Editing Spherical trigonometry

{{Short description|Geometry of figures on the surface of a sphere}}
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{{Use British English|date=March 2018}}
[[File:Triangle trirectangle.png|thumb|The [[octant of a sphere]] is a spherical triangle with three right angles.]]

'''Spherical trigonometry''' is the branch of [[spherical geometry]] that deals with the metrical relationships between the [[edge (geometry)|sides]] and [[angle]]s of '''spherical triangles''', traditionally expressed using [[trigonometric function]]s. On the [[sphere]], [[geodesics]] are [[great circle]]s. Spherical trigonometry is of great importance for calculations in [[astronomy]], [[geodesy]], and [[navigation]].

The origins of spherical trigonometry in [[Greek mathematics]] and the major developments in Islamic mathematics are discussed fully in [[History of trigonometry]] and [[Mathematics in medieval Islam]]. The subject came to fruition in Early Modern times with important developments by [[John Napier]], [[Jean Baptiste Joseph Delambre|Delambre]] and others, and attained an essentially complete form by the end of the nineteenth century with the publication of Todhunter's textbook ''Spherical trigonometry for the use of colleges and Schools''.<ref name=todhunter>{{cite book
|last = Todhunter
|first = I.
|author-link = Isaac Todhunter
|title = Spherical Trigonometry
|year = 1886
|publisher = MacMillan
|edition = 5th
|url = http://www.gutenberg.org/ebooks/19770
|access-date = 2013-07-28
|archive-date = 2020-04-14
|archive-url = https://web.archive.org/web/20200414233849/http://www.gutenberg.org/ebooks/19770
|url-status = live
}}</ref>
Since then, significant developments have been the application of vector methods, quaternion methods, and the use of numerical methods.
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(Todhunter<ref name=todhunter/>,Art.33) -->
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==Preliminaries==
[[File:Spherical trigonometry Intersecting circles.svg|right|thumb|200px|Eight spherical triangles defined by the intersection of three great circles.]]

===Spherical polygons===
A '''spherical polygon''' is a ''[[polygon]]'' on the surface of the sphere. Its sides are [[Circular arc|arc]]s of [[great circle]]s—the spherical geometry equivalent of [[line segment]]s in [[Euclidean geometry|plane geometry]].

Such polygons may have any number of sides greater than 1. Two-sided spherical polygons—''[[spherical lune|lune]]s'', also called ''[[digon]]s'' or ''bi-angles''—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.

One spherical polygon with interesting properties is the [[pentagramma mirificum]], a 5-sided spherical [[star polygon]] with a right angle at every vertex.

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

===Notation===
[[File:Spherical trigonometry basic triangle.svg|thumb|right|200px|The basic triangle on a unit sphere.]]
*Both vertices and angles at the vertices of a triangle are denoted by the same upper case letters {{mvar|A}}, {{mvar|B}}, and {{mvar|C}}.
*Sides are denoted by lower-case letters: {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}. The sphere has a radius of 1, and so the side lengths and lower case angles are equivalent (see [[arc length]]). 
*The ''angle'' {{mvar|A}} (respectively, {{mvar|B}} and {{mvar|C}}) may be regarded either as the [[dihedral angle]] between the two planes that intersect the sphere at the ''[[vertex (geometry)|vertex]]'' {{mvar|A}}, or, equivalently, as the angle between the [[tangent]]s of the great circle arcs where they meet at the vertex.
*Angles are expressed in [[radian]]s. The angles of ''proper'' spherical triangles are (by convention) less than {{pi}}, so that <math display=block>
\pi < A + B + C < 3\pi
</math>(Todhunter,<ref name=todhunter/> Art.22,32). 
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 {{pi}} radians.
*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 ''proper'' spherical triangles are less than {{pi}}, so that <math display=block>0 < a + b + c < 2\pi
</math>(Todhunter,<ref name=todhunter/> Art.22,32).
*The sphere's radius is taken as unity. For specific practical problems on a sphere of radius {{mvar|R}} the measured lengths of the sides must be divided by {{mvar|R}} before using the identities given below. Likewise, after a calculation on the unit sphere the sides {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} must be multiplied by&nbsp;{{mvar|R}}.

