Spherical trigonometry

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case, which is discussed below. For three given elements there are six cases: three sides, two sides and an included or opposite angle, two angles and an included or opposite side, or three angles. (The last case has no analogue in planar trigonometry.) No single method solves all cases. The figure below shows the seven non-trivial cases: in each case the given sides are marked with a cross-bar and the given angles with an arc. (The given elements are also listed below the triangle). In the summary notation here such as ASA, A refers to a given angle and S refers to a given side, and the sequence of A's and S's in the notation refers to the corresponding sequence in the triangle.

6185: 2930: 5069: 4283: 7800:{\displaystyle {\begin{alignedat}{4}&{\text{(Q1)}}&\qquad \cos C&=-\cos A\,\cos B,&\qquad \qquad &{\text{(Q6)}}&\qquad \tan B&=-\cos a\,\tan C,\\&{\text{(Q2)}}&\sin A&=\sin a\,\sin C,&&{\text{(Q7)}}&\tan A&=-\cos b\,\tan C,\\&{\text{(Q3)}}&\sin B&=\sin b\,\sin C,&&{\text{(Q8)}}&\cos a&=\sin b\,\cos A,\\&{\text{(Q4)}}&\tan A&=\tan a\,\sin B,&&{\text{(Q9)}}&\cos b&=\sin a\,\cos B,\\&{\text{(Q5)}}&\tan B&=\tan b\,\sin A,&&{\text{(Q10)}}&\cos C&=-\cot a\,\cot b.\end{alignedat}}} 7202:{\displaystyle {\begin{alignedat}{4}&{\text{(R1)}}&\qquad \cos c&=\cos a\,\cos b,&\qquad \qquad &{\text{(R6)}}&\qquad \tan b&=\cos A\,\tan c,\\&{\text{(R2)}}&\sin a&=\sin A\,\sin c,&&{\text{(R7)}}&\tan a&=\cos B\,\tan c,\\&{\text{(R3)}}&\sin b&=\sin B\,\sin c,&&{\text{(R8)}}&\cos A&=\sin B\,\cos a,\\&{\text{(R4)}}&\tan a&=\tan A\,\sin b,&&{\text{(R9)}}&\cos B&=\sin A\,\cos b,\\&{\text{(R5)}}&\tan b&=\tan B\,\sin a,&&{\text{(R10)}}&\cos c&=\cot A\,\cot B.\end{alignedat}}} 3833:{\displaystyle {\begin{alignedat}{5}{\text{(CT1)}}&&\qquad \cos b\,\cos C&=\cot a\,\sin b-\cot A\,\sin C\qquad &&(aCbA)\\{\text{(CT2)}}&&\cos b\,\cos A&=\cot c\,\sin b-\cot C\,\sin A&&(CbAc)\\{\text{(CT3)}}&&\cos c\,\cos A&=\cot b\,\sin c-\cot B\,\sin A&&(bAcB)\\{\text{(CT4)}}&&\cos c\,\cos B&=\cot a\,\sin c-\cot A\,\sin B&&(AcBa)\\{\text{(CT5)}}&&\cos a\,\cos B&=\cot c\,\sin a-\cot C\,\sin B&&(cBaC)\\{\text{(CT6)}}&&\cos a\,\cos C&=\cot b\,\sin a-\cot B\,\sin C&&(BaCb)\end{alignedat}}} 8818: 7217: 123: 5627: 6384: 3198:{\displaystyle \cos \!{\Bigl (}{\begin{smallmatrix}{\text{inner}}\\{\text{side}}\end{smallmatrix}}{\Bigr )}\cos \!{\Bigl (}{\begin{smallmatrix}{\text{inner}}\\{\text{angle}}\end{smallmatrix}}{\Bigr )}=\cot \!{\Bigl (}{\begin{smallmatrix}{\text{outer}}\\{\text{side}}\end{smallmatrix}}{\Bigr )}\sin \!{\Bigl (}{\begin{smallmatrix}{\text{inner}}\\{\text{side}}\end{smallmatrix}}{\Bigr )}-\cot \!{\Bigl (}{\begin{smallmatrix}{\text{outer}}\\{\text{angle}}\end{smallmatrix}}{\Bigr )}\sin \!{\Bigl (}{\begin{smallmatrix}{\text{inner}}\\{\text{angle}}\end{smallmatrix}}{\Bigr )},} 8684: 438: 207: 5607:{\displaystyle {\begin{aligned}{\frac {\sin {\tfrac {1}{2}}(A+B)}{\cos {\tfrac {1}{2}}C}}={\frac {\cos {\tfrac {1}{2}}(a-b)}{\cos {\tfrac {1}{2}}c}}&\qquad \qquad &{\frac {\sin {\tfrac {1}{2}}(A-B)}{\cos {\tfrac {1}{2}}C}}={\frac {\sin {\tfrac {1}{2}}(a-b)}{\sin {\tfrac {1}{2}}c}}\\{\frac {\cos {\tfrac {1}{2}}(A+B)}{\sin {\tfrac {1}{2}}C}}={\frac {\cos {\tfrac {1}{2}}(a+b)}{\cos {\tfrac {1}{2}}c}}&\qquad &{\frac {\cos {\tfrac {1}{2}}(A-B)}{\sin {\tfrac {1}{2}}C}}={\frac {\sin {\tfrac {1}{2}}(a+b)}{\sin {\tfrac {1}{2}}c}}\end{aligned}}} 4908:{\displaystyle {\begin{alignedat}{5}\sin {\tfrac {1}{2}}A&={\sqrt {\frac {\sin(s-b)\sin(s-c)}{\sin b\sin c}}}&\qquad \qquad \sin {\tfrac {1}{2}}a&={\sqrt {\frac {-\cos S\cos(S-A)}{\sin B\sin C}}}\\\cos {\tfrac {1}{2}}A&={\sqrt {\frac {\sin s\sin(s-a)}{\sin b\sin c}}}&\cos {\tfrac {1}{2}}a&={\sqrt {\frac {\cos(S-B)\cos(S-C)}{\sin B\sin C}}}\\\tan {\tfrac {1}{2}}A&={\sqrt {\frac {\sin(s-b)\sin(s-c)}{\sin s\sin(s-a)}}}&\tan {\tfrac {1}{2}}a&={\sqrt {\frac {-\cos S\cos(S-A)}{\cos(S-B)\cos(S-C)}}}\end{alignedat}}} 1243: 10856:"One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical discipline in its own right rather than as just a tool for astronomical applications. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. This work is really the first in history on trigonometry as an independent branch of pure mathematics and the first in which all six cases for a right-angled spherical triangle are set forth" 6180:{\displaystyle {\begin{aligned}\tan {\tfrac {1}{2}}(A+B)={\frac {\cos {\tfrac {1}{2}}(a-b)}{\cos {\tfrac {1}{2}}(a+b)}}\cot {\tfrac {1}{2}}C&\qquad &\tan {\tfrac {1}{2}}(a+b)={\frac {\cos {\tfrac {1}{2}}(A-B)}{\cos {\tfrac {1}{2}}(A+B)}}\tan {\tfrac {1}{2}}c\\\tan {\tfrac {1}{2}}(A-B)={\frac {\sin {\tfrac {1}{2}}(a-b)}{\sin {\tfrac {1}{2}}(a+b)}}\cot {\tfrac {1}{2}}C&\qquad &\tan {\tfrac {1}{2}}(a-b)={\frac {\sin {\tfrac {1}{2}}(A-B)}{\sin {\tfrac {1}{2}}(A+B)}}\tan {\tfrac {1}{2}}c\end{aligned}}} 2425: 180:—are bounded by two great-circle arcs: a familiar example is the curved outward-facing surface of a segment of an orange. Three arcs serve to define a spherical triangle, the principal subject of this article. Polygons with higher numbers of sides (4-sided spherical quadrilaterals, 5-sided spherical pentagons, etc.) are defined in similar manner. Analogously to their plane counterparts, spherical polygons with more than 3 sides can always be treated as the composition of spherical triangles. 8952: 1994: 33: 4163: 2420:{\displaystyle {\begin{aligned}\sin ^{2}A&=1-\left({\frac {\cos a-\cos b\cos c}{\sin b\sin c}}\right)^{2}\\&={\frac {(1-\cos ^{2}b)(1-\cos ^{2}c)-(\cos a-\cos b\cos c)^{2}}{\sin ^{2}\!b\,\sin ^{2}\!c}}\\{\frac {\sin A}{\sin a}}&={\frac {\sqrt {1-\cos ^{2}\!a-\cos ^{2}\!b-\cos ^{2}\!c+2\cos a\cos b\cos c}}{\sin a\sin b\sin c}}.\end{aligned}}} 702: 8059: 2882: 6691: 10234: 1615: 963: 3850: 2456:

The derivation of the cosine rule presented above has the merits of simplicity and directness and the derivation of the sine rule emphasises the fact that no separate proof is required other than the cosine rule. However, the above geometry may be used to give an independent proof of the sine rule.

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gives four different proofs of the cosine rule. Text books on geodesy and spherical astronomy give different proofs and the online resources of MathWorld provide yet more. There are even more exotic derivations, such as that of Banerjee who derives the formulae using the linear algebra of projection

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then we can immediately derive a second identity by applying the first identity to the polar triangle by making the above substitutions. This is how the supplemental cosine equations are derived from the cosine equations. Similarly, the identities for a quadrantal triangle can be derived from those

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The solution methods listed here are not the only possible choices: many others are possible. In general it is better to choose methods that avoid taking an inverse sine because of the possible ambiguity between an angle and its supplement. The use of half-angle formulae is often advisable because

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The solution of triangles is the principal purpose of spherical trigonometry: given three, four or five elements of the triangle, determine the others. The case of five given elements is trivial, requiring only a single application of the sine rule. For four given elements there is one non-trivial

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The key for remembering which trigonometric function goes with which part is to look at the first vowel of the kind of part: middle parts take the sine, adjacent parts take the tangent, and opposite parts take the cosine. For an example, starting with the sector containing

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There are many ways of deriving the fundamental cosine and sine rules and the other rules developed in the following sections. For example, Todhunter gives two proofs of the cosine rule (Articles 37 and 60) and two proofs of the sine rule (Articles 40 and 42). The page on

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The spherical cosine formulae were originally proved by elementary geometry and the planar cosine rule (Todhunter, Art.37). He also gives a derivation using simple coordinate geometry and the planar cosine rule (Art.60). The approach outlined here uses simpler

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often small: for example the triangles of geodetic survey typically have a spherical excess much less than 1' of arc. On the Earth the excess of an equilateral triangle with sides 21.3 km (and area 393 km) is approximately 1 arc second.

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Sides are also expressed in radians. A side (regarded as a great circle arc) is measured by the angle that it subtends at the centre. On the unit sphere, this radian measure is numerically equal to the arc length. By convention, the sides of

8444: 10400: 8303: 4158:{\displaystyle {\begin{aligned}\cos a&=\cos b\cos c+\sin b\sin c\cos A\\&=\cos b\ (\cos a\cos b+\sin a\sin b\cos C)+\sin b\sin C\sin a\cot A\\\cos a\sin ^{2}b&=\cos b\sin a\sin b\cos C+\sin b\sin C\sin a\cot A.\end{aligned}}} 9338: 8658: 1222: 9606: 466:. This great circle is defined by the intersection of a diametral plane with the surface. Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane as 6211: 9868: 6431:

First, write the six parts of the triangle (three vertex angles, three arc angles for the sides) in the order they occur around any circuit of the triangle: for the triangle shown above left, going clockwise starting with

9000: 1740: 6488: 8536: 3213: 10302: 1908: 8155: 1999: 739: 8054:{\displaystyle {\begin{aligned}\cos a&=(\cos a\,\cos c+\sin a\,\sin c\,\cos B)\cos c+\sin b\,\sin c\,\cos A\\\cos a\,\sin ^{2}c&=\sin a\,\cos c\,\sin c\,\cos B+\sin b\,\sin c\,\cos A\end{aligned}}} 5632: 5074: 2877:{\displaystyle {\begin{aligned}\cos A&=-\cos B\,\cos C+\sin B\,\sin C\,\cos a,\\\cos B&=-\cos C\,\cos A+\sin C\,\sin A\,\cos b,\\\cos C&=-\cos A\,\cos B+\sin A\,\sin B\,\cos c.\end{aligned}}} 7822: 6686:{\displaystyle {\begin{aligned}\sin a&=\tan({\tfrac {\pi }{2}}-B)\,\tan b\\&=\cos({\tfrac {\pi }{2}}-c)\,\cos({\tfrac {\pi }{2}}-A)\\&=\cot B\,\tan b\\&=\sin c\,\sin A.\end{aligned}}} 6460:

from the list. The remaining parts can then be drawn as five ordered, equal slices of a pentagram, or circle, as shown in the above figure (right). For any choice of three contiguous parts, one (the

1820: 10229:{\displaystyle \tan {\tfrac {1}{2}}E_{4}={\frac {\sin {\tfrac {1}{2}}(\varphi _{2}+\varphi _{1})}{\cos {\tfrac {1}{2}}(\varphi _{2}-\varphi _{1})}}\tan {\tfrac {1}{2}}(\lambda _{2}-\lambda _{1}).} 5054: 4982: 4288: 7240:/2 on the unit sphere the equations governing the remaining sides and angles may be obtained by applying the rules for the right spherical triangle of the previous section to the polar triangle 3855: 2640: 1397: 9129: 1610:{\displaystyle {\begin{aligned}{\vec {OA}}:&\quad (0,\,0,\,1)\\{\vec {OB}}:&\quad (\sin c,\,0,\,\cos c)\\{\vec {OC}}:&\quad (\sin b\cos A,\,\sin b\sin A,\,\cos b).\end{aligned}}} 10672: 9241: 1096: 10405:

The area of a polygon can be calculated from individual quadrangles of the above type, from (analogously) individual triangle bounded by a segment of the polygon and two meridians, by a

10034: 9988: 9557: 9601: 1982: 958:{\displaystyle {\begin{aligned}\cos a&=\cos b\cos c+\sin b\sin c\cos A,\\\cos b&=\cos c\cos a+\sin c\sin a\cos B,\\\cos c&=\cos a\cos b+\sin a\sin b\cos C.\end{aligned}}} 8315: 331: 8177: 10780: 697:{\displaystyle {\begin{alignedat}{3}A'&=\pi -a,&\qquad B'&=\pi -b,&\qquad C'&=\pi -c,\\a'&=\pi -A,&b'&=\pi -B,&c'&=\pi -C.\end{alignedat}}} 399: 337:

In particular, the sum of the angles of a spherical triangle is strictly greater than the sum of the angles of a triangle defined on the Euclidean plane, which is always exactly

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Robert G. Chamberlain, William H. Duquette, Jet Propulsion Laboratory. The paper develops and explains many useful formulae, perhaps with a focus on navigation and cartography.

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for the ten independent equations: the mnemonic is called Napier's circle or Napier's pentagon (when the circle in the above figure, right, is replaced by a pentagon).

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Since the area of a triangle cannot be negative the spherical excess is always positive. It is not necessarily small, because the sum of the angles may attain 5

7287: 6701: 4989: 4176:. Similar techniques with the other two cosine rules give CT3 and CT5. The other three equations follow by applying rules 1, 3 and 5 to the polar triangle. 731:

The cosine rule is the fundamental identity of spherical trigonometry: all other identities, including the sine rule, may be derived from the cosine rule:

1825: 8071: 9496:{\displaystyle \tan {\tfrac {1}{4}}E={\sqrt {\tan {\tfrac {1}{2}}s\,\tan {\tfrac {1}{2}}(s-a)\,\tan {\tfrac {1}{2}}(s-b)\,\tan {\tfrac {1}{2}}(s-c)}}} 11018: 9748:{\displaystyle \tan {\tfrac {1}{2}}E={\frac {\tan {\frac {1}{2}}a\tan {\frac {1}{2}}b\sin C}{1+\tan {\frac {1}{2}}a\tan {\frac {1}{2}}b\cos C}}.} 11021:

by Sudipto Banerjee. The paper derives the spherical law of cosines and law of sines using elementary linear algebra and projection matrices.

6369:{\displaystyle {\frac {\tan {\tfrac {1}{2}}(A-B)}{\tan {\tfrac {1}{2}}(A+B)}}={\frac {\tan {\tfrac {1}{2}}(a-b)}{\tan {\tfrac {1}{2}}(a+b)}}} 1745: 511: 10885: 8449: 114:. Since then, significant developments have been the application of vector methods, quaternion methods, and the use of numerical methods. 10729: 10241: 8159:

Similar substitutions in the other cosine and supplementary cosine formulae give a large variety of 5-part rules. They are rarely used.

