4712:⟨ 4704:⁡ 4638:θ 4632:⁡ 4623:⟩ 4607:⟨ 4585:⟩ 4582:⋅ 4576:⋅ 4573:⟨ 4495:θ 4489:⁡ 4475:⋅ 4381:cissoidal 4215:congruent 4159:clockwise 4104:rotations 4092:clockwise 4025:1 rad = 1 3935:θ 3929:θ 3926:η 3897:⋯ 3870:θ 3867:η 3858:− 3834:θ 3831:η 3798:θ 3795:η 3786:− 3783:θ 3780:η 3774:⋯ 3746:− 3696:− 3681:⁡ 3672:θ 3669:⁡ 3658:becomes: 3569:(rad/s), 3494:1 rad = 1 3488:equal to 3356:≈0.95493° 2854:⁠16 2758:radians. 2728:surveying 2701:kilometre 2588:arcsecond 2576:1,296,000 2572:arcsecond 2484:arcminute 2464:arcminute 2448:astronomy 2212:⋅ 2206:π 2191:θ 2090:θ 2068:compasses 1991:congruent 1919:collinear 1904:bisectors 1739:conjugate 1735:explement 1601:cotangent 1574:⁡ 1562:⁡ 1550:⁡ 1538:⁡ 1518:⁡ 1499:⁡ 1474:⁡ 1455:⁡ 1289:∠ 1269:∠ 1249:∠ 1111:or angle 1069:would be 1058:would be 764:, 1) turn 318:^ 261:variables 152:The word 112:magnitude 8199:Category 7698:Geometry 7415:26 April 7409:Archived 6764:(1946). 6740:(1947). 6717:Archived 6687:Archived 6596:Archived 6566:archived 5923:See also 5672:altitude 5670:such as 5618:latitude 5598:(1748). 5554:argument 5359:. 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4242:θ 4229:. 4170:θ 4166:θ 4151:y 4147:x 4139:y 4033:η 4019:η 3939:/ 3932:= 3923:= 3920:x 3900:, 3894:+ 3888:! 3885:7 3878:7 3874:) 3864:( 3852:! 3849:5 3842:5 3838:) 3828:( 3822:+ 3816:! 3813:3 3806:3 3802:) 3792:( 3777:= 3771:+ 3765:! 3762:7 3756:7 3752:x 3740:! 3737:5 3731:5 3727:x 3721:+ 3715:! 3712:3 3706:3 3702:x 3693:x 3690:= 3687:x 3675:= 3656:θ 3639:s 3629:s 3623:0 3620:ε 3613:η 3604:r 3602:π 3550:ω 3545:r 3543:/ 3541:v 3537:ω 3528:θ 3524:r 3520:θ 3516:r 3507:y 3490:1 3481:r 3479:/ 3477:A 3473:θ 3456:r 3452:s 3447:r 3443:s 3439:θ 3434:r 3432:/ 3430:s 3425:θ 3417:. 3371:/ 3368:1 3340:2 3337:/ 3334:1 3330:+ 3315:° 3310:2 3307:/ 3304:1 3300:+ 3259:π 3252:6 3249:/ 3246:1 3216:6 3201:4 3198:/ 3195:1 3185:2 3182:/ 3178:π 3157:4 3154:/ 3151:1 3136:4 3113:π 3107:π 3097:2 3092:π 3077:/ 3074:1 3065:n 3017:/ 3014:1 3001:/ 2997:π 2995:2 2990:( 2981:/ 2978:1 2969:" 2944:π 2926:8 2923:/ 2920:1 2907:/ 2904:1 2866:3 2863:/ 2860:2 2856:+ 2844:/ 2841:1 2831:6 2828:/ 2825:1 2812:/ 2808:π 2786:/ 2783:1 2752:π 2740:1 2720:n 2718:( 2654:/ 2638:/ 2635:7 2628:n 2618:/ 2615:1 2602:/ 2599:1 2564:n 2546:/ 2530:/ 2514:/ 2511:1 2498:/ 2495:1 2456:n 2454:( 2391:π 2387:/ 2377:π 2369:π 2365:n 2351:π 2349:2 2307:n 2303:n 2251:r 2249:/ 2247:s 2242:θ 2225:. 2220:r 2217:s 2203:2 2199:k 2194:= 2169:k 2164:k 2156:π 2154:2 2151:/ 2147:k 2118:. 2114:d 2111:a 2108:r 2101:r 2098:s 2093:= 2076:r 2072:s 2059:θ 2052:. 2044:r 2040:/ 2036:s 2026:θ 1935:( 1921:. 1853:2 1850:/ 1847:1 1840:n 1836:n 1832:n 1828:π 1824:n 1820:n 1812:π 1798:2 1795:/ 1792:1 1785:π 1721:π 1660:π 1653:2 1650:/ 1647:1 1631:b 1627:a 1614:" 1577:B 1568:= 1565:A 1553:B 1544:= 1541:A 1527:1 1524:= 1521:B 1513:2 1505:+ 1502:A 1494:2 1483:1 1480:= 1477:B 1469:2 1461:+ 1458:A 1450:2 1430:B 1426:A 1394:2 1391:/ 1387:π 1376:4 1373:/ 1370:1 1354:b 1350:a 1346:a 1338:b 1336:( 1334:b 1330:a 1299:C 1296:O 1293:B 1286:m 1283:+ 1279:B 1276:O 1273:A 1266:m 1263:= 1259:C 1256:O 1253:A 1246:m 1159:B 1155:A 1144:x 1140:B 1136:A 1131:x 1127:x 1123:x 1117:B 1113:D 1109:C 1105:B 1101:D 1097:C 1093:B 1088:x 1082:D 1078:C 1073:x 1067:D 1062:x 1056:C 1052:x 1048:A 994:. 899:π 896:2 888:π 882:π 879:( 871:π 861:π 855:π 848:2 845:/ 842:1 837:( 829:π 822:2 819:/ 816:1 804:π 797:2 794:/ 791:1 759:2 756:/ 753:1 748:( 738:2 735:/ 732:1 718:2 715:/ 712:1 702:4 699:/ 696:1 691:( 681:4 678:/ 675:1 661:4 658:/ 655:1 578:c 574:b 570:a 536:. 529:. 511:π 506:. 499:. 493:π 486:2 483:/ 480:1 465:. 436:2 433:/ 429: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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