===Polar triangles===
[[File:Spherical trigonometry polar triangle.svg|right|thumb|200px|The polar triangle {{math|△''A'B'C' ''}}]]
The '''polar triangle''' associated with a triangle {{math|△''ABC''}} is defined as follows. Consider the great circle that contains the side&nbsp;{{mvar|BC}}. 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 {{mvar|A}} is (conventionally) termed the pole of {{mvar|A}} and it is denoted by {{mvar|A'}}. The points {{mvar|B'}} and {{mvar|C'}} are defined similarly.

The triangle {{math|△''A'B'C' ''}} is the polar triangle corresponding to triangle&nbsp;{{math|△''ABC''}}. The angles and sides of the polar triangle are
given by (Todhunter,<ref name=todhunter/> Art.27) 
<math display=block>\begin{alignat}{3}
  A' &= \pi - a, &\qquad B' &= \pi - b , &\qquad C' &= \pi - c, \\
  a' &= \pi - A, & b' &= \pi - B , & c' &= \pi - C .
\end{alignat}</math>
Therefore, if any identity is proved for {{math|△''ABC''}} 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 for a right-angled triangle. The polar triangle of a polar triangle is the original triangle.

If the {{math|3 × 3}} matrix {{mvar|M}} has the positions {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} as its columns then the rows of the matrix inverse {{math|''M''{{isup|−1}}}}, if normalized to unit length, are the positions {{mvar|A′}}, {{mvar|B′}}, and {{mvar|C′}}.  In particular, when {{math|△''A′B′C′''}} is the polar triangle of {{math|△''ABC''}} then {{math|△''ABC''}} is the polar triangle of {{math|△''A′B′C′''}}.

==Cosine rules and sine rules==

===Cosine rules===
{{main|Spherical law of cosines}}

The cosine rule is the fundamental identity of spherical trigonometry: all other identities, including the sine rule, may be derived from the cosine rule:
<math display=block>\begin{align}
  \cos a &= \cos b \cos c + \sin b \sin c \cos A, \\[2pt]
  \cos b &= \cos c \cos a + \sin c \sin a \cos B, \\[2pt]
  \cos c &= \cos a \cos b + \sin a \sin b \cos C.
\end{align}</math>

These identities generalize the cosine rule of plane [[trigonometry]], to which they are asymptotically equivalent
in the limit of small interior angles. (On the unit sphere, if <math>a, b, c \rightarrow 0</math>  set <math> \sin a \approx a </math> and <math> \cos a \approx 1 - \frac{a^2}{2} </math> etc.; see [[Spherical law of cosines]].)

===Sine rules===
{{main|Spherical law of sines}}

The spherical [[Law of sines#Curvature|law of sines]] is given by the formula
<math display=block>\frac{\sin A}{\sin a} = \frac{\sin B}{\sin b} = \frac{\sin C}{\sin c}.</math>
These identities approximate the sine rule of plane [[trigonometry]] when the sides are much smaller than the radius of the sphere.

===Derivation of the cosine rule ===
{{main|Spherical law of cosines}}

[[File:Spherical trigonometry vectors.svg|thumb|right|200px]]
The spherical cosine formulae were originally proved by elementary geometry and the planar cosine rule (Todhunter,<ref name=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 [[Euclidean vector|vector]] methods. (These methods are also discussed at [[Spherical law of cosines]].)