6398:/2 the various identities given above are considerably simplified. There are ten identities relating three elements chosen from the set 110:

and others, and attained an essentially complete form by the end of the nineteenth century with the publication of Todhunter's textbook

9102:{\displaystyle {{\text{Area of polygon}} \atop {\text{(on the unit sphere)}}}\equiv E_{N}=\left(\sum _{n=1}^{N}A_{n}\right)-(N-2)\pi .} 94:

The origins of spherical trigonometry in Greek mathematics and the major developments in Islamic mathematics are discussed fully in

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The Delambre analogies (also called Gauss analogies) were published independently by Delambre, Gauss, and Mollweide in 1807–1809.

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A free software to solve the spherical triangles, configurable to different practical applications and configured for gnomonic

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Not all of the rules obtained are numerically robust in extreme examples, for example when an angle approaches zero or

289: 10760: 8933:. Problems and solutions may have to be examined carefully, particularly when writing code to solve an arbitrary triangle. 8829:

Another approach is to split the triangle into two right-angled triangles. For example, take the Case 3 example where

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A quadrantal spherical triangle is defined to be a spherical triangle in which one of the sides subtends an angle of

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There are many formulae for the excess. For example, Todhunter, (Art.101—103) gives ten examples including that of

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angles). For example, an octant of a sphere is a spherical triangle with three right angles, so that the excess is

503:. A very important theorem (Todhunter, Art.27) proves that the angles and sides of the polar triangle are given by 1044: 11064: 9993: 9947: 9332: 8814:

was the first to list the six distinct cases (2-7 in the diagram) of a right triangle in spherical trigonometry.

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There is a full discussion of the solution of oblique triangles in Todhunter. See also the discussion in Ross.

9506: 242:. The sphere has a radius of 1, and so the side lengths and lower case angles are equivalent (see arc length). 6200: 1929: 971:, to which they are asymptotically equivalent in the limit of small interior angles. (On the unit sphere, if 107: 99: 9564: 194:

From this point in the article, discussion will be restricted to spherical triangles, referred to simply as

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part) will be adjacent to two parts and opposite the other two parts. The ten Napier's Rules are given by

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as commonly done in GIS. The other algorithms can still be used with the side lengths calculated using a

9196:{\displaystyle {{\text{Area of triangle}} \atop {\text{(on the unit sphere)}}}\equiv E=E_{3}=A+B+C-\pi ,} 974: 10842: 9887:

The spherical excess of a spherical quadrangle bounded by the equator, the two meridians of longitudes

9603:), it is often better to use the formula for the excess in terms of two edges and their included angle 5617:

Proved by expanding the numerators and using the half angle formulae. (Todhunter, Art.54 and Delambre)

11040:, a manuscript in Arabic that dates back to 1740 and talks about spherical trigonometry, with diagrams 9298:{\displaystyle A+B+C=\pi +{\frac {4\pi \times {\text{Area of triangle}}}{\text{Area of the sphere}}}.} 3840:

To prove the first formula start from the first cosine rule and on the right-hand side substitute for

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before using the identities given below. Likewise, after a calculation on the unit sphere the sides

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This equation can be re-arranged to give explicit expressions for the angle in terms of the sides:

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Both vertices and angles at the vertices of a triangle are denoted by the same upper case letters

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whenever the area of the triangle is small relative to the surface area of the entire Earth; see

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the third is a quotient and the remainder follow by applying the results to the polar triangle.

4187: 8817: 8439:{\displaystyle \cos a\cos A=-\cos B\,\cos C\,\cos a+\sin B\,\sin C-\sin B\,\sin C\,\sin ^{2}a.} 1921: 1111: 64: 10723: 10395:{\textstyle E_{4}\approx {\frac {1}{2}}(\varphi _{2}+\varphi _{1})(\lambda _{2}-\lambda _{1})} 8298:{\displaystyle \cos a\cos A=\cos b\,\cos c\,\cos A+\sin b\,\sin c-\sin b\,\sin c\,\sin ^{2}A.} 7216: 712:

for a right-angled triangle. The polar triangle of a polar triangle is the original triangle.

260:) may be regarded either as the angle between the two planes that intersect the sphere at the 122: 10692:

Todhunter, Isaac (1873). "Note on the history of certain formulæ in spherical trigonometry".

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matrices and also quotes methods in differential geometry and the group theory of rotations.

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The sphere's radius is taken as unity. For specific practical problems on a sphere of radius

184: 10977: 10607: 10466: 10435: 8811: 8803:/2 and therefore free from ambiguity. There is a full discussion in Todhunter. The article 8653:{\displaystyle \sin b\,\sin c+\cos b\,\cos c\,\cos A=\sin B\,\sin C-\cos B\,\cos C\,\cos a} 6203: 1217:{\displaystyle {\frac {\sin A}{\sin a}}={\frac {\sin B}{\sin b}}={\frac {\sin C}{\sin c}}.} 8: 10838: 9876: 8955: 8683: 437: 206: 164:

Such polygons may have any number of sides greater than 1. Two-sided spherical polygons—

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A quadrantal spherical triangle together with Napier's circle for use in his mnemonics

2895:). The cotangent, or four-part, formulae relate two sides and two angles forming four 10994: 10974: 10668: 10636: 10604: 10560: 1117: 1290:

drawn from the origin to the vertices of the triangle (on the unit sphere). The arc

102:. The subject came to fruition in Early Modern times with important developments by 10701: 10652: 10648: 10589: 10568: 10456: 8967: 8946: 1251: 6192:

These identities follow by division of the Delambre formulae. (Todhunter, Art.52)

2564:. Therefore, the invariance of the triple product under cyclic permutations gives 10872: 10797: 10784: 10754: 10522: 9863:{\displaystyle \tan {\tfrac {1}{2}}E=\tan {\tfrac {1}{2}}a\tan {\tfrac {1}{2}}b.} 1242: 52: 9219:. An earlier proof was derived, but not published, by the English mathematician 3163: 3121: 3076: 3034: 2989: 2947: 10694:

The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science

10430: 9872: 9220: 8997:-th interior angle. The area of such a polygon is given by (Todhunter, Art.99) 7228:/2 radians at the centre of the sphere: on the unit sphere the side has length 166: 10866: 10705: 8762:

and then we have Case 7 (rotated). There are either one or two solutions.

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The full set of rules for the right spherical triangle is (Todhunter, Art.62)

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in terms of the sides and replacing the sum of two cosines by a product. (See

11058: 11043: 10912: 10451: 10406: 9216: 1735:{\displaystyle {\vec {OB}}\cdot {\vec {OC}}=\sin c\sin b\cos A+\cos c\cos b.} 462:

is defined as follows. Consider the great circle that contains the side

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Eight spherical triangles defined by the intersection of three great circles.

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The proof (Todhunter, Art.49) of the first formula starts from the identity

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This result is obtained from one of Napier's analogies. In the limit where

9233:. The definition of the excess is independent of the radius of the sphere. 7814:

Substituting the second cosine rule into the first and simplifying gives:

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sine of the middle part = the product of the tangents of the adjacent parts

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and the six possible equations are (with the relevant set shown at right):

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Applying the cosine rules to the polar triangle gives (Todhunter, Art.47),

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sine of the middle part = the product of the cosines of the opposite parts

2907:). In such a set there are inner and outer parts: for example in the set ( 10750: 8942: 6421: 103: 10526: 9944:

and the great-circle arc between two points with longitude and latitude

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Because some triangles are badly characterized by their edges (e.g., if

8446:

Subtracting the two and noting that it follows from the sine rules that

32: 10660: 8807:

presents variants on these methods with a slightly different notation.

88: 10588:(6th ed.). Cambridge University Press. Chapter 1 – via the 11002: 10982: 10612: 8951: 8660:

which is a relation between the six parts of the spherical triangle.

8531:{\displaystyle \sin b\,\sin c\,\sin ^{2}A=\sin B\,\sin C\,\sin ^{2}a} 2427:

Since the right hand side is invariant under a cyclic permutation of

80: 72: 11037: 8726:

and then we have Case 7. There are either one or two solutions.

79:. Spherical trigonometry is of great importance for calculations in 10719: 10297:{\displaystyle \varphi _{1},\varphi _{2},\lambda _{2}-\lambda _{1}} 6425: 2583:

which is the first of the sine rules. See curved variations of the

1926:

This derivation is given in Todhunter, (Art.40). From the identity

1903:{\displaystyle \cos A={\frac {\cos a-\cos b\cos c}{\sin b\sin c}}.} 8150:{\displaystyle \cos a\sin c=\sin a\,\cos c\,\cos B+\sin b\,\cos A} 10941:"Surface area of polygon on sphere or ellipsoid – MATLAB areaint" 8706:

but, to avoid ambiguities, the half angle formulae are preferred.

268: 141: 84: 10917:. Association of American Geographers Annual Meeting. NASA JPL. 8780:

but, to avoid ambiguities, the half-side formulae are preferred.

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The six parts of a triangle may be written in cyclic order as (

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when the sides are much smaller than the radius of the sphere.

275: 68: 11019:"Revisiting Spherical Trigonometry with Orthogonal Projectors" 10911:

Chamberlain, Robert G.; Duquette, William H. (17 April 2007).

10637:"Revisiting Spherical Trigonometry with Orthogonal Projectors" 10304:

are all small, this reduces to the familiar trapezoidal area,

8305:

Similarly multiplying the first supplementary cosine rule by

11038:"The Book of Instruction on Deviant Planes and Simple Planes" 2506:

in the basis shown. Similarly, in a basis oriented with the

172: 56: 1912:

The other cosine rules are obtained by cyclic permutations.

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The supplemental cosine rule may be used to give the sides

2927:. The cotangent rule may be written as (Todhunter, Art.44) 7211: 1742:

Equating the two expressions for the scalar product gives

112:

Spherical trigonometry for the use of colleges and Schools

9215:

of the triangle. This theorem is named after its author,

183:

One spherical polygon with interesting properties is the

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a more thorough list of identities, with some derivation

10989:

a more thorough list of identities, with some derivation

10802:. Philadelphia: J. B. Lippincott & Co. p. 165. 8670:

Solution of triangles § Solving spherical triangles

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Another twelve identities follow by cyclic permutation.

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Case 7: two angles and two opposite sides given (SSAA).

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Another eight identities follow by cyclic permutation.

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Another eight identities follow by cyclic permutation.

1815:{\displaystyle \cos a=\cos b\cos c+\sin b\sin c\cos A.} 271:

of the great circle arcs where they meet at the vertex.

51:

that deals with the metrical relationships between the

10310: 10183: 10133: 10084: 10050: 9843: 9822: 9798: 9617: 9567: 9517: 9465: 9431: 9397: 9375: 9349: 6598: 6567: 6517: 6337: 6302: 6260: 6225: 6159: 6123: 6088: 6049: 6023: 5987: 5952: 5913: 5888: 5852: 5817: 5778: 5752: 5716: 5681: 5642: 5583: 5548: 5518: 5483: 5451: 5416: 5386: 5351: 5320: 5285: 5255: 5220: 5187: 5152: 5122: 5087: 5014: 4942: 4795: 4683: 4581: 4493: 4400: 4298: 10774:

The Construction of the Wonderful Canon of Logarithms

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both of the above area expressions are multiplied by

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is the amount by which the sum of the angles exceeds

9132: 9003: 8824: 8544: 8452: 8318: 8180: 8074: 7820: 7285: 6699: 6486: 6214: 5630: 5072: 5049:{\displaystyle 2\cos ^{2}\!{\tfrac {A}{2}}=1+\cos A,} 4998: 4977:{\displaystyle 2\sin ^{2}\!{\tfrac {A}{2}}=1-\cos A,} 4926: 4286: 4237: 4190: 3853: 3211: 2933: 2638: 1997: 1932: 1828: 1748: 1640: 1395: 1126: 1047: 1015: 977: 967:

These identities generalize the cosine rule of plane

737: 509: 496:

is the polar triangle corresponding to triangle

408:

the measured lengths of the sides must be divided by

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Case 5: two angles and an opposite side given (AAS).

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Case 4: two angles and an included side given (ASA).

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Case 3: two sides and an opposite angle given (SSA).

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Case 2: two sides and an included angle given (SAS).

4179: 1224:

These identities approximate the sine rule of plane

10910: 10647:(5), Mathematical Association of America: 375–381, 10865:Another proof of Girard's theorem may be found at 10394: 10296: 10228: 10028: 9982: 9936: 9906: 9862: 9747: 9595: 9551: 9495: 9297: 9195: 9101: 8652: 8530: 8438: 8297: 8149: 8053: 7799: 7201: 6685: 6440:. Next replace the parts that are not adjacent to 6368: 6179: 5606: 5048: 4976: 4907: 4273: 4223: 4157: 3832: 3197: 2876: 2419: 1976: 1902: 1814: 1734: 1609: 1216: 1090: 1033: 1001: 957: 696: 393: 325: 282:spherical triangles are (by convention) less than 10836: 9111:For the case of a spherical triangle with angles 8958:: the triangles of constant area on a fixed base 8805:Solution of triangles#Solving spherical triangles 6456:) by their complements and then delete the angle 5012: 4940: 3187: 3156: 3153: 3145: 3114: 3111: 3100: 3069: 3066: 3058: 3027: 3024: 3013: 2982: 2979: 2971: 2940: 2937: 2341: 2324: 2307: 2244: 2229: 11056: 8883:. Then use Napier's rules to solve the triangle 1387:. Therefore, the three vectors have components: 1231: 40:is a spherical triangle with three right angles. 8694:The cosine rule may be used to give the angles 4992:.) The second formula starts from the identity 2886: 10799:A Treatise on Plane and Spherical Trigonometry 9882: 1915: 1254:methods. (These methods are also discussed at 715: 27:Geometry of figures on the surface of a sphere 2439:the spherical sine rule follows immediately. 267:, or, equivalently, as the angle between the 10517: 10515: 10513: 10511: 10509: 10507: 8936: 6379:Napier's rules for right spherical triangles 2595: 1091:{\displaystyle (\cos a-\cos b)^{2}\approx 0} 145:on the surface of the sphere. Its sides are 10505: 10503: 10501: 10499: 10497: 10495: 10493: 10491: 10489: 10487: 10029:{\displaystyle (\lambda _{2},\varphi _{2})} 9983:{\displaystyle (\lambda _{1},\varphi _{1})} 8924: 8853:. Use Napier's rules to solve the triangle 8841:are given. Construct the great circle from 8732:The four-part cotangent formulae for sets ( 10756:Mirifici Logarithmorum Canonis Constructio 2442: 11050:Online computation of spherical triangles 10886:Legendre's theorem on spherical triangles 10795: 10691: 10521: 9457: 9423: 9389: 8640: 8630: 8608: 8586: 8576: 8554: 8511: 8501: 8472: 8462: 8416: 8406: 8384: 8362: 8352: 8275: 8265: 8243: 8221: 8211: 8137: 8115: 8105: 8037: 8027: 8005: 7995: 7985: 7952: 7929: 7919: 7885: 7875: 7853: 7780: 7729: 7679: 7631: 7581: 7533: 7483: 7432: 7382: 7327: 7182: 7134: 7084: 7036: 6986: 6938: 6888: 6840: 6790: 6738: 6666: 6637: 6587: 6537: 3793: 3771: 3745: 3691: 3669: 3643: 3589: 3567: 3541: 3487: 3465: 3439: 3385: 3363: 3337: 3282: 3260: 3234: 2857: 2847: 2825: 2780: 2770: 2748: 2703: 2693: 2671: 2233: 1587: 1565: 1499: 1492: 1441: 1434: 704:Therefore, if any identity is proved for 230:Sides are denoted by lower-case letters: 11044:Some Algorithms for Polygons on a Sphere 10914:Some algorithms for polygons on a sphere 10718: 10634: 10484: 9764:is a right triangle with right angle at 9552:{\displaystyle s={\tfrac {1}{2}}(a+b+c)} 8950: 8816: 8682: 8663: 7215: 6382: 2899:parts around the triangle, for example ( 1241: 436: 205: 121: 31: 10848:MacTutor History of Mathematics Archive 10628: 9596:{\textstyle a=b\approx {\frac {1}{2}}c} 8794:; or, use Case 3 (SSA) or case 5 (AAS). 7212:Napier's rules for quadrantal triangles 1977:{\displaystyle \sin ^{2}A=1-\cos ^{2}A} 470:is (conventionally) termed the pole of 14: 11057: 10889: 10878: 10749: 10552: 9236:The converse result may be written as 8167:Multiplying the first cosine rule by 6394:, of a spherical triangle is equal to 326:{\displaystyle \pi <A+B+C<3\pi } 153:—the spherical geometry equivalent of 10993: 10973: 10603: 10581: 8162: 6195:Taking quotients of these yields the 5620: 5059: 130: 11026:"A Visual Proof of Girard's Theorem" 8716:and then we are back to Case 1. 8674: 210:The basic triangle on a unit sphere. 191:with a right angle at every vertex. 2587:to see details of this derivation. 1324:. Introduce a Cartesian basis with 394:{\displaystyle 0<a+b+c<2\pi } 24: 9134: 9005: 8825:Solution by right-angled triangles 7809: 4165:The result follows on dividing by 1002:{\displaystyle a,b,c\rightarrow 0} 432: 350:spherical triangles are less than 25: 11076: 10966: 10921:from the original on 22 July 2020 9323:/2. In practical applications it 8986:-sided spherical polygon and let 8766:Case 6: three angles given (AAA). 4984:using the cosine rule to express 4180:Half-angle and half-side formulae 3162: 3120: 3075: 3033: 2988: 2946: 10585:Text-Book on Spherical Astronomy 8970:through the points antipodal to 8692:Case 1: three sides given (SSS). 1984:and the explicit expression for 117: 63:, traditionally expressed using 10951:from the original on 2021-05-01 10933: 10904: 10859: 10830: 10817: 10806:from the original on 2021-07-11 10776:is available as en e-book from 10763:from the original on 2013-04-30 10732:from the original on 2020-07-22 10675:from the original on 2020-07-22 10641:The College Mathematics Journal 10553:Clarke, Alexander Ross (1880). 10535:from the original on 2020-04-14 7353: 7343: 7342: 7298: 7232:/2. In the case that the side 6764: 6754: 6753: 6712: 6039: 5768: 5470: 5207: 5206: 4392: 4391: 3292: 3224: 1540: 1476: 1424: 1296:subtends an angle of magnitude 1034:{\displaystyle \sin a\approx a} 720: 573: 543: 11030:Wolfram Demonstrations Project 10789: 10743: 10712: 10685: 10653:10.1080/07468342.2004.11922099 10596: 10575: 10546: 10389: 10363: 10360: 10334: 10220: 10194: 10170: 10144: 10121: 10095: 10023: 9997: 9977: 9951: 9546: 9528: 9488: 9476: 9454: 9442: 9420: 9408: 9090: 9078: 8799:half-angles will be less than 7895: 7841: 6615: 6594: 6584: 6563: 6534: 6513: 6360: 6348: 6325: 6313: 6283: 6271: 6248: 6236: 6146: 6134: 6111: 6099: 6072: 6060: 6010: 5998: 5975: 5963: 5936: 5924: 5875: 5863: 5840: 5828: 5801: 5789: 5739: 5727: 5704: 5692: 5665: 5653: 5571: 5559: 5506: 5494: 5439: 5427: 5374: 5362: 5308: 5296: 5243: 5231: 5175: 5163: 5110: 5098: 4894: 4882: 4873: 4861: 4850: 4838: 4779: 4767: 4747: 4735: 4726: 4714: 4645: 4633: 4624: 4612: 4545: 4533: 4455: 4443: 4362: 4350: 4341: 4329: 4265: 4247: 4218: 4200: 3995: 3944: 3823: 3808: 3721: 3706: 3619: 3604: 3517: 3502: 3415: 3400: 3313: 3298: 2208: 2174: 2168: 2143: 2140: 2115: 1672: 1652: 1634:in terms of the components is 1597: 1541: 1527: 1509: 1477: 1463: 1445: 1425: 1411: 1073: 1048: 993: 13: 1: 10477: 9937:{\displaystyle \lambda _{2},} 2590: 1375:-plane and the angle between 1232:Derivation of the cosine rule 1105: 100:Mathematics in medieval Islam 9907:{\displaystyle \lambda _{1}} 8538:produces Cagnoli's equation 6390:When one of the angles, say 3847:from the third cosine rule: 2887:Cotangent four-part formulae 1300:at the centre and therefore 1261:Consider three unit vectors 7: 10796:Chauvenet, William (1867). 10559:. Oxford: Clarendon Press. 10531:(5th ed.). MacMillan. 10424: 9883:From latitude and longitude 8845:that is normal to the side 8786:Use Napier's analogies for 8752:follows from the sine rule. 4274:{\displaystyle 2S=(A+B+C),} 1916:Derivation of the sine rule 716:Cosine rules and sine rules 455:associated with a triangle 424:must be multiplied by 201: 10: 11081: 10725:Connaissance des Tems 1809 10635:Banerjee, Sudipto (2004), 8940: 8667: 4224:{\displaystyle 2s=(a+b+c)} 1919: 1235: 1109: 724: 10825:Master Math: Trigonometry 10706:10.1080/14786447308640820 10472:Triangulation (surveying) 9875:is defined similarly for 8937:Area and spherical excess 8061:Cancelling the factor of 4990:sum-to-product identities 2596:Supplemental cosine rules 10978:"Spherical Trigonometry" 10853:University of St Andrews 10608:"Spherical Trigonometry" 10441:Ellipsoidal trigonometry 9223:. On a sphere of radius 8925:Numerical considerations 2450:Spherical law of cosines 1991:given immediately above 1256:Spherical law of cosines 1238:Spherical law of cosines 1120:is given by the formula 1100:Spherical law of cosines 727:Spherical law of cosines 274:Angles are expressed in 8962:have their free vertex 2443:Alternative derivations 1350:-plane making an angle 486:are defined similarly. 401:(Todhunter, Art.22,32). 333:(Todhunter, Art.22,32). 96:History of trigonometry 65:trigonometric functions 11065:Spherical trigonometry 10891:Clarke, Alexander Ross 10843:"Nasir al-Din al-Tusi" 10528:Spherical Trigonometry 10396: 10298: 10230: 10030: 9984: 9938: 9908: 9864: 9749: 9597: 9553: 9497: 9299: 9197: 9103: 9059: 8979: 8821: 8712:The cosine rule gives 8687: 8654: 8532: 8440: 8299: 8151: 8055: 7801: 7279:etc. The results are: 7221: 7203: 6687: 6387: 6370: 6181: 5608: 5050: 4978: 4909: 4275: 4225: 4159: 3834: 3199: 2878: 2421: 1978: 1922:Spherical law of sines 1904: 1816: 1736: 1611: 1246: 1218: 1112:Spherical law of sines 1092: 1035: 1003: 959: 698: 448: 395: 327: 211: 187:, a 5-sided spherical 127: 45:Spherical trigonometry 41: 10827:, Career Press, 2002. 10448:or spherical distance 10446:Great-circle distance 10419:great-circle distance 10415:equal-area projection 10397: 10299: 10231: 10031: 9985: 9939: 9909: 9865: 9784:, so this reduces to 9750: 9598: 9554: 9498: 9300: 9198: 9104: 9039: 8954: 8820: 8686: 8664:Solution of triangles 8655: 8533: 8441: 8300: 8152: 8056: 7802: 7219: 7204: 6688: 6386: 6371: 6201:Persian mathematician 6182: 5609: 5051: 4979: 4910: 4276: 4226: 4160: 3835: 3200: 2919:, the outer angle is 2911:) the inner angle is 2879: 2519:, the triple product 2459:scalar triple product 2422: 1979: 1905: 1817: 1737: 1612: 1245: 1219: 1093: 1036: 1004: 960: 699: 474:and it is denoted by 440: 396: 328: 209: 185:pentagramma mirificum 125: 35: 10998:"Spherical Triangle" 10839:Robertson, Edmund F. 10772:An 1889 translation 10582:Smart, W.M. (1977). 10467:Spherical polyhedron 10436:Celestial navigation 10308: 10242: 10040: 9994: 9948: 9918: 9891: 9788: 9607: 9565: 9507: 9339: 9242: 9211:radians, called the 9142:(on the unit sphere) 9130: 9013:(on the unit sphere) 9001: 8921:follow by addition. 8812:Nasir al-Din al-Tusi 8758:The sine rule gives 8722:The sine rule gives 8542: 8450: 8316: 8178: 8072: 7818: 7283: 6697: 6484: 6424:provided an elegant 6212: 6204:Nasir al-Din al-Tusi 5628: 5070: 4996: 4924: 4284: 4235: 4188: 3851: 3209: 2931: 2923:, the outer side is 2915:, the inner side is 2636: 1995: 1930: 1826: 1746: 1638: 1393: 1124: 1045: 1013: 975: 735: 507: 358: 290: 10901:(Chapters 2 and 9). 10837:O'Connor, John J.; 9877:hyperbolic geometry 1619:The scalar product 441:The polar triangle 61:spherical triangles 10995:Weisstein, Eric W. 10975:Weisstein, Eric W. 10899:. Clarendon Press. 