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Consider three unit vectors {{math|''{{vec|OA}}'', ''{{vec|OB}}'', ''{{vec|OC}}''}} drawn from the origin to the vertices of the triangle (on the unit sphere). The arc {{mvar|{{overarc|BC}}}} subtends an angle of magnitude {{mvar|a}} at the centre and therefore {{math|1={{vec|''OB''}} · ''{{vec|OC}}'' = cos ''a''}}. Introduce a Cartesian basis with {{mvar|{{vec|OA}}}} along the {{mvar|z}}-axis and {{mvar|{{vec|OB}}}} in the {{mvar|xz}}-plane making an angle {{mvar|c}} with the {{mvar|z}}-axis. The vector {{mvar|{{vec|OC}}}} projects to {{mvar|ON}} in the {{mvar|xy}}-plane and the angle between {{mvar|ON}} and the {{mvar|x}}-axis is {{mvar|A}}. Therefore, the three vectors have components:

<math display=block>\begin{align}
  \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{align}</math>

The scalar product {{mvar|{{vec|OB}} · {{vec|OC}}}} in terms of the components is
<math display=block>\vec{OB} \cdot \vec{OC} =\sin c \sin b \cos A  +  \cos c \cos b.</math>
Equating the two expressions for the scalar product gives
<math display=block>\cos a  =  \cos b \cos c  +  \sin b \sin c \cos A.</math>
This equation can be re-arranged to give explicit expressions for the angle in terms of the sides:
<math display=block>\cos A = \frac{\cos a-\cos b\cos c}{\sin b \sin c}.</math>

The other cosine rules are obtained by cyclic permutations.

===Derivation of the sine rule ===
{{main|Spherical law of sines}}

This derivation is given in Todhunter,<ref name=todhunter/> (Art.40). From the identity <math>\sin^2 A=1-\cos^2 A</math> and the explicit expression for {{math|cos ''A''}} given immediately above
<math display=block>
\begin{align}
   \sin^2 A &= 1 - \left(\frac{\cos a - \cos b \cos c}{\sin b \sin c}\right)^2 \\[5pt]
   &= \frac{(1-\cos^2 b)(1-\cos^2 c)-(\cos a  - \cos b\cos c)^2}{\sin^2\!b \,\sin^2\!c} \\[5pt]
 \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{align}</math>
Since the right hand side is invariant under a cyclic permutation of {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} the spherical sine rule follows immediately.

===Alternative derivations===
There are many ways of deriving the fundamental cosine and sine rules and the other rules developed in the following sections. For example, Todhunter<ref name=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 [[Spherical law of cosines]] gives four different proofs of the cosine rule. Text books on geodesy<ref>{{cite book|year=1880|last=Clarke|first=Alexander Ross|title=Geodesy|url=https://archive.org/details/in.ernet.dli.2015.42772|publisher=Clarendon Press|location=Oxford|oclc=2484948|via=the [[Internet Archive]]}}</ref> and spherical astronomy<ref>{{cite book |year=1977|last=Smart|first=W.M.|title=Text-Book on Spherical Astronomy|publisher=Cambridge University Press|edition=6th|url=https://archive.org/details/textbookonspheri0000smar|at=Chapter 1|via=the [[Internet Archive]]}}</ref> give different proofs and the online resources of [[MathWorld]] provide yet more.<ref>{{MathWorld|title=Spherical Trigonometry|id=SphericalTrigonometry|access-date=8 April 2018}}</ref> There are even more exotic derivations, such as that of Banerjee<ref name="banerjee">{{Citation | last = Banerjee | first = Sudipto | date = 2004 | title = Revisiting Spherical Trigonometry with Orthogonal Projectors | journal = The College Mathematics Journal | volume = 35 | issue = 5 | pages = 375–381 | publisher = Mathematical Association of America | url = https://www.researchgate.net/publication/228849546 | doi = 10.1080/07468342.2004.11922099 | jstor = 4146847 | s2cid = 122277398 | access-date = 2016-01-10 | archive-date = 2020-07-22 | archive-url = https://web.archive.org/web/20200722071405/https://www.researchgate.net/publication/228849546_Revisiting_Spherical_Trigonometry_with_Orthogonal_Projectors | url-status = live }}</ref> who derives the formulae using the linear algebra of projection matrices and also quotes methods in [[differential geometry]] and the group theory of rotations.