10884:This follows from 10871:2012-10-31 at the 10823:Ross, Debra Anne. 10783:2020-03-03 at the 10720:Delambre, J. B. J. 10605:Weisstein, Eric W. 10462:Spherical geometry 10392: 10294: 10226: 10192: 10142: 10093: 10059: 10026: 9980: 9934: 9904: 9860: 9852: 9831: 9807: 9745: 9626: 9593: 9549: 9526: 9493: 9474: 9440: 9406: 9384: 9358: 9295: 9288:Area of the sphere 9193: 9099: 8980: 8868:to find the sides 8822: 8688: 8650: 8528: 8436: 8295: 8163:Cagnoli's Equation 8147: 8051: 8049: 7797: 7795: 7222: 7199: 7197: 6683: 6681: 6607: 6576: 6526: 6388: 6366: 6346: 6311: 6269: 6234: 6199:, first stated by 6177: 6175: 6168: 6132: 6097: 6058: 6032: 5996: 5961: 5922: 5897: 5861: 5826: 5787: 5761: 5725: 5690: 5651: 5621:Napier's analogies 5604: 5602: 5592: 5557: 5527: 5492: 5460: 5425: 5395: 5360: 5329: 5294: 5264: 5229: 5196: 5161: 5131: 5096: 5060:Delambre analogies 5046: 5023: 4974: 4951: 4905: 4903: 4804: 4692: 4590: 4502: 4409: 4307: 4271: 4221: 4155: 4153: 3830: 3828: 3195: 3183: 3182: 3141: 3140: 3096: 3095: 3054: 3053: 3009: 3008: 2967: 2966: 2874: 2872: 2417: 2415: 1974: 1900: 1812: 1732: 1607: 1605: 1358:-axis. The vector 1247: 1214: 1088: 1031: 999: 955: 953: 694: 692: 449: 391: 323: 212: 131:Spherical polygons 128: 49:spherical geometry 42: 38:octant of a sphere 18:Spherical triangle 10945:www.mathworks.com 10332: 10191: 10174: 10141: 10092: 10058: 9851: 9830: 9806: 9740: 9725: 9706: 9670: 9651: 9625: 9588: 9525: 9491: 9473: 9439: 9405: 9383: 9357: 9290: 9289: 9284: 9145: 9143: 9138: 9016: 9014: 9009: 8898:to find the side 8675:Oblique triangles 7748: 7700: 7650: 7602: 7552: 7504: 7451: 7403: 7349: 7294: 7153: 7105: 7055: 7007: 6957: 6909: 6859: 6811: 6760: 6708: 6606: 6575: 6525: 6364: 6345: 6310: 6287: 6268: 6233: 6167: 6150: 6131: 6096: 6057: 6031: 6014: 5995: 5960: 5921: 5896: 5879: 5860: 5825: 5786: 5760: 5743: 5724: 5689: 5650: 5598: 5591: 5556: 5533: 5526: 5491: 5466: 5459: 5424: 5401: 5394: 5359: 5335: 5328: 5293: 5270: 5263: 5228: 5202: 5195: 5160: 5137: 5130: 5095: 5022: 4950: 4899: 4898: 4803: 4784: 4783: 4691: 4670: 4669: 4589: 4570: 4569: 4501: 4480: 4479: 4408: 4387: 4386: 4306: 3943: 3731: 3629: 3527: 3425: 3323: 3219: 3178: 3169: 3136: 3127: 3091: 3082: 3049: 3040: 3004: 2995: 2962: 2953: 2544: 2535: 2526: 2516: 2486: 2477: 2468: 2408: 2378: 2279: 2249: 2090: 1895: 1675: 1655: 1631: 1625: 1530: 1466: 1414: 1364: 1343: 1330: 1316: 1308: 1286: 1277: 1268: 1209: 1180: 1151: 137:spherical polygon 47:is the branch of 16:(Redirected from 11072: 11033: 11008: 11007: 10988: 10987: 10960: 10959: 10957: 10956: 10937: 10931: 10930: 10928: 10926: 10908: 10902: 10900: 10882: 10876: 10863: 10857: 10855: 10834: 10828: 10821: 10815: 10814: 10812: 10811: 10793: 10787: 10771: 10769: 10768: 10747: 10741: 10740: 10738: 10737: 10716: 10710: 10709: 10689: 10683: 10682: 10681: 10680: 10632: 10626: 10625: 10624: 10622: 10620: 10600: 10594: 10593: 10590:Internet Archive 10579: 10573: 10572: 10569:Internet Archive 10567:– via the 10550: 10544: 10543: 10541: 10540: 10519: 10457:Schwarz triangle 10401: 10399: 10398: 10393: 10388: 10387: 10375: 10374: 10359: 10358: 10346: 10345: 10333: 10325: 10320: 10319: 10303: 10301: 10300: 10295: 10293: 10292: 10280: 10279: 10267: 10266: 10254: 10253: 10235: 10233: 10232: 10227: 10219: 10218: 10206: 10205: 10193: 10184: 10175: 10173: 10169: 10168: 10156: 10155: 10143: 10134: 10124: 10120: 10119: 10107: 10106: 10094: 10085: 10075: 10070: 10069: 10060: 10051: 10035: 10033: 10032: 10027: 10022: 10021: 10009: 10008: 9989: 9987: 9986: 9981: 9976: 9975: 9963: 9962: 9943: 9941: 9940: 9935: 9930: 9929: 9913: 9911: 9910: 9905: 9903: 9902: 9869: 9867: 9866: 9861: 9853: 9844: 9832: 9823: 9808: 9799: 9783: 9775: 9767: 9763: 9754: 9752: 9751: 9746: 9741: 9739: 9726: 9718: 9707: 9699: 9684: 9671: 9663: 9652: 9644: 9635: 9627: 9618: 9602: 9600: 9599: 9594: 9589: 9581: 9558: 9556: 9555: 9550: 9527: 9518: 9502: 9500: 9499: 9494: 9492: 9475: 9466: 9441: 9432: 9407: 9398: 9385: 9376: 9367: 9359: 9350: 9322: 9314: 9310: 9304: 9302: 9301: 9296: 9291: 9287: 9286: 9285: 9283:Area of triangle 9282: 9270: 9232: 9226: 9213:spherical excess 9210: 9206: 9202: 9200: 9199: 9194: 9165: 9164: 9146: 9144: 9141: 9139: 9137:Area of triangle 9136: 9125:Girard's theorem 9123:this reduces to 9122: 9118: 9114: 9108: 9106: 9105: 9100: 9074: 9070: 9069: 9068: 9058: 9053: 9030: 9029: 9017: 9015: 9012: 9010: 9007: 8996: 8992: 8985: 8977: 8973: 8965: 8961: 8956:Lexell's theorem 8947:Geodesic polygon 8932: 8920: 8916: 8912: 8905: 8901: 8897: 8893: 8889: 8882: 8875: 8871: 8867: 8863: 8859: 8852: 8848: 8844: 8840: 8836: 8832: 8802: 8793: 8789: 8779: 8775: 8771: 8761: 8751: 8747: 8743: 8739: 8735: 8725: 8715: 8705: 8701: 8697: 8659: 8657: 8656: 8651: 8537: 8535: 8534: 8529: 8521: 8520: 8482: 8481: 8445: 8443: 8442: 8437: 8426: 8425: 8311: 8304: 8302: 8301: 8296: 8285: 8284: 8173: 8156: 8154: 8153: 8148: 8067: 8060: 8058: 8057: 8052: 8050: 7962: 7961: 7806: 7804: 7803: 7798: 7796: 7749: 7746: 7743: 7701: 7698: 7695: 7651: 7648: 7645: 7603: 7600: 7597: 7553: 7550: 7547: 7505: 7502: 7499: 7452: 7449: 7446: 7404: 7401: 7398: 7350: 7347: 7295: 7292: 7289: 7278: 7273: 7264: 7259: 7250: 7246: 7239: 7235: 7231: 7227: 7208: 7206: 7205: 7200: 7198: 7154: 7151: 7148: 7106: 7103: 7100: 7056: 7053: 7050: 7008: 7005: 7002: 6958: 6955: 6952: 6910: 6907: 6904: 6860: 6857: 6854: 6812: 6809: 6806: 6761: 6758: 6709: 6706: 6703: 6692: 6690: 6689: 6684: 6682: 6650: 6621: 6608: 6599: 6577: 6568: 6550: 6527: 6518: 6479: 6459: 6455: 6451: 6447: 6443: 6439: 6435: 6417: 6413: 6409: 6405: 6401: 6397: 6393: 6375: 6373: 6372: 6367: 6365: 6363: 6347: 6338: 6328: 6312: 6303: 6293: 6288: 6286: 6270: 6261: 6251: 6235: 6226: 6216: 6186: 6184: 6183: 6178: 6176: 6169: 6160: 6151: 6149: 6133: 6124: 6114: 6098: 6089: 6079: 6059: 6050: 6033: 6024: 6015: 6013: 5997: 5988: 5978: 5962: 5953: 5943: 5923: 5914: 5898: 5889: 5880: 5878: 5862: 5853: 5843: 5827: 5818: 5808: 5788: 5779: 5762: 5753: 5744: 5742: 5726: 5717: 5707: 5691: 5682: 5672: 5652: 5643: 5613: 5611: 5610: 5605: 5603: 5599: 5597: 5593: 5584: 5574: 5558: 5549: 5539: 5534: 5532: 5528: 5519: 5509: 5493: 5484: 5474: 5467: 5465: 5461: 5452: 5442: 5426: 5417: 5407: 5402: 5400: 5396: 5387: 5377: 5361: 5352: 5342: 5336: 5334: 5330: 5321: 5311: 5295: 5286: 5276: 5271: 5269: 5265: 5256: 5246: 5230: 5221: 5211: 5203: 5201: 5197: 5188: 5178: 5162: 5153: 5143: 5138: 5136: 5132: 5123: 5113: 5097: 5088: 5078: 5055: 5053: 5052: 5047: 5024: 5015: 5011: 5010: 4987: 4983: 4981: 4980: 4975: 4952: 4943: 4939: 4938: 4914: 4912: 4911: 4906: 4904: 4900: 4897: 4853: 4818: 4817: 4805: 4796: 4785: 4782: 4750: 4706: 4705: 4693: 4684: 4671: 4668: 4648: 4604: 4603: 4591: 4582: 4571: 4568: 4548: 4516: 4515: 4503: 4494: 4481: 4478: 4458: 4423: 4422: 4410: 4401: 4388: 4385: 4365: 4321: 4320: 4308: 4299: 4280: 4278: 4277: 4272: 4230: 4228: 4227: 4222: 4175: 4164: 4162: 4161: 4156: 4154: 4059: 4058: 3941: 3925: 3846: 3839: 3837: 3836: 3831: 3829: 3804: 3734: 3732: 3729: 3702: 3632: 3630: 3627: 3600: 3530: 3528: 3525: 3498: 3428: 3426: 3423: 3396: 3326: 3324: 3321: 3294: 3222: 3220: 3217: 3204: 3202: 3201: 3196: 3191: 3190: 3184: 3179: 3176: 3170: 3167: 3160: 3159: 3149: 3148: 3142: 3137: 3134: 3128: 3125: 3118: 3117: 3104: 3103: 3097: 3092: 3089: 3083: 3080: 3073: 3072: 3062: 3061: 3055: 3050: 3047: 3041: 3038: 3031: 3030: 3017: 3016: 3010: 3005: 3002: 2996: 2993: 2986: 2985: 2975: 2974: 2968: 2963: 2960: 2954: 2951: 2944: 2943: 2926: 2922: 2918: 2914: 2910: 2906: 2902: 2894: 2883: 2881: 2880: 2875: 2873: 2631: 2626: 2621: 2617: 2612: 2607: 2582: 2563: 2548: 2545: 2542: 2536: 2533: 2527: 2524: 2518: 2517: 2514: 2509: 2505: 2490: 2487: 2484: 2478: 2475: 2469: 2466: 2438: 2434: 2430: 2426: 2424: 2423: 2418: 2416: 2409: 2407: 2340: 2339: 2323: 2322: 2306: 2305: 2290: 2289: 2280: 2278: 2267: 2256: 2250: 2248: 2243: 2242: 2228: 2227: 2217: 2216: 2215: 2161: 2160: 2133: 2132: 2113: 2105: 2101: 2100: 2095: 2091: 2089: 2069: 2037: 2011: 2010: 1990: 1983: 1981: 1980: 1975: 1967: 1966: 1942: 1941: 1909: 1907: 1906: 1901: 1896: 1894: 1874: 1842: 1821: 1819: 1818: 1813: 1741: 1739: 1738: 1733: 1677: 1676: 1671: 1663: 1657: 1656: 1651: 1643: 1633: 1632: 1629: 1626: 1623: 1616: 1614: 1613: 1608: 1606: 1532: 1531: 1526: 1518: 1468: 1467: 1462: 1454: 1416: 1415: 1410: 1402: 1386: 1382: 1378: 1374: 1370: 1366: 1365: 1362: 1357: 1353: 1349: 1345: 1344: 1341: 1336: 1332: 1331: 1328: 1323: 1317: 1314: 1309: 1306: 1299: 1295: 1294: 1289: 1287: 1284: 1278: 1275: 1269: 1266: 1223: 1221: 1220: 1215: 1210: 1208: 1197: 1186: 1181: 1179: 1168: 1157: 1152: 1150: 1139: 1128: 1097: 1095: 1094: 1089: 1081: 1080: 1040: 1038: 1037: 1032: 1008: 1006: 1005: 1000: 964: 962: 961: 956: 954: 710: 703: 701: 700: 695: 693: 670: 641: 612: 581: 551: 521: 502: 495: 485: 481: 477: 473: 469: 465: 461: 447: 427: 423: 419: 415: 411: 407: 400: 398: 397: 392: 353: 340: 332: 330: 329: 324: 285: 278:. The angles of 266: 259: 255: 251: 241: 237: 233: 226: 222: 218: 21: 11080: 11079: 11075: 11074: 11073: 11071: 11070: 11069: 11055: 11054: 11024: 10969: 10964: 10963: 10954: 10952: 10939: 10938: 10934: 10924: 10922: 10909: 10905: 10883: 10879: 10873:Wayback Machine 10864: 10860: 10835: 10831: 10822: 10818: 10809: 10807: 10794: 10790: 10785:Wayback Machine 10766: 10764: 10748: 10744: 10735: 10733: 10728:. p. 445. 10717: 10713: 10700:(298): 98–100. 10690: 10686: 10678: 10676: 10633: 10629: 10618: 10616: 10601: 10597: 10580: 10576: 10551: 10547: 10538: 10536: 10520: 10485: 10480: 10427: 10411:Green's theorem 10383: 10379: 10370: 10366: 10354: 10350: 10341: 10337: 10324: 10315: 10311: 10309: 10306: 10305: 10288: 10284: 10275: 10271: 10262: 10258: 10249: 10245: 10243: 10240: 10239: 10214: 10210: 10201: 10197: 10182: 10164: 10160: 10151: 10147: 10132: 10125: 10115: 10111: 10102: 10098: 10083: 10076: 10074: 10065: 10061: 10049: 10041: 10038: 10037: 10017: 10013: 10004: 10000: 9995: 9992: 9991: 9971: 9967: 9958: 9954: 9949: 9946: 9945: 9925: 9921: 9919: 9916: 9915: 9898: 9894: 9892: 9889: 9888: 9885: 9842: 9821: 9797: 9789: 9786: 9785: 9777: 9769: 9765: 9758: 9717: 9698: 9685: 9662: 9643: 9636: 9634: 9616: 9608: 9605: 9604: 9580: 9566: 9563: 9562: 9516: 9508: 9505: 9504: 9464: 9430: 9396: 9374: 9366: 9348: 9340: 9337: 9336: 9320: 9312: 9308: 9281: 9271: 9269: 9243: 9240: 9239: 9228: 9224: 9208: 9204: 9160: 9156: 9140: 9135: 9133: 9131: 9128: 9127: 9120: 9116: 9112: 9064: 9060: 9054: 9043: 9038: 9034: 9025: 9021: 9011: 9008:Area of polygon 9006: 9004: 9002: 8999: 8998: 8994: 8991: 8987: 8983: 8975: 8971: 8963: 8959: 8949: 8939: 8930: 8927: 8918: 8914: 8907: 8903: 8902:and the angles 8899: 8895: 8891: 8884: 8877: 8873: 8869: 8865: 8861: 8854: 8850: 8846: 8842: 8838: 8834: 8830: 8827: 8800: 8791: 8787: 8777: 8773: 8769: 8759: 8749: 8745: 8741: 8737: 8733: 8723: 8713: 8703: 8699: 8695: 8677: 8672: 8666: 8543: 8540: 8539: 8516: 8512: 8477: 8473: 8451: 8448: 8447: 8421: 8417: 8317: 8314: 8313: 8306: 8280: 8276: 8179: 8176: 8175: 8168: 8165: 8073: 8070: 8069: 8062: 8048: 8047: 7969: 7957: 7953: 7940: 7939: 7834: 7821: 7819: 7816: 7815: 7812: 7810:Five-part rules 7794: 7793: 7761: 7750: 7745: 7742: 7713: 7702: 7697: 7693: 7692: 7663: 7652: 7647: 7644: 7615: 7604: 7599: 7595: 7594: 7565: 7554: 7549: 7546: 7517: 7506: 7501: 7497: 7496: 7464: 7453: 7448: 7445: 7416: 7405: 7400: 7396: 7395: 7363: 7351: 7346: 7344: 7340: 7308: 7296: 7291: 7286: 7284: 7281: 7280: 7271: 7266: 7257: 7252: 7248: 7241: 7237: 7233: 7229: 7225: 7214: 7196: 7195: 7166: 7155: 7150: 7147: 7118: 7107: 7102: 7098: 7097: 7068: 7057: 7052: 7049: 7020: 7009: 7004: 7000: 6999: 6970: 6959: 6954: 6951: 6922: 6911: 6906: 6902: 6901: 6872: 6861: 6856: 6853: 6824: 6813: 6808: 6804: 6803: 6774: 6762: 6757: 6755: 6751: 6722: 6710: 6705: 6700: 6698: 6695: 6694: 6680: 6679: 6648: 6647: 6619: 6618: 6597: 6566: 6548: 6547: 6516: 6500: 6487: 6485: 6482: 6481: 6477: 6457: 6453: 6449: 6445: 6441: 6437: 6433: 6415: 6411: 6407: 6403: 6399: 6395: 6391: 6381: 6336: 6329: 6301: 6294: 6292: 6259: 6252: 6224: 6217: 6215: 6213: 6210: 6209: 6197:law of tangents 6174: 6173: 6158: 6122: 6115: 6087: 6080: 6078: 6048: 6040: 6037: 6022: 5986: 5979: 5951: 5944: 5942: 5912: 5903: 5902: 5887: 5851: 5844: 5816: 5809: 5807: 5777: 5769: 5766: 5751: 5715: 5708: 5680: 5673: 5671: 5641: 5631: 5629: 5626: 5625: 5623: 5601: 5600: 5582: 5575: 5547: 5540: 5538: 5517: 5510: 5482: 5475: 5473: 5471: 5468: 5450: 5443: 5415: 5408: 5406: 5385: 5378: 5350: 5343: 5341: 5338: 5337: 5319: 5312: 5284: 5277: 5275: 5254: 5247: 5219: 5212: 5210: 5208: 5204: 5186: 5179: 5151: 5144: 5142: 5121: 5114: 5086: 5079: 5077: 5073: 5071: 5068: 5067: 5062: 5013: 5006: 5002: 4997: 4994: 4993: 4985: 4941: 4934: 4930: 4925: 4922: 4921: 4902: 4901: 4854: 4819: 4816: 4809: 4794: 4786: 4751: 4707: 4704: 4697: 4682: 4673: 4672: 4649: 4605: 4602: 4595: 4580: 4572: 4549: 4517: 4514: 4507: 4492: 4483: 4482: 4459: 4424: 4421: 4414: 4399: 4389: 4366: 4322: 4319: 4312: 4297: 4287: 4285: 4282: 4281: 4236: 4233: 4232: 4189: 4186: 4185: 4182: 4166: 4152: 4151: 4066: 4054: 4050: 4038: 4037: 3923: 3922: 3867: 3854: 3852: 3849: 3848: 3841: 3827: 3826: 3803: 3755: 3733: 3728: 3725: 3724: 3701: 3653: 3631: 3626: 3623: 3622: 3599: 3551: 3529: 3524: 3521: 3520: 3497: 3449: 3427: 3422: 3419: 3418: 3395: 3347: 3325: 3320: 3317: 3316: 3293: 3244: 3221: 3216: 3212: 3210: 3207: 3206: 3186: 3185: 3181: 3180: 3175: 3172: 3171: 3166: 3161: 3155: 3154: 3144: 3143: 3139: 3138: 3133: 3130: 3129: 3124: 3119: 3113: 3112: 3099: 3098: 3094: 3093: 3088: 3085: 3084: 3079: 3074: 3068: 3067: 3057: 3056: 3052: 3051: 3046: 3043: 3042: 3037: 3032: 3026: 3025: 3012: 3011: 3007: 3006: 3001: 2998: 2997: 2992: 2987: 2981: 2980: 2970: 2969: 2965: 2964: 2959: 2956: 2955: 2950: 2945: 2939: 2938: 2932: 2929: 2928: 2924: 2920: 2916: 2912: 2908: 2904: 2900: 2892: 2889: 2871: 2870: 2806: 2794: 2793: 2729: 2717: 2716: 2652: 2639: 2637: 2634: 2633: 2624: 2623: 2619: 2610: 2609: 2605: 2598: 2593: 2565: 2550: 2549:, evaluates to 2541: 2532: 2523: 2520: 2513: 2511: 2507: 2492: 2483: 2474: 2465: 2462: 2445: 2436: 2432: 2428: 2414: 2413: 2379: 2335: 2331: 2318: 2314: 2301: 2297: 2288: 2281: 2268: 2257: 2255: 2252: 2251: 2238: 2234: 2223: 2219: 2218: 2211: 2207: 2156: 2152: 2128: 2124: 2114: 2112: 2103: 2102: 2096: 2070: 2038: 2036: 2032: 2031: 2018: 2006: 2002: 1998: 1996: 1993: 1992: 1985: 1962: 1958: 1937: 1933: 1931: 1928: 1927: 1924: 1918: 1875: 1843: 1841: 1827: 1824: 1823: 1747: 1744: 1743: 1664: 1662: 1661: 1644: 1642: 1641: 1639: 1636: 1635: 1628: 1622: 1620: 1604: 1603: 1536: 1519: 1517: 1516: 1513: 1512: 1472: 1455: 1453: 1452: 1449: 1448: 1420: 1403: 1401: 1400: 1396: 1394: 1391: 1390: 1384: 1380: 1376: 1372: 1368: 1361: 1359: 1355: 1351: 1347: 1340: 1338: 1334: 1327: 1325: 1313: 1305: 1301: 1297: 1292: 1291: 1283: 1274: 1265: 1262: 1240: 1234: 1198: 1187: 1185: 1169: 1158: 1156: 1140: 1129: 1127: 1125: 1122: 1121: 1114: 1108: 1076: 1072: 1046: 1043: 1042: 1014: 1011: 1010: 976: 973: 972: 952: 951: 893: 881: 880: 822: 810: 809: 751: 738: 736: 733: 732: 729: 723: 718: 705: 691: 690: 671: 663: 661: 642: 634: 632: 613: 605: 602: 601: 582: 574: 571: 552: 544: 541: 522: 514: 510: 508: 505: 504: 497: 490: 483: 479: 475: 471: 467: 463: 456: 442: 435: 433:Polar triangles 425: 421: 417: 413: 409: 405: 359: 356: 355: 351: 338: 291: 288: 287: 283: 264: 257: 253: 252:(respectively, 249: 239: 235: 231: 224: 220: 216: 204: 133: 120: 28: 23: 22: 15: 12: 11: 5: 11078: 11068: 11067: 11053: 11052: 11047: 11041: 11035: 11022: 11016: 11010: 10990: 10968: 10967:External links 10965: 10962: 10961: 10932: 10903: 10877: 10858: 10829: 10816: 10788: 10759:. p. 50. 10742: 10711: 10684: 10627: 10595: 10574: 10545: 10482: 10481: 10479: 10476: 10475: 10474: 10469: 10464: 10459: 10454: 10449: 10443: 10438: 10433: 10431:Air navigation 10426: 10423: 10391: 10386: 10382: 10378: 10373: 10369: 10365: 10362: 10357: 10353: 10349: 10344: 10340: 10336: 10331: 10328: 10323: 10318: 10314: 10291: 10287: 10283: 10278: 10274: 10270: 10265: 10261: 10257: 10252: 10248: 10225: 10222: 10217: 10213: 10209: 10204: 10200: 10196: 10190: 10187: 10181: 10178: 10172: 10167: 10163: 10159: 10154: 10150: 10146: 10140: 10137: 10131: 10128: 10123: 10118: 10114: 10110: 10105: 10101: 10097: 10091: 10088: 10082: 10079: 10073: 10068: 10064: 10057: 10054: 10048: 10045: 10025: 10020: 10016: 10012: 10007: 10003: 9999: 9979: 9974: 9970: 9966: 9961: 9957: 9953: 9933: 9928: 9924: 9901: 9897: 9884: 9881: 9859: 9856: 9850: 9847: 9841: 9838: 9835: 9829: 9826: 9820: 9817: 9814: 9811: 9805: 9802: 9796: 9793: 9757:When triangle 9744: 9738: 9735: 9732: 9729: 9724: 9721: 9716: 9713: 9710: 9705: 9702: 9697: 9694: 9691: 9688: 9683: 9680: 9677: 9674: 9669: 9666: 9661: 9658: 9655: 9650: 9647: 9642: 9639: 9633: 9630: 9624: 9621: 9615: 9612: 9592: 9587: 9584: 9579: 9576: 9573: 9570: 9548: 9545: 9542: 9539: 9536: 9533: 9530: 9524: 9521: 9515: 9512: 9490: 9487: 9484: 9481: 9478: 9472: 9469: 9463: 9460: 9456: 9453: 9450: 9447: 9444: 9438: 9435: 9429: 9426: 9422: 9419: 9416: 9413: 9410: 9404: 9401: 9395: 9392: 9388: 9382: 9379: 9373: 9370: 9365: 9362: 9356: 9353: 9347: 9344: 9294: 9280: 9277: 9274: 9268: 9265: 9262: 9259: 9256: 9253: 9250: 9247: 9221:Thomas Harriot 9192: 9189: 9186: 9183: 9180: 9177: 9174: 9171: 9168: 9163: 9159: 9155: 9152: 9149: 9098: 9095: 9092: 9089: 9086: 9083: 9080: 9077: 9073: 9067: 9063: 9057: 9052: 9049: 9046: 9042: 9037: 9033: 9028: 9024: 9020: 8989: 8938: 8935: 8926: 8923: 8890:: that is use 8876:and the angle 8826: 8823: 8796: 8795: 8781: 8763: 8753: 8727: 8717: 8707: 8676: 8673: 8668:Main article: 8665: 8662: 8649: 8646: 8643: 8639: 8636: 8633: 8629: 8626: 8623: 8620: 8617: 8614: 8611: 8607: 8604: 8601: 8598: 8595: 8592: 8589: 8585: 8582: 8579: 8575: 8572: 8569: 8566: 8563: 8560: 8557: 8553: 8550: 8547: 8527: 8524: 8519: 8515: 8510: 8507: 8504: 8500: 8497: 8494: 8491: 8488: 8485: 8480: 8476: 8471: 8468: 8465: 8461: 8458: 8455: 8435: 8432: 8429: 8424: 8420: 8415: 8412: 8409: 8405: 8402: 8399: 8396: 8393: 8390: 8387: 8383: 8380: 8377: 8374: 8371: 8368: 8365: 8361: 8358: 8355: 8351: 8348: 8345: 8342: 8339: 8336: 8333: 8330: 8327: 8324: 8321: 8294: 8291: 8288: 8283: 8279: 8274: 8271: 8268: 8264: 8261: 8258: 8255: 8252: 8249: 8246: 8242: 8239: 8236: 8233: 8230: 8227: 8224: 8220: 8217: 8214: 8210: 8207: 8204: 8201: 8198: 8195: 8192: 8189: 8186: 8183: 8164: 8161: 8146: 8143: 8140: 8136: 8133: 8130: 8127: 8124: 8121: 8118: 8114: 8111: 8108: 8104: 8101: 8098: 8095: 8092: 8089: 8086: 8083: 8080: 8077: 8046: 8043: 8040: 8036: 8033: 8030: 8026: 8023: 8020: 8017: 8014: 8011: 8008: 8004: 8001: 7998: 7994: 7991: 7988: 7984: 7981: 7978: 7975: 7972: 7970: 7968: 7965: 7960: 7956: 7951: 7948: 7945: 7942: 7941: 7938: 7935: 7932: 7928: 7925: 7922: 7918: 7915: 7912: 7909: 7906: 7903: 7900: 7897: 7894: 7891: 7888: 7884: 7881: 7878: 7874: 7871: 7868: 7865: 7862: 7859: 7856: 7852: 7849: 7846: 7843: 7840: 7837: 7835: 7833: 7830: 7827: 7824: 7823: 7811: 7808: 7792: 7789: 7786: 7783: 7779: 7776: 7773: 7770: 7767: 7764: 7762: 7760: 7757: 7754: 7751: 7744: 7741: 7738: 7735: 7732: 7728: 7725: 7722: 7719: 7716: 7714: 7712: 7709: 7706: 7703: 7696: 7694: 7691: 7688: 7685: 7682: 7678: 7675: 7672: 7669: 7666: 7664: 7662: 7659: 7656: 7653: 7646: 7643: 7640: 7637: 7634: 7630: 7627: 7624: 7621: 7618: 7616: 7614: 7611: 7608: 7605: 7598: 7596: 7593: 7590: 7587: 7584: 7580: 7577: 7574: 7571: 7568: 7566: 7564: 7561: 7558: 7555: 7548: 7545: 7542: 7539: 7536: 7532: 7529: 7526: 7523: 7520: 7518: 7516: 7513: 7510: 7507: 7500: 7498: 7495: 7492: 7489: 7486: 7482: 7479: 7476: 7473: 7470: 7467: 7465: 7463: 7460: 7457: 7454: 7447: 7444: 7441: 7438: 7435: 7431: 7428: 7425: 7422: 7419: 7417: 7415: 7412: 7409: 7406: 7399: 7397: 7394: 7391: 7388: 7385: 7381: 7378: 7375: 7372: 7369: 7366: 7364: 7362: 7359: 7356: 7352: 7345: 7341: 7339: 7336: 7333: 7330: 7326: 7323: 7320: 7317: 7314: 7311: 7309: 7307: 7304: 7301: 7297: 7290: 7288: 7213: 7210: 7194: 7191: 7188: 7185: 7181: 7178: 7175: 7172: 7169: 7167: 7165: 7162: 7159: 7156: 7149: 7146: 7143: 7140: 7137: 7133: 7130: 7127: 7124: 7121: 7119: 7117: 7114: 7111: 7108: 7101: 7099: 7096: 7093: 7090: 7087: 7083: 7080: 7077: 7074: 7071: 7069: 7067: 7064: 7061: 7058: 7051: 7048: 7045: 7042: 7039: 7035: 7032: 7029: 7026: 7023: 7021: 7019: 7016: 7013: 7010: 7003: 7001: 6998: 6995: 6992: 6989: 6985: 6982: 6979: 6976: 6973: 6971: 6969: 6966: 6963: 6960: 6953: 6950: 6947: 6944: 6941: 6937: 6934: 6931: 6928: 6925: 6923: 6921: 6918: 6915: 6912: 6905: 6903: 6900: 6897: 6894: 6891: 6887: 6884: 6881: 6878: 6875: 6873: 6871: 6868: 6865: 6862: 6855: 6852: 6849: 6846: 6843: 6839: 6836: 6833: 6830: 6827: 6825: 6823: 6820: 6817: 6814: 6807: 6805: 6802: 6799: 6796: 6793: 6789: 6786: 6783: 6780: 6777: 6775: 6773: 6770: 6767: 6763: 6756: 6752: 6750: 6747: 6744: 6741: 6737: 6734: 6731: 6728: 6725: 6723: 6721: 6718: 6715: 6711: 6704: 6702: 6678: 6675: 6672: 6669: 6665: 6662: 6659: 6656: 6653: 6651: 6649: 6646: 6643: 6640: 6636: 6633: 6630: 6627: 6624: 6622: 6620: 6617: 6614: 6611: 6605: 6602: 6596: 6593: 6590: 6586: 6583: 6580: 6574: 6571: 6565: 6562: 6559: 6556: 6553: 6551: 6549: 6546: 6543: 6540: 6536: 6533: 6530: 6524: 6521: 6515: 6512: 6509: 6506: 6503: 6501: 6499: 6496: 6493: 6490: 6489: 6473: 6472: 6469: 6380: 6377: 6362: 6359: 6356: 6353: 6350: 6344: 6341: 6335: 6332: 6327: 6324: 6321: 6318: 6315: 6309: 6306: 6300: 6297: 6291: 6285: 6282: 6279: 6276: 6273: 6267: 6264: 6258: 6255: 6250: 6247: 6244: 6241: 6238: 6232: 6229: 6223: 6220: 6172: 6166: 6163: 6157: 6154: 6148: 6145: 6142: 6139: 6136: 6130: 6127: 6121: 6118: 6113: 6110: 6107: 6104: 6101: 6095: 6092: 6086: 6083: 6077: 6074: 6071: 6068: 6065: 6062: 6056: 6053: 6047: 6044: 6041: 6038: 6036: 6030: 6027: 6021: 6018: 6012: 6009: 6006: 6003: 6000: 5994: 5991: 5985: 5982: 5977: 5974: 5971: 5968: 5965: 5959: 5956: 5950: 5947: 5941: 5938: 5935: 5932: 5929: 5926: 5920: 5917: 5911: 5908: 5905: 5904: 5901: 5895: 5892: 5886: 5883: 5877: 5874: 5871: 5868: 5865: 5859: 5856: 5850: 5847: 5842: 5839: 5836: 5833: 5830: 5824: 5821: 5815: 5812: 5806: 5803: 5800: 5797: 5794: 5791: 5785: 5782: 5776: 5773: 5770: 5767: 5765: 5759: 5756: 5750: 5747: 5741: 5738: 5735: 5732: 5729: 5723: 5720: 5714: 5711: 5706: 5703: 5700: 5697: 5694: 5688: 5685: 5679: 5676: 5670: 5667: 5664: 5661: 5658: 5655: 5649: 5646: 5640: 5637: 5634: 5633: 5622: 5619: 5596: 5590: 5587: 5581: 5578: 5573: 5570: 5567: 5564: 5561: 5555: 5552: 5546: 5543: 5537: 5531: 5525: 5522: 5516: 5513: 5508: 5505: 5502: 5499: 5496: 5490: 5487: 5481: 5478: 5472: 5469: 5464: 5458: 5455: 5449: 5446: 5441: 5438: 5435: 5432: 5429: 5423: 5420: 5414: 5411: 5405: 5399: 5393: 5390: 5384: 5381: 5376: 5373: 5370: 5367: 5364: 5358: 5355: 5349: 5346: 5340: 5339: 5333: 5327: 5324: 5318: 5315: 5310: 5307: 5304: 5301: 5298: 5292: 5289: 5283: 5280: 5274: 5268: 5262: 5259: 5253: 5250: 5245: 5242: 5239: 5236: 5233: 5227: 