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.  The [[scalar triple product]], {{math|''{{vec|OA}}'' · (''{{vec|OB}}'' × ''{{vec|OC}}'')}} evaluates to {{math|sin ''b'' sin ''c'' sin ''A''}} in the basis shown. Similarly, in a basis oriented with  the {{mvar|z}}-axis along {{mvar|{{vec|OB}}}}, the triple product {{math|''{{vec|OB}}'' · (''{{vec|OC}}'' × ''{{vec|OA}}'')}}, evaluates to {{math|sin ''c'' sin ''a'' sin ''B''}}. Therefore, the invariance of the triple product under cyclic permutations gives {{math|1=sin ''b'' sin ''A'' = sin ''a'' sin ''B''}} which is the first of the sine rules. See curved variations of the [[law of sines]] to see details of this derivation.

=== Differential variations ===
When any three of the differentials ''da'', ''db'', ''dc'', ''dA'', ''dB'', ''dC'' are known, the following equations, which are found by differentiating the cosine rule and using the sine rule, can be used to calculate the other three by elimination:<ref>{{cite book |title=A Treatise on Plane and Spherical Trigonometry |author1=William Chauvenet |edition=9th |publisher=J.B. Lippincott Company |year=1887 |isbn=978-3-382-17783-6 |page=240 |url=https://books.google.com/books?id=nWu4EAAAQBAJ}}</ref>

<math display=block>\begin{align}
 da = \cos C \ db + \cos B \ dc + \sin b \ \sin C \ dA, \\
 db = \cos A \ dc + \cos C \ da + \sin c \ \sin A \ dB, \\
 dc = \cos B \ da + \cos A \ db + \sin a \ \sin B \ dC. \\
\end{align}</math>

==Identities==

===Supplemental cosine rules===
Applying the cosine rules to the polar triangle gives (Todhunter,<ref name=todhunter/> Art.47), ''i.e.'' replacing {{mvar|A}} by {{math|{{pi}} – ''a''}}, {{mvar|a}} by {{math|{{pi}} – ''A''}} etc.,
<math display=block>\begin{align}
\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{align}</math>

===Cotangent four-part formulae===
The six parts of a triangle may be written in cyclic order as ({{mvar|aCbAcB}}). The cotangent, or four-part, formulae relate two sides and two angles forming four ''consecutive'' parts around the triangle, for example ({{mvar|aCbA}}) or {{mvar|BaCb}}). In such a set there are inner and outer parts: for example in the set ({{mvar|BaCb}}) the inner angle is {{mvar|C}}, the inner side is {{mvar|a}}, the outer angle is {{mvar|B}}, the outer side is {{mvar|b}}. The cotangent rule may be written as (Todhunter,<ref name=todhunter/> Art.44)
<math display=block>
  \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),
</math>
and the six possible equations are (with the relevant set shown at right):
<math display=block>\begin{alignat}{5}
  \text{(CT1)}&& \qquad \cos b\,\cos C &= \cot a\,\sin b - \cot A \,\sin C \qquad&&(aCbA)\\[0ex]
  \text{(CT2)}&& \cos b\,\cos A &= \cot c\,\sin b - \cot C \,\sin A &&(CbAc)\\[0ex]
  \text{(CT3)}&& \cos c\,\cos A &= \cot b\,\sin c - \cot B \,\sin A &&(bAcB)\\[0ex]
  \text{(CT4)}&& \cos c\,\cos B &= \cot a\,\sin c - \cot A \,\sin B &&(AcBa)\\[0ex]
  \text{(CT5)}&& \cos a\,\cos B &= \cot c\,\sin a - \cot C \,\sin B &&(cBaC)\\[0ex]
  \text{(CT6)}&& \cos a\,\cos C &= \cot b\,\sin a - \cot B \,\sin C &&(BaCb)
\end{alignat}</math>
To prove the first formula start from the first cosine rule and on the right-hand side substitute for {{math|cos ''c''}} from the third cosine rule:
<math display=block>\begin{align}
  \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{align}</math>
The result follows on dividing by {{math|sin ''a'' sin ''b''}}. 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.

===Half-angle and half-side formulae===
With <math>2s=(a+b+c)</math> and <math>2S=(A+B+C),</math>
<math display="block">
\begin{alignat}{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}} \\[2ex]
 \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}} \\[2ex]
  \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{alignat}
</math>

Another twelve identities follow by cyclic permutation.