5224: 5218: 5215: 5209: 5205: 5200: 5194: 5191: 5185: 5182: 5177: 5174: 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1837: 1834: 1831: 1811: 1808: 1805: 1802: 1799: 1796: 1793: 1790: 1787: 1784: 1781: 1778: 1775: 1772: 1769: 1766: 1763: 1760: 1757: 1754: 1751: 1731: 1728: 1725: 1722: 1719: 1716: 1713: 1710: 1707: 1704: 1701: 1698: 1695: 1692: 1689: 1686: 1683: 1680: 1674: 1670: 1667: 1660: 1654: 1650: 1647: 1602: 1599: 1596: 1593: 1590: 1586: 1583: 1580: 1577: 1574: 1571: 1568: 1564: 1561: 1558: 1555: 1552: 1549: 1546: 1543: 1539: 1537: 1535: 1529: 1525: 1522: 1515: 1514: 1511: 1508: 1505: 1502: 1498: 1495: 1491: 1488: 1485: 1482: 1479: 1475: 1473: 1471: 1465: 1461: 1458: 1451: 1450: 1447: 1444: 1440: 1437: 1433: 1430: 1427: 1423: 1421: 1419: 1413: 1409: 1406: 1399: 1398: 1236:Main article: 1233: 1230: 1213: 1207: 1204: 1201: 1196: 1193: 1190: 1184: 1178: 1175: 1172: 1167: 1164: 1161: 1155: 1149: 1146: 1143: 1138: 1135: 1132: 1116:The spherical 1110:Main article: 1107: 1104: 1087: 1084: 1079: 1075: 1071: 1068: 1065: 1062: 1059: 1056: 1053: 1050: 1030: 1027: 1024: 1021: 1018: 998: 995: 992: 989: 986: 983: 980: 950: 947: 944: 941: 938: 935: 932: 929: 926: 923: 920: 917: 914: 911: 908: 905: 902: 899: 896: 894: 892: 889: 886: 883: 882: 879: 876: 873: 870: 867: 864: 861: 858: 855: 852: 849: 846: 843: 840: 837: 834: 831: 828: 825: 823: 821: 818: 815: 812: 811: 808: 805: 802: 799: 796: 793: 790: 787: 784: 781: 778: 775: 772: 769: 766: 763: 760: 757: 754: 752: 750: 747: 744: 741: 740: 725:Main article: 722: 719: 717: 714: 689: 686: 683: 680: 677: 674: 672: 669: 666: 662: 660: 657: 654: 651: 648: 645: 643: 640: 637: 633: 631: 628: 625: 622: 619: 616: 614: 611: 608: 604: 603: 600: 597: 594: 591: 588: 585: 583: 580: 577: 572: 570: 567: 564: 561: 558: 555: 553: 550: 547: 542: 540: 537: 534: 531: 528: 525: 523: 520: 517: 513: 512: 453:polar triangle 434: 431: 430: 429: 402: 390: 387: 384: 381: 378: 375: 372: 369: 366: 363: 335: 334: 322: 319: 316: 313: 310: 307: 304: 301: 298: 295: 272: 243: 228: 203: 200: 170:, also called 159:plane geometry 132: 129: 119: 116: 26: 9: 6: 4: 3: 2: 11077: 11066: 11063: 11062: 11060: 11051: 11048: 11045: 11042: 11039: 11036: 11031: 11027: 11023: 11020: 11017: 11014: 11011: 11005: 11004: 10999: 10996: 10991: 10985: 10984: 10979: 10976: 10971: 10970: 10950: 10946: 10942: 10936: 10920: 10916: 10915: 10907: 10898: 10897: 10892: 10887: 10881: 10874: 10870: 10867: 10862: 10854: 10850: 10849: 10844: 10840: 10833: 10826: 10820: 10805: 10801: 10800: 10792: 10786: 10782: 10779: 10775: 10762: 10758: 10757: 10752: 10746: 10731: 10727: 10726: 10721: 10715: 10707: 10703: 10699: 10695: 10688: 10674: 10670: 10666: 10662: 10658: 10654: 10650: 10646: 10642: 10638: 10631: 10615: 10614: 10609: 10606: 10599: 10591: 10587: 10586: 10578: 10570: 10566: 10562: 10558: 10557: 10549: 10534: 10530: 10529: 10524: 10523:Todhunter, I. 10518: 10516: 10514: 10512: 10510: 10508: 10506: 10504: 10502: 10500: 10498: 10496: 10494: 10492: 10490: 10488: 10483: 10473: 10470: 10468: 10465: 10463: 10460: 10458: 10455: 10453: 10452:Lenart sphere 10450: 10447: 10444: 10442: 10439: 10437: 10434: 10432: 10429: 10428: 10422: 10420: 10416: 10412: 10408: 10407:line integral 10403: 10384: 10380: 10376: 10371: 10367: 10355: 10351: 10347: 10342: 10338: 10329: 10326: 10321: 10316: 10312: 10289: 10285: 10281: 10276: 10272: 10268: 10263: 10259: 10255: 10250: 10246: 10236: 10223: 10215: 10211: 10207: 10202: 10198: 10188: 10185: 10179: 10176: 10165: 10161: 10157: 10152: 10148: 10138: 10135: 10129: 10126: 10116: 10112: 10108: 10103: 10099: 10089: 10086: 10080: 10077: 10071: 10066: 10062: 10055: 10052: 10046: 10043: 10018: 10014: 10010: 10005: 10001: 9972: 9968: 9964: 9959: 9955: 9931: 9926: 9922: 9899: 9895: 9880: 9878: 9874: 9873:Angle deficit 9870: 9857: 9854: 9848: 9845: 9839: 9836: 9833: 9827: 9824: 9818: 9815: 9812: 9809: 9803: 9800: 9794: 9791: 9781: 9773: 9762: 9755: 9742: 9736: 9733: 9730: 9727: 9722: 9719: 9714: 9711: 9708: 9703: 9700: 9695: 9692: 9689: 9686: 9681: 9678: 9675: 9672: 9667: 9664: 9659: 9656: 9653: 9648: 9645: 9640: 9637: 9631: 9628: 9622: 9619: 9613: 9610: 9590: 9585: 9582: 9577: 9574: 9571: 9568: 9559: 9543: 9540: 9537: 9534: 9531: 9522: 9519: 9513: 9510: 9485: 9482: 9479: 9470: 9467: 9461: 9458: 9451: 9448: 9445: 9436: 9433: 9427: 9424: 9417: 9414: 9411: 9402: 9399: 9393: 9390: 9386: 9380: 9377: 9371: 9368: 9363: 9360: 9354: 9351: 9345: 9342: 9334: 9329: 9326: 9318: 9305: 9292: 9278: 9275: 9272: 9266: 9263: 9260: 9257: 9254: 9251: 9248: 9245: 9237: 9234: 9231: 9222: 9218: 9217:Albert Girard 9214: 9190: 9187: 9184: 9181: 9178: 9175: 9172: 9169: 9166: 9161: 9157: 9153: 9150: 9147: 9126: 9109: 9096: 9093: 9087: 9084: 9081: 9075: 9071: 9065: 9061: 9055: 9050: 9047: 9044: 9040: 9035: 9031: 9026: 9022: 9018: 8982:Consider an 8969: 8957: 8953: 8948: 8944: 8934: 8922: 8911: 8888: 8881: 8858: 8849:at the point 8819: 8815: 8813: 8808: 8806: 8785: 8782: 8767: 8764: 8757: 8754: 8731: 8728: 8721: 8718: 8711: 8708: 8693: 8690: 8689: 8685: 8681: 8671: 8661: 8647: 8644: 8641: 8637: 8634: 8631: 8627: 8624: 8621: 8618: 8615: 8612: 8609: 8605: 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6497: 6494: 6491: 6470: 6467: 6466: 6465: 6463: 6429: 6427: 6423: 6419: 6385: 6376: 6357: 6354: 6351: 6342: 6339: 6333: 6330: 6322: 6319: 6316: 6307: 6304: 6298: 6295: 6289: 6280: 6277: 6274: 6265: 6262: 6256: 6253: 6245: 6242: 6239: 6230: 6227: 6221: 6218: 6207: 6206:(1201–1274), 6205: 6202: 6198: 6193: 6190: 6187: 6170: 6164: 6161: 6155: 6152: 6143: 6140: 6137: 6128: 6125: 6119: 6116: 6108: 6105: 6102: 6093: 6090: 6084: 6081: 6075: 6069: 6066: 6063: 6054: 6051: 6045: 6042: 6034: 6028: 6025: 6019: 6016: 6007: 6004: 6001: 5992: 5989: 5983: 5980: 5972: 5969: 5966: 5957: 5954: 5948: 5945: 5939: 5933: 5930: 5927: 5918: 5915: 5909: 5906: 5899: 5893: 5890: 5884: 5881: 5872: 5869: 5866: 5857: 5854: 5848: 5845: 5837: 5834: 5831: 5822: 5819: 5813: 5810: 5804: 5798: 5795: 5792: 5783: 5780: 5774: 5771: 5763: 5757: 5754: 5748: 5745: 5736: 5733: 5730: 5721: 5718: 5712: 5709: 5701: 5698: 5695: 5686: 5683: 5677: 5674: 5668: 5662: 5659: 5656: 5647: 5644: 5638: 5635: 5618: 5615: 5594: 5588: 5585: 5579: 5576: 5568: 5565: 5562: 5553: 5550: 5544: 5541: 5535: 5529: 5523: 5520: 5514: 5511: 5503: 5500: 5497: 5488: 5485: 5479: 5476: 5462: 5456: 5453: 5447: 5444: 5436: 5433: 5430: 5421: 5418: 5412: 5409: 5403: 5397: 5391: 5388: 5382: 5379: 5371: 5368: 5365: 5356: 5353: 5347: 5344: 5331: 5325: 5322: 5316: 5313: 5305: 5302: 5299: 5290: 5287: 5281: 5278: 5272: 5266: 5260: 5257: 5251: 5248: 5240: 5237: 5234: 5225: 5222: 5216: 5213: 5198: 5192: 5189: 5183: 5180: 5172: 5169: 5166: 5157: 5154: 5148: 5145: 5139: 5133: 5127: 5124: 5118: 5115: 5107: 5104: 5101: 5092: 5089: 5083: 5080: 5065: 5057: 5043: 5040: 5037: 5034: 5031: 5028: 5025: 5019: 5016: 5007: 5003: 4999: 4991: 4971: 4968: 4965: 4962: 4959: 4956: 4953: 4947: 4944: 4935: 4931: 4927: 4918: 4915: 4891: 4888: 4885: 4879: 4876: 4870: 4867: 4864: 4858: 4855: 4847: 4844: 4841: 4835: 4832: 4829: 4826: 4823: 4820: 4813: 4811: 4806: 4800: 4797: 4791: 4788: 4776: 4773: 4770: 4764: 4761: 4758: 4755: 4752: 4744: 4741: 4738: 4732: 4729: 4723: 4720: 4717: 4711: 4708: 4701: 4699: 4694: 4688: 4685: 4679: 4676: 4665: 4662: 4659: 4656: 4653: 4650: 4642: 4639: 4636: 4630: 4627: 4621: 4618: 4615: 4609: 4606: 4599: 4597: 4592: 4586: 4583: 4577: 4574: 4565: 4562: 4559: 4556: 4553: 4550: 4542: 4539: 4536: 4530: 4527: 4524: 4521: 4518: 4511: 4509: 4504: 4498: 4495: 4489: 4486: 4475: 4472: 4469: 4466: 4463: 4460: 4452: 4449: 4446: 4440: 4437: 4434: 4431: 4428: 4425: 4418: 4416: 4411: 4405: 4402: 4396: 4393: 4382: 4379: 4376: 4373: 4370: 4367: 4359: 4356: 4353: 4347: 4344: 4338: 4335: 4332: 4326: 4323: 4316: 4314: 4309: 4303: 4300: 4294: 4291: 4268: 4262: 4259: 4256: 4253: 4250: 4244: 4241: 4238: 4215: 4212: 4209: 4206: 4203: 4197: 4194: 4191: 4177: 4174: 4170: 4148: 4145: 4142: 4139: 4136: 4133: 4130: 4127: 4124: 4121: 4118: 4115: 4112: 4109: 4106: 4103: 4100: 4097: 4094: 4091: 4088: 4085: 4082: 4079: 4076: 4073: 4070: 4068: 4063: 4060: 4055: 4051: 4047: 4044: 4041: 4034: 4031: 4028: 4025: 4022: 4019: 4016: 4013: 4010: 4007: 4004: 4001: 3998: 3992: 3989: 3986: 3983: 3980: 3977: 3974: 3971: 3968: 3965: 3962: 3959: 3956: 3953: 3950: 3947: 3938: 3935: 3932: 3929: 3927: 3919: 3916: 3913: 3910: 3907: 3904: 3901: 3898: 3895: 3892: 3889: 3886: 3883: 3880: 3877: 3874: 3871: 3869: 3864: 3861: 3858: 3845: 3820: 3817: 3814: 3811: 3806: 3800: 3797: 3794: 3790: 3787: 3784: 3781: 3778: 3775: 3772: 3768: 3765: 3762: 3759: 3757: 3752: 3749: 3746: 3742: 3739: 3736: 3718: 3715: 3712: 3709: 3704: 3698: 3695: 3692: 3688: 3685: 3682: 3679: 3676: 3673: 3670: 3666: 3663: 3660: 3657: 3655: 3650: 3647: 3644: 3640: 3637: 3634: 3616: 3613: 3610: 3607: 3602: 3596: 3593: 3590: 3586: 3583: 3580: 3577: 3574: 3571: 3568: 3564: 3561: 3558: 3555: 3553: 3548: 3545: 3542: 3538: 3535: 3532: 3514: 3511: 3508: 3505: 3500: 3494: 3491: 3488: 3484: 3481: 3478: 3475: 3472: 3469: 3466: 3462: 3459: 3456: 3453: 3451: 3446: 3443: 3440: 3436: 3433: 3430: 3412: 3409: 3406: 3403: 3398: 3392: 3389: 3386: 3382: 3379: 3376: 3373: 3370: 3367: 3364: 3360: 3357: 3354: 3351: 3349: 3344: 3341: 3338: 3334: 3331: 3328: 3310: 3307: 3304: 3301: 3296: 3289: 3286: 3283: 3279: 3276: 3273: 3270: 3267: 3264: 3261: 3257: 3254: 3251: 3248: 3246: 3241: 3238: 3235: 3231: 3228: 3225: 3192: 3150: 3108: 3105: 3063: 3021: 3018: 2976: 2934: 2898: 2884: 2867: 2864: 2861: 2858: 2854: 2851: 2848: 2844: 2841: 2838: 2835: 2832: 2829: 2826: 2822: 2819: 2816: 2813: 2810: 2808: 2803: 2800: 2797: 2790: 2787: 2784: 2781: 2777: 2774: 2771: 2767: 2764: 2761: 2758: 2755: 2752: 2749: 2745: 2742: 2739: 2736: 2733: 2731: 2726: 2723: 2720: 2713: 2710: 2707: 2704: 2700: 2697: 2694: 2690: 2687: 2684: 2681: 2678: 2675: 2672: 2668: 2665: 2662: 2659: 2656: 2654: 2649: 2646: 2643: 2630: 2616: 2603: 2588: 2586: 2581: 2577: 2573: 2569: 2562: 2558: 2554: 2546: 2537: 2528: 2504: 2500: 2496: 2491:evaluates to 2488: 2479: 2470: 2460: 2454: 2451: 2440: 2410: 2404: 2401: 2398: 2395: 2392: 2389: 2386: 2383: 2380: 2375: 2372: 2369: 2366: 2363: 2360: 2357: 2354: 2351: 2348: 2345: 2342: 2336: 2332: 2328: 2325: 2319: 2315: 2311: 2308: 2302: 2298: 2294: 2291: 2285: 2283: 2275: 2272: 2269: 2264: 2261: 2258: 2245: 2239: 2235: 2230: 2224: 2220: 2212: 2204: 2201: 2198: 2195: 2192: 2189: 2186: 2183: 2180: 2177: 2171: 2165: 2162: 2157: 2153: 2149: 2146: 2137: 2134: 2129: 2125: 2121: 2118: 2109: 2107: 2097: 2092: 2086: 2083: 2080: 2077: 2074: 2071: 2066: 2063: 2060: 2057: 2054: 2051: 2048: 2045: 2042: 2039: 2033: 2028: 2025: 2022: 2020: 2015: 2012: 2007: 2003: 1989: 1971: 1968: 1963: 1959: 1955: 1952: 1949: 1946: 1943: 1938: 1934: 1923: 1913: 1910: 1897: 1891: 1888: 1885: 1882: 1879: 1876: 1871: 1868: 1865: 1862: 1859: 1856: 1853: 1850: 1847: 1844: 1838: 1835: 1832: 1829: 1809: 1806: 1803: 1800: 1797: 1794: 1791: 1788: 1785: 1782: 1779: 1776: 1773: 1770: 1767: 1764: 1761: 1758: 1755: 1752: 1749: 1729: 1726: 1723: 1720: 1717: 1714: 1711: 1708: 1705: 1702: 1699: 1696: 1693: 1690: 1687: 1684: 1681: 1678: 1668: 1665: 1658: 1648: 1645: 1617: 1600: 1594: 1591: 1588: 1584: 1581: 1578: 1575: 1572: 1569: 1566: 1562: 1559: 1556: 1553: 1550: 1547: 1544: 1538: 1533: 1523: 1520: 1506: 1503: 1500: 1496: 1493: 1489: 1486: 1483: 1480: 1474: 1469: 1459: 1456: 1442: 1438: 1435: 1431: 1428: 1422: 1417: 1407: 1404: 1388: 1322: 1318: 1304: 1288: 1279: 1270: 1259: 1257: 1253: 1244: 1239: 1229: 1227: 1211: 1205: 1202: 1199: 1194: 1191: 1188: 1182: 1176: 1173: 1170: 1165: 1162: 1159: 1153: 1147: 1144: 1141: 1136: 1133: 1130: 1119: 1113: 1103: 1101: 1085: 1082: 1077: 1069: 1066: 1063: 1060: 1057: 1054: 1051: 1028: 1025: 1022: 1019: 1016: 996: 990: 987: 984: 981: 978: 970: 965: 948: 945: 942: 939: 936: 933: 930: 927: 924: 921: 918: 915: 912: 909: 906: 903: 900: 897: 895: 890: 887: 884: 877: 874: 871: 868: 865: 862: 859: 856: 853: 850: 847: 844: 841: 838: 835: 832: 829: 826: 824: 819: 816: 813: 806: 803: 800: 797: 794: 791: 788: 785: 782: 779: 776: 773: 770: 767: 764: 761: 758: 755: 753: 748: 745: 742: 728: 713: 709: 687: 684: 681: 678: 675: 673: 667: 664: 658: 655: 652: 649: 646: 644: 638: 635: 629: 626: 623: 620: 617: 615: 609: 606: 598: 595: 592: 589: 586: 584: 578: 575: 568: 565: 562: 559: 556: 554: 548: 545: 538: 535: 532: 529: 526: 524: 518: 515: 501: 494: 489:The triangle 487: 478:. The points 460: 454: 446: 439: 403: 388: 385: 382: 379: 376: 373: 370: 367: 364: 361: 349: 344: 343: 342: 320: 317: 314: 311: 308: 305: 302: 299: 296: 293: 281: 277: 273: 270: 263: 248: 244: 229: 214: 213: 208: 199: 197: 192: 190: 186: 181: 179: 175: 174: 169: 168: 162: 160: 156: 155:line segments 152: 151:great circles 148: 144: 143: 138: 124: 118:Preliminaries 115: 113: 109: 105: 101: 97: 92: 90: 86: 82: 78: 77:great circles 74: 70: 66: 62: 58: 54: 50: 46: 39: 34: 30: 19: 11034:by Okay Arik 11001: 10981: 10953:. Retrieved 10944: 10935: 10923:. Retrieved 10913: 10906: 10895: 10880: 10861: 10846: 10832: 10824: 10819: 10808:. Retrieved 10798: 10791: 10773: 10765:. Retrieved 10755: 10745: 10734:. Retrieved 10724: 10714: 10697: 10693: 10687: 10677:, retrieved 10644: 10640: 10630: 10617:. Retrieved 10611: 10598: 10584: 10577: 10555: 10548: 10537:. Retrieved 10527: 10413:, or via an 10404: 10237: 9886: 9871: 9779: 9771: 9760: 9756: 9560: 9330: 9324: 9316: 9306: 9238: 9235: 9229: 9212: 9124: 9110: 8981: 8968:small circle 8928: 8913:. The angle 8909: 8886: 8879: 8856: 8828: 8809: 8797: 8783: 8765: 8755: 8729: 8719: 8709: 8691: 8678: 8308: 8170: 8166: 8158: 8064: 7813: 7275: 7267: 7261: 7253: 7243: 7223: 6474: 6461: 6430: 6426:mnemonic aid 6420: 6389: 6208: 6194: 6191: 6188: 5624: 5616: 5066: 5063: 4919: 4916: 4183: 4172: 4168: 3843: 2896: 2890: 2628: 2614: 2601: 2599: 2585:law of sines 2579: 2575: 2571: 2567: 2560: 2556: 2552: 2539: 2530: 2521: 2510:-axis along 2502: 2498: 2494: 2481: 2472: 2463: 2455: 2446: 1987: 1925: 1911: 1618: 1389: 1367:projects to 1320: 1311: 1302: 1281: 1272: 1263: 1260: 1248: 1226:trigonometry 1118:law of sines 1115: 969:trigonometry 966: 730: 721:Cosine rules 707: 499: 492: 488: 458: 452: 450: 444: 347: 336: 279: 261: 246: 195: 193: 189:star polygon 182: 177: 171: 165: 163: 140: 136: 134: 111: 93: 60: 44: 43: 29: 8993:denote the 8943:Solid angle 7247:with sides 7236:has length 2897:consecutive 104:John Napier 10955:2021-05-01 10810:2021-07-11 10767:2016-05-14 10736:2016-05-14 10679:2016-01-10 10539:2013-07-28 10478:References 8941:See also: 7251:such that 7249:a', b', c' 2604:replacing 2591:Identities 1337:-axis and 1333:along the 1106:Sine rules 1098:etc.; see 354:, so that 286:, so that 89:navigation 11003:MathWorld 10983:MathWorld 10778:Abe Books 10751:Napier, J 10669:122277398 10613:MathWorld 10421:formula. 10381:λ 10377:− 10368:λ 10352:φ 10339:φ 10322:≈ 10286:λ 10282:− 10273:λ 10260:φ 10247:φ 10212:λ 10208:− 10199:λ 10180:⁡ 10162:φ 10158:− 10149:φ 10130:⁡ 10113:φ 10100:φ 10081:⁡ 10047:⁡ 10015:φ 10002:λ 9969:φ 9956:λ 9923:λ 9896:λ 9840:⁡ 9819:⁡ 9795:⁡ 9734:⁡ 9715:⁡ 9696:⁡ 9679:⁡ 9660:⁡ 9641:⁡ 9614:⁡ 9578:≈ 9483:− 9462:⁡ 9449:− 9428:⁡ 9415:− 9394:⁡ 9372:⁡ 9346:⁡ 9333:L'Huilier 9279:× 9276:π 9264:π 9188:π 9185:− 9148:≡ 9094:π 9085:− 9076:− 9041:∑ 9019:≡ 8917:and side 8645:⁡ 8635:⁡ 8625:⁡ 8619:− 8613:⁡ 8603:⁡ 8591:⁡ 8581:⁡ 8571:⁡ 8559:⁡ 8549:⁡ 8523:⁡ 8506:⁡ 8496:⁡ 8484:⁡ 8467:⁡ 8457:⁡ 8428:⁡ 8411:⁡ 8401:⁡ 8395:− 8389:⁡ 8379:⁡ 8367:⁡ 8357:⁡ 8347:⁡ 8341:− 8332:⁡ 8323:⁡ 8287:⁡ 8270:⁡ 8260:⁡ 8254:− 8248:⁡ 8238:⁡ 8226:⁡ 8216:⁡ 8206:⁡ 8194:⁡ 8185:⁡ 8142:⁡ 8132:⁡ 8120:⁡ 8110:⁡ 8100:⁡ 8088:⁡ 8079:⁡ 8042:⁡ 8032:⁡ 8022:⁡ 8010:⁡ 8000:⁡ 7990:⁡ 7980:⁡ 7964:⁡ 7947:⁡ 7934:⁡ 7924:⁡ 7914:⁡ 7902:⁡ 7890:⁡ 7880:⁡ 7870:⁡ 7858:⁡ 7848:⁡ 7829:⁡ 7785:⁡ 7775:⁡ 7769:− 7756:⁡ 7734:⁡ 7724:⁡ 7708:⁡ 7684:⁡ 7674:⁡ 7658:⁡ 7636:⁡ 7626:⁡ 7610:⁡ 7586:⁡ 7576:⁡ 7560:⁡ 7538:⁡ 7528:⁡ 7512:⁡ 7488:⁡ 7478:⁡ 7472:− 7459:⁡ 7437:⁡ 7427:⁡ 7411:⁡ 7387:⁡ 7377:⁡ 7371:− 7358:⁡ 7332:⁡ 7322:⁡ 7316:− 7303:⁡ 7187:⁡ 7177:⁡ 7161:⁡ 7139:⁡ 7129:⁡ 7113:⁡ 7089:⁡ 7079:⁡ 7063:⁡ 7041:⁡ 7031:⁡ 7015:⁡ 6991:⁡ 6981:⁡ 6965:⁡ 6943:⁡ 6933:⁡ 6917:⁡ 6893:⁡ 6883:⁡ 6867:⁡ 6845:⁡ 6835:⁡ 6819:⁡ 6795:⁡ 6785:⁡ 6769:⁡ 6743:⁡ 6733:⁡ 6717:⁡ 6671:⁡ 6661:⁡ 6642:⁡ 6632:⁡ 6610:− 6601:π 6592:⁡ 6579:− 6570:π 6561:⁡ 6542:⁡ 6529:− 6520:π 6511:⁡ 6495:⁡ 6480:we have: 6444:(that is 6334:⁡ 6320:− 6299:⁡ 6257:⁡ 6243:− 6222:⁡ 6156:⁡ 6120:⁡ 6106:− 6085:⁡ 6067:− 6046:⁡ 6020:⁡ 5984:⁡ 5970:− 5949:⁡ 5931:− 5910:⁡ 5885:⁡ 5849:⁡ 5835:− 5814:⁡ 5775:⁡ 5749:⁡ 5713:⁡ 5699:− 5678:⁡ 5639:⁡ 5580:⁡ 5545:⁡ 5515:⁡ 5501:− 5480:⁡ 5448:⁡ 5413:⁡ 5383:⁡ 5348:⁡ 5317:⁡ 5303:− 5282:⁡ 5252:⁡ 5238:− 5217:⁡ 5184:⁡ 5170:− 5149:⁡ 5119:⁡ 5084:⁡ 5038:⁡ 4966:⁡ 4960:− 4889:− 4880:⁡ 4868:− 4859:⁡ 4845:− 4836:⁡ 4827:⁡ 4821:− 4792:⁡ 4774:− 4765:⁡ 4756:⁡ 4742:− 4733:⁡ 4721:− 4712:⁡ 4680:⁡ 4663:⁡ 4654:⁡ 4640:− 4631:⁡ 4619:− 4610:⁡ 4578:⁡ 4563:⁡ 4554:⁡ 4540:− 4531:⁡ 4522:⁡ 4490:⁡ 4473:⁡ 4464:⁡ 4450:− 4441:⁡ 4432:⁡ 4426:− 4397:⁡ 4380:⁡ 4371:⁡ 4357:− 4348:⁡ 4336:− 4327:⁡ 4295:⁡ 4143:⁡ 4134:⁡ 4125:⁡ 4116:⁡ 4104:⁡ 4095:⁡ 4086:⁡ 4077:⁡ 4061:⁡ 4045:⁡ 4032:⁡ 4023:⁡ 4014:⁡ 4005:⁡ 3990:⁡ 3981:⁡ 3972:⁡ 3960:⁡ 3951:⁡ 3936:⁡ 3917:⁡ 3908:⁡ 3899:⁡ 3887:⁡ 3878:⁡ 3862:⁡ 3798:⁡ 3788:⁡ 3782:− 3776:⁡ 3766:⁡ 3750:⁡ 3740:⁡ 3696:⁡ 3686:⁡ 3680:− 3674:⁡ 3664:⁡ 3648:⁡ 3638:⁡ 3594:⁡ 3584:⁡ 3578:− 3572:⁡ 3562:⁡ 3546:⁡ 3536:⁡ 3492:⁡ 3482:⁡ 3476:− 3470:⁡ 3460:⁡ 3444:⁡ 3434:⁡ 3390:⁡ 3380:⁡ 3374:− 3368:⁡ 3358:⁡ 3342:⁡ 3332:⁡ 3287:⁡ 3277:⁡ 3271:− 3265:⁡ 3255:⁡ 3239:⁡ 3229:⁡ 3106:− 2862:⁡ 2852:⁡ 2842:⁡ 2830:⁡ 2820:⁡ 2814:− 2801:⁡ 2785:⁡ 2775:⁡ 2765:⁡ 2753:⁡ 2743:⁡ 2737:− 2724:⁡ 2708:⁡ 2698:⁡ 2688:⁡ 2676:⁡ 2666:⁡ 2660:− 2647:⁡ 2402:⁡ 2393:⁡ 2384:⁡ 2373:⁡ 2364:⁡ 2355:⁡ 2329:− 2312:− 2295:− 2273:⁡ 2262:⁡ 2202:⁡ 2193:⁡ 2187:− 2181:⁡ 2172:− 2163:⁡ 2150:− 2135:⁡ 2122:− 2084:⁡ 2075:⁡ 2064:⁡ 2055:⁡ 2049:− 2043:⁡ 2029:− 2013:⁡ 1969:⁡ 1956:− 1944:⁡ 1889:⁡ 1880:⁡ 1869:⁡ 1860:⁡ 1854:− 1848:⁡ 1833:⁡ 1804:⁡ 1795:⁡ 1786:⁡ 1774:⁡ 1765:⁡ 1753:⁡ 1724:⁡ 1715:⁡ 1703:⁡ 1694:⁡ 1685:⁡ 1673:→ 1659:⋅ 1653:→ 1592:⁡ 1579:⁡ 1570:⁡ 1557:⁡ 1548:⁡ 1528:→ 1504:⁡ 1484:⁡ 1464:→ 1412:→ 1383:-axis is 1354:with the 1203:⁡ 1192:⁡ 1174:⁡ 1163:⁡ 1145:⁡ 1134:⁡ 1083:≈ 1067:⁡ 1061:− 1055:⁡ 1026:≈ 1020:⁡ 994:→ 943:⁡ 934:⁡ 925:⁡ 913:⁡ 904:⁡ 888:⁡ 872:⁡ 863:⁡ 854:⁡ 842:⁡ 833:⁡ 817:⁡ 801:⁡ 792:⁡ 783:⁡ 771:⁡ 762:⁡ 746:⁡ 682:− 679:π 653:− 650:π 624:− 621:π 593:− 590:π 563:− 560:π 533:− 530:π 389:π 341:radians. 321:π 294:π 196:triangles 178:bi-angles 81:astronomy 73:geodesics 67:. On the 11059:Category 10949:Archived 10925:7 August 10919:Archived 10893:(1880). 10869:Archived 10804:Archived 10781:Archived 10761:Archived 10753:(1614). 10730:Archived 10722:(1807). 10673:archived 10533:Archived 10525:(1886). 10425:See also 8966:along a 1379:and the 668:′ 639:′ 610:′ 579:′ 549:′ 519:′ 269:tangents 202:Notation 108:Delambre 10896:Geodesy 10661:4146847 10619:8 April 10565:2484948 10556:Geodesy 9768:, then 8748:, then 8740:) give 8736:) and ( 8312:yields 7244:A'B'C' 1371:in the 1346:in the 493:A'B'C' 445:A'B'C' 276:radians 142:polygon 85:geodesy 11013:TriSph 10667: 10659: 10563: 9503:where 9317:proper 9203:where 9119:, and 8860:: use 8837:, and 8776:, and 8702:, and 8174:gives 8068:gives 6462:middle 6452:, and 6438:aCbAcB 6436:gives 6422:Napier 6414:, and 3942: 2893:aCbAcB 2632:etc., 2574:= sin 2435:, and 1319:= cos 1252:vector 420:, and 348:proper 280:proper 262:vertex 238:, and 223:, and 173:digons 87:, and 69:sphere 57:angles 10665:S2CID 10657:JSTOR 10409:with 9914:and 7747:(Q10) 7152:(R10) 4184:With 3730:(CT6) 3628:(CT5) 3526:(CT4) 3424:(CT3) 3322:(CT2) 3218:(CT1) 3177:angle 3168:inner 3135:angle 3126:outer 3081:inner 3039:outer 3003:angle 2994:inner 2952:inner 2903:) or 247:angle 167:lunes 139:is a 53:sides 10927:2020 10621:2018 10561:OCLC 9990:and 9778:sin 9776:and 9770:cos 9315:for 8974:and 8945:and 8906:and 8894:and 8872:and 8864:and 8790:and 8744:and 8738:BaCb 8734:cBaC 8307:cos 8169:cos 8063:sin 7699:(Q5) 7649:(Q9) 7601:(Q4) 7551:(Q8) 7503:(Q3) 7450:(Q7) 7402:(Q2) 7348:(Q6) 7293:(Q1) 7104:(R5) 7054:(R9) 7006:(R4) 6956:(R8) 6908:(R3) 6858:(R7) 6810:(R2) 6759:(R6) 6707:(R1) 4231:and 4171:sin 4167:sin 3842:cos 3090:side 3048:side 2961:side 2909:BaCb 2905:BaCb 2901:aCbA 2602:i.e. 2578:sin 2570:sin 2566:sin 2559:sin 2555:sin 2551:sin 2501:sin 2497:sin 2493:sin 2457:The 1986:cos 1627:· OC 1041:and 1009:set 482:and 451:The 383:< 365:< 315:< 297:< 256:and 245:The 147:arcs 98:and 75:are 55:and 36:The 10702:doi 10649:doi 10177:tan 10127:cos 10078:sin 10044:tan 10036:is 9837:tan 9816:tan 9792:tan 9782:= 1 9774:= 0 9761:ABC 9731:cos 9712:tan 9693:tan 9676:sin 9657:tan 9638:tan 9611:tan 9459:tan 9425:tan 9391:tan 9369:tan 9343:tan 8910:DAC 8887:ACD 8880:BAD 8857:ABD 8642:cos 8632:cos 8622:cos 8610:sin 8600:sin 8588:cos 8578:cos 8568:cos 8556:sin 8546:sin 8514:sin 8503:sin 8493:sin 8475:sin 8464:sin 8454:sin 8419:sin 8408:sin 8398:sin 8386:sin 8376:sin 8364:cos 8354:cos 8344:cos 8329:cos 8320:cos 8278:sin 8267:sin 8257:sin 8245:sin 8235:sin 8223:cos 8213:cos 8203:cos 8191:cos 8182:cos 8139:cos 8129:sin 8117:cos 8107:cos 8097:sin 8085:sin 8076:cos 8039:cos 8029:sin 8019:sin 8007:cos 7997:sin 7987:cos 7977:sin 7955:sin 7944:cos 7931:cos 7921:sin 7911:sin 7899:cos 7887:cos 7877:sin 7867:sin 7855:cos 7845:cos 7826:cos 7782:cot 7772:cot 7753:cos 7731:sin 7721:tan 7705:tan 7681:cos 7671:sin 7655:cos 7633:sin 7623:tan 7607:tan 7583:cos 7573:sin 7557:cos 7535:sin 7525:sin 7509:sin 7485:tan 7475:cos 7456:tan 7434:sin 7424:sin 7408:sin 7384:tan 7374:cos 7355:tan 7329:cos 7319:cos 7300:cos 7268:a' 7254:A' 7184:cot 7174:cot 7158:cos 7136:sin 7126:tan 7110:tan 7086:cos 7076:sin 7060:cos 7038:sin 7028:tan 7012:tan 6988:cos 6978:sin 6962:cos 6940:sin 6930:sin 6914:sin 6890:tan 6880:cos 6864:tan 6842:sin 6832:sin 6816:sin 6792:tan 6782:cos 6766:tan 6740:cos 6730:cos 6714:cos 6668:sin 6658:sin 6639:tan 6629:cot 6589:cos 6558:cos 6539:tan 6508:tan 6492:sin 6331:tan 6296:tan 6254:tan 6219:tan 6153:tan 6117:sin 6082:sin 6043:tan 6017:cot 5981:sin 5946:sin 5907:tan 5882:tan 5846:cos 5811:cos 5772:tan 5746:cot 5710:cos 5675:cos 5636:tan 5577:sin 5542:sin 5512:sin 5477:cos 5445:cos 5410:cos 5380:sin 5345:cos 5314:sin 5279:sin 5249:cos 5214:sin 5181:cos 5146:cos 5116:cos 5081:sin 5035:cos 5004:cos 4963:cos 4932:sin 4877:cos 4856:cos 4833:cos 4824:cos 4789:tan 4762:sin 4753:sin 4730:sin 4709:sin 4677:tan 4660:sin 4651:sin 4628:cos 4607:cos 4575:cos 4560:sin 4551:sin 4528:sin 4519:sin 4487:cos 4470:sin 4461:sin 4438:cos 4429:cos 4394:sin 4377:sin 4368:sin 4345:sin 4324:sin 4292:sin 4140:cot 4131:sin 4122:sin 4113:sin 4101:cos 4092:sin 4083:sin 4074:cos 4052:sin 4042:cos 4029:cot 4020:sin 4011:sin 4002:sin 3987:cos 3978:sin 3969:sin 3957:cos 3948:cos 3933:cos 3914:cos 3905:sin 3896:sin 3884:cos 3875:cos 3859:cos 3795:sin 3785:cot 3773:sin 3763:cot 3747:cos 3737:cos 3693:sin 3683:cot 3671:sin 3661:cot 3645:cos 3635:cos 3591:sin 3581:cot 3569:sin 3559:cot 3543:cos 3533:cos 3489:sin 3479:cot 3467:sin 3457:cot 3441:cos 3431:cos 3387:sin 3377:cot 3365:sin 3355:cot 3339:cos 3329:cos 3284:sin 3274:cot 3262:sin 3252:cot 3236:cos 3226:cos 3151:sin 3109:cot 3064:sin 3022:cot 2977:cos 2935:cos 2859:cos 2849:sin 2839:sin 2827:cos 2817:cos 2798:cos 2782:cos 2772:sin 2762:sin 2750:cos 2740:cos 2721:cos 2705:cos 2695:sin 2685:sin 2673:cos 2663:cos 2644:cos 2622:by 2608:by 2529:· ( 2471:· ( 2399:sin 2390:sin 2381:sin 2370:cos 2361:cos 2352:cos 2333:cos 2316:cos 2299:cos 2270:sin 2259:sin 2236:sin 2221:sin 2199:cos 2190:cos 2178:cos 2154:cos 2126:cos 2081:sin 2072:sin 2061:cos 2052:cos 2040:cos 2004:sin 1960:cos 1935:sin 1886:sin 1877:sin 1866:cos 1857:cos 1845:cos 1830:cos 1801:cos 1792:sin 1783:sin 1771:cos 1762:cos 1750:cos 1721:cos 1712:cos 1700:cos 1691:sin 1682:sin 1589:cos 1576:sin 1567:sin 1554:cos 1545:sin 1501:cos 1481:sin 1258:.) 1200:sin 1189:sin 1171:sin 1160:sin 1142:sin 1131:sin 1102:.) 1064:cos 1052:cos 1017:sin 940:cos 931:sin 922:sin 910:cos 901:cos 885:cos 869:cos 860:sin 851:sin 839:cos 830:cos 814:cos 798:cos 789:sin 780:sin 768:cos 759:cos 743:cos 708:ABC 500:ABC 459:ABC 176:or 157:in 149:of 59:of 11061:: 11028:. 11000:. 10980:. 10947:. 10943:. 10851:, 10845:, 10841:, 10698:45 10696:. 10671:, 10663:, 10655:, 10645:35 10643:, 10639:, 10610:. 10486:^ 10402:. 9879:. 9335:: 9325:is 9311:(3 9115:, 8960:AB 8900:DC 8892:AD 8874:BD 8870:AD 8847:BC 8833:, 8772:, 8698:, 7274:− 7270:= 7265:, 7260:− 7256:= 6448:, 6418:. 6410:, 6406:, 6402:, 2627:– 2618:, 2613:– 2540:OA 2538:× 2531:OC 2522:OB 2512:OB 2482:OC 2480:× 2473:OB 2464:OA 2461:, 2431:, 1621:OB 1377:ON 1373:xy 1369:ON 1360:OC 1348:xz 1339:OB 1326:OA 1312:OC 1310:· 1303:OB 1293:BC 1282:OC 1280:, 1273:OB 1271:, 1264:OA 484:C' 480:B' 476:A' 464:BC 416:, 234:, 219:, 198:. 161:. 135:A 106:, 91:. 83:, 71:, 11032:. 11006:. 10986:. 10958:. 10929:. 10875:. 10813:. 10770:. 10739:. 10708:. 10704:: 10651:: 10623:. 10592:. 10571:. 10542:. 10390:) 10385:1 10372:2 10364:( 10361:) 10356:1 10348:+ 10343:2 10335:( 10330:2 10327:1 10317:4 10313:E 10290:1 10277:2 10269:, 10264:2 10256:, 10251:1 10224:. 10221:) 10216:1 10203:2 10195:( 10189:2 10186:1 10171:) 10166:1 10153:2 10145:( 10139:2 10136:1 10122:) 10117:1 10109:+ 10104:2 10096:( 10090:2 10087:1 10072:= 10067:4 10063:E 10056:2 10053:1 10024:) 10019:2 10011:, 10006:2 9998:( 9978:) 9973:1 9965:, 9960:1 9952:( 9932:, 9927:2 9900:1 9858:. 9855:b 9849:2 9846:1 9834:a 9828:2 9825:1 9813:= 9810:E 9804:2 9801:1 9780:C 9772:C 9766:C 9759:△ 9743:. 9737:C 9728:b 9723:2 9720:1 9709:a 9704:2 9701:1 9690:+ 9687:1 9682:C 9673:b 9668:2 9665:1 9654:a 9649:2 9646:1 9632:= 9629:E 9623:2 9620:1 9591:c 9586:2 9583:1 9575:b 9572:= 9569:a 9547:) 9544:c 9541:+ 9538:b 9535:+ 9532:a 9529:( 9523:2 9520:1 9514:= 9511:s 9489:) 9486:c 9480:s 9477:( 9471:2 9468:1 9455:) 9452:b 9446:s 9443:( 9437:2 9434:1 9421:) 9418:a 9412:s 9409:( 9403:2 9400:1 9387:s 9381:2 9378:1 9364:= 9361:E 9355:4 9352:1 9321:π 9313:π 9309:π 9293:. 9273:4 9267:+ 9261:= 9258:C 9255:+ 9252:B 9249:+ 9246:A 9230:R 9225:R 9209:π 9205:E 9191:, 9182:C 9179:+ 9176:B 9173:+ 9170:A 9167:= 9162:3 9158:E 9154:= 9151:E 9121:C 9117:B 9113:A 9097:. 9091:) 9088:2 9082:N 9079:( 9072:) 9066:n 9062:A 9056:N 9051:1 9048:= 9045:n 9036:( 9032:= 9027:N 9023:E 8995:n 8990:n 8988:A 8984:N 8978:. 8976:B 8972:A 8964:C 8931:π 8919:a 8915:A 8908:∠ 8904:C 8896:b 8885:△ 8878:∠ 8866:B 8862:c 8855:△ 8851:D 8843:A 8839:B 8835:c 8831:b 8801:π 8792:A 8788:a 8778:c 8774:b 8770:a 8760:b 8750:A 8746:b 8742:c 8724:C 8714:a 8704:C 8700:B 8696:A 8648:a 8638:C 8628:B 8616:C 8606:B 8597:= 8594:A 8584:c 8574:b 8565:+ 8562:c 8552:b 8526:a 8518:2 8509:C 8499:B 8490:= 8487:A 8479:2 8470:c 8460:b 8434:. 8431:a 8423:2 8414:C 8404:B 8392:C 8382:B 8373:+ 8370:a 8360:C 8350:B 8338:= 8335:A 8326:a 8309:a 8293:. 8290:A 8282:2 8273:c 8263:b 8251:c 8241:b 8232:+ 8229:A 8219:c 8209:b 8200:= 8197:A 8188:a 8171:A 8145:A 8135:b 8126:+ 8123:B 8113:c 8103:a 8094:= 8091:c 8082:a 8065:c 8045:A 8035:c 8025:b 8016:+ 8013:B 8003:c 7993:c 7983:a 7974:= 7967:c 7959:2 7950:a 7937:A 7927:c 7917:b 7908:+ 7905:c 7896:) 7893:B 7883:c 7873:a 7864:+ 7861:c 7851:a 7842:( 7839:= 7832:a 7791:. 7788:b 7778:a 7766:= 7759:C 7740:, 7737:A 7727:b 7718:= 7711:B 7690:, 7687:B 7677:a 7668:= 7661:b 7642:, 7639:B 7629:a 7620:= 7613:A 7592:, 7589:A 7579:b 7570:= 7563:a 7544:, 7541:C 7531:b 7522:= 7515:B 7494:, 7491:C 7481:b 7469:= 7462:A 7443:, 7440:C 7430:a 7421:= 7414:A 7393:, 7390:C 7380:a 7368:= 7361:B 7338:, 7335:B 7325:A 7313:= 7306:C 7276:A 7272:π 7262:a 7258:π 7242:△ 7238:π 7234:c 7230:π 7226:π 7193:. 7190:B 7180:A 7171:= 7164:c 7145:, 7142:a 7132:B 7123:= 7116:b 7095:, 7092:b 7082:A 7073:= 7066:B 7047:, 7044:b 7034:A 7025:= 7018:a 6997:, 6994:a 6984:B 6975:= 6968:A 6949:, 6946:c 6936:B 6927:= 6920:b 6899:, 6896:c 6886:B 6877:= 6870:a 6851:, 6848:c 6838:A 6829:= 6822:a 6801:, 6798:c 6788:A 6779:= 6772:b 6749:, 6746:b 6736:a 6727:= 6720:c 6677:. 6674:A 6664:c 6655:= 6645:b 6635:B 6626:= 6616:) 6613:A 6604:2 6595:( 6585:) 6582:c 6573:2 6564:( 6555:= 6545:b 6535:) 6532:B 6523:2 6514:( 6505:= 6498:a 6478:a 6458:C 6454:B 6450:c 6446:A 6442:C 6434:a 6416:B 6412:A 6408:c 6404:b 6400:a 6396:π 6392:C 6361:) 6358:b 6355:+ 6352:a 6349:( 6343:2 6340:1 6326:) 6323:b 6317:a 6314:( 6308:2 6305:1 6290:= 6284:) 6281:B 6278:+ 6275:A 6272:( 6266:2 6263:1 6249:) 6246:B 6240:A 6237:( 6231:2 6228:1 6171:c 6165:2 6162:1 6147:) 6144:B 6141:+ 6138:A 6135:( 6129:2 6126:1 6112:) 6109:B 6103:A 6100:( 6094:2 6091:1 6076:= 6073:) 6070:b 6064:a 6061:( 6055:2 6052:1 6035:C 6029:2 6026:1 6011:) 6008:b 6005:+ 6002:a 5999:( 5993:2 5990:1 5976:) 5973:b 5967:a 5964:( 5958:2 5955:1 5940:= 5937:) 5934:B 5928:A 5925:( 5919:2 5916:1 5900:c 5894:2 5891:1 5876:) 5873:B 5870:+ 5867:A 5864:( 5858:2 5855:1 5841:) 5838:B 5832:A 5829:( 5823:2 5820:1 5805:= 5802:) 5799:b 5796:+ 5793:a 5790:( 5784:2 5781:1 5764:C 5758:2 5755:1 5740:) 5737:b 5734:+ 5731:a 5728:( 5722:2 5719:1 5705:) 5702:b 5696:a 5693:( 5687:2 5684:1 5669:= 5666:) 5663:B 5660:+ 5657:A 5654:( 5648:2 5645:1 5595:c 5589:2 5586:1 5572:) 5569:b 5566:+ 5563:a 5560:( 5554:2 5551:1 5536:= 5530:C 5524:2 5521:1 5507:) 5504:B 5498:A 5495:( 5489:2 5486:1 5463:c 5457:2 5454:1 5440:) 5437:b 5434:+ 5431:a 5428:( 5422:2 5419:1 5404:= 5398:C 5392:2 5389:1 5375:) 5372:B 5369:+ 5366:A 5363:( 5357:2 5354:1 5332:c 5326:2 5323:1 5309:) 5306:b 5300:a 5297:( 5291:2 5288:1 5273:= 5267:C 5261:2 5258:1 5244:) 5241:B 5235:A 5232:( 5226:2 5223:1 5199:c 5193:2 5190:1 5176:) 5173:b 5167:a 5164:( 5158:2 5155:1 5140:= 5134:C 5128:2 5125:1 5111:) 5108:B 5105:+ 5102:A 5099:( 5093:2 5090:1 5044:, 5041:A 5032:+ 5029:1 5026:= 5020:2 5017:A 5008:2 5000:2 4986:A 4972:, 4969:A 4957:1 4954:= 4948:2 4945:A 4936:2 4928:2 4895:) 4892:C 4886:S 4883:( 4874:) 4871:B 4865:S 4862:( 4851:) 4848:A 4842:S 4839:( 4830:S 4814:= 4807:a 4801:2 4798:1 4780:) 4777:a 4771:s 4768:( 4759:s 4748:) 4745:c 4739:s 4736:( 4727:) 4724:b 4718:s 4715:( 4702:= 4695:A 4689:2 4686:1 4666:C 4657:B 4646:) 4643:C 4637:S 4634:( 4625:) 4622:B 4616:S 4613:( 4600:= 4593:a 4587:2 4584:1 4566:c 4557:b 4546:) 4543:a 4537:s 4534:( 4525:s 4512:= 4505:A 4499:2 4496:1 4476:C 4467:B 4456:) 4453:A 4447:S 4444:( 4435:S 4419:= 4412:a 4406:2 4403:1 4383:c 4374:b 4363:) 4360:c 4354:s 4351:( 4342:) 4339:b 4333:s 4330:( 4317:= 4310:A 4304:2 4301:1 4269:, 4266:) 4263:C 4260:+ 4257:B 4254:+ 4251:A 4248:( 4245:= 4242:S 4239:2 4219:) 4216:c 4213:+ 4210:b 4207:+ 4204:a 4201:( 4198:= 4195:s 4192:2 4173:b 4169:a 4149:. 4146:A 4137:a 4128:C 4119:b 4110:+ 4107:C 4098:b 4089:a 4080:b 4071:= 4064:b 4056:2 4048:a 4035:A 4026:a 4017:C 4008:b 3999:+ 3996:) 3993:C 3984:b 3975:a 3966:+ 3963:b 3954:a 3945:( 3939:b 3930:= 3920:A 3911:c 3902:b 3893:+ 3890:c 3881:b 3872:= 3865:a 3844:c 3824:) 3821:b 3818:C 3815:a 3812:B 3809:( 3801:C 3791:B 3779:a 3769:b 3760:= 3753:C 3743:a 3722:) 3719:C 3716:a 3713:B 3710:c 3707:( 3699:B 3689:C 3677:a 3667:c 3658:= 3651:B 3641:a 3620:) 3617:a 3614:B 3611:c 3608:A 3605:( 3597:B 3587:A 3575:c 3565:a 3556:= 3549:B 3539:c 3518:) 3515:B 3512:c 3509:A 3506:b 3503:( 3495:A 3485:B 3473:c 3463:b 3454:= 3447:A 3437:c 3416:) 3413:c 3410:A 3407:b 3404:C 3401:( 3393:A 3383:C 3371:b 3361:c 3352:= 3345:A 3335:b 3314:) 3311:A 3308:b 3305:C 3302:a 3299:( 3290:C 3280:A 3268:b 3258:a 3249:= 3242:C 3232:b 3193:, 3188:) 3157:( 3146:) 3115:( 3101:) 3070:( 3059:) 3028:( 3019:= 3014:) 2983:( 2972:) 2941:( 2925:b 2921:B 2917:a 2913:C 2868:. 2865:c 2855:B 2845:A 2836:+ 2833:B 2823:A 2811:= 2804:C 2791:, 2788:b 2778:A 2768:C 2759:+ 2756:A 2746:C 2734:= 2727:B 2714:, 2711:a 2701:C 2691:B 2682:+ 2679:C 2669:B 2657:= 2650:A 2629:A 2625:π 2620:a 2615:a 2611:π 2606:A 2580:B 2576:a 2572:A 2568:b 2561:B 2557:a 2553:c 2547:) 2543:→ 2534:→ 2525:→ 2515:→ 2508:z 2503:A 2499:c 2495:b 2489:) 2485:→ 2476:→ 2467:→ 2437:c 2433:b 2429:a 2411:. 2405:c 2396:b 2387:a 2376:c 2367:b 2358:a 2349:2 2346:+ 2343:c 2337:2 2326:b 2320:2 2309:a 2303:2 2292:1 2286:= 2276:a 2265:A 2246:c 2240:2 2231:b 2225:2 2213:2 2209:) 2205:c 2196:b 2184:a 2175:( 2169:) 2166:c 2158:2 2147:1 2144:( 2141:) 2138:b 2130:2 2119:1 2116:( 2110:= 2098:2 2093:) 2087:c 2078:b 2067:c 2058:b 2046:a 2034:( 2026:1 2023:= 2016:A 2008:2 1988:A 1972:A 1964:2 1953:1 1950:= 1947:A 1939:2 1898:. 1892:c 1883:b 1872:c 1863:b 1851:a 1839:= 1836:A 1810:. 1807:A 1798:c 1789:b 1780:+ 1777:c 1768:b 1759:= 1756:a 1730:. 1727:b 1718:c 1709:+ 1706:A 1697:b 1688:c 1679:= 1669:C 1666:O 1649:B 1646:O 1630:→ 1624:→ 1601:. 1598:) 1595:b 1585:, 1582:A 1573:b 1563:, 1560:A 1551:b 1542:( 1534:: 1524:C 1521:O 1510:) 1507:c 1497:, 1494:0 1490:, 1487:c 1478:( 1470:: 1460:B 1457:O 1446:) 1443:1 1439:, 1436:0 1432:, 1429:0 1426:( 1418:: 1408:A 1405:O 1385:A 1381:x 1363:→ 1356:z 1352:c 1342:→ 1335:z 1329:→ 1321:a 1315:→ 1307:→ 1298:a 1285:→ 1276:→ 1267:→ 1212:. 1206:c 1195:C 1183:= 1177:b 1166:B 1154:= 1148:a 1137:A 1086:0 1078:2 1074:) 1070:b 1058:a 1049:( 1029:a 1023:a 997:0 991:c 988:, 985:b 982:, 979:a 949:. 946:C 937:b 928:a 919:+ 916:b 907:a 898:= 891:c 878:, 875:B 866:a 857:c 848:+ 845:a 836:c 827:= 820:b 807:, 804:A 795:c 786:b 777:+ 774:c 765:b 756:= 749:a 706:△ 688:. 685:C 676:= 665:c 659:, 656:B 647:= 636:b 630:, 627:A 618:= 607:a 599:, 596:c 587:= 576:C 569:, 566:b 557:= 546:B 539:, 536:a 527:= 516:A 498:△ 491:△ 472:A 468:A 457:△ 443:△ 428:. 426:R 422:c 418:b 414:a 410:R 406:R 386:2 380:c 377:+ 374:b 371:+ 368:a 362:0 352:π 339:π 318:3 312:C 309:+ 306:B 303:+ 300:A 284:π 265:A 258:C 254:B 250:A 240:c 236:b 232:a 227:. 225:C 221:B 217:A 20:)

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Knowledge

Spherical trigonometry

Index