The proof (Todhunter,<ref name=todhunter/> Art.49) of the first formula starts from the identity <math>2\sin^2\!\tfrac{A}{2} = 1 - \cos A,</math> using the cosine rule to express {{mvar|A}} in terms of the sides and replacing the sum of two cosines by a product. (See [[List of trigonometric identities#Product-to-sum and sum-to-product identities|sum-to-product identities]].) The second formula starts from the identity <math>2\cos^2\!\tfrac{A}{2} = 1 + \cos A,</math> the third is a quotient and the remainder follow by applying the results to the polar triangle.

===Delambre analogies===
The Delambre analogies (also called Gauss analogies) were published independently by Delambre, Gauss, and Mollweide in 1807–1809.<ref>{{cite journal |last=Todhunter |first=Isaac |year=1873 |title=Note on the history of certain formulæ in spherical trigonometry |journal=The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science| volume=45 |issue=298 |pages=98–100|doi=10.1080/14786447308640820 }}</ref>

<math display="block">
\begin{align}
 \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}
\\[2ex]
 \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{align}
</math>
Another eight identities follow by cyclic permutation.

Proved by expanding the numerators and using the half angle formulae. (Todhunter,<ref name=todhunter/> Art.54 and Delambre<ref>{{cite book
|last = Delambre
|first = J. B. J.
|author-link = Delambre
|title = Connaissance des Tems 1809
|year = 1807
|page = 445
|url = https://books.google.com/books?id=M8Mi6hU5tR0C&pg=PA445
|access-date = 2016-05-14
|archive-date = 2020-07-22
|archive-url = https://web.archive.org/web/20200722071420/https://books.google.com/books?id=M8Mi6hU5tR0C&pg=PA445
|url-status = live
}}</ref>)

===Napier's analogies===

<math display="block">\begin{align}
  \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
\\[2ex]
  \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{align}</math>

Another eight identities follow by cyclic permutation.

These identities follow by division of the Delambre formulae. (Todhunter,<ref name=todhunter/> Art.52)

Taking quotients of these yields the [[Law of tangents#Spherical version|law of tangents]], first stated by [[Mathematics in medieval Islam|Persian mathematician]] [[Nasir al-Din al-Tusi]] (1201–1274),

<math display=block>
\frac{\tan\tfrac12(A-B)}{\tan\tfrac12(A+B)}
= \frac{\tan\tfrac12(a-b)}{\tan\tfrac12(a+b)} 
</math>

===Napier's rules for right spherical triangles===
[[File:Spherical trigonometry Napier right-angled.svg|center|thumb|300px]]

When one of the angles, say {{math|C}}, of a spherical triangle is equal to {{pi}}/2 the various identities given above are considerably simplified. There are ten identities relating three elements chosen from the set {{mvar|a}}, {{mvar|b}}, {{mvar|c}}, {{mvar|A}}, and {{mvar|B}}.

[[John Napier|Napier]]<ref>{{cite book
|last = Napier
|first = J
|author-link = John Napier
|title = Mirifici Logarithmorum Canonis Constructio
|year = 1614
|page = 50
|url = https://books.google.com/books?id=VukHAQAAIAAJ
|access-date = 2016-05-14
|archive-date = 2013-04-30
|archive-url = https://web.archive.org/web/20130430105056/http://books.google.com/books?id=VukHAQAAIAAJ
|url-status = live
}}
An 1889 translation ''The Construction of the Wonderful Canon of Logarithms'' is available as en e-book from [https://www.abebooks.co.uk/servlet/SearchResults?tn=Construction+Wonderful+Canon+Logarithms Abe Books] {{Webarchive|url=https://web.archive.org/web/20200303190642/https://www.abebooks.co.uk/servlet/SearchResults%3Ftn%3DConstruction%2BWonderful%2BCanon%2BLogarithms |date=2020-03-03 }}</ref> provided an elegant [[mnemonic|mnemonic aid]] 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).

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 {{mvar|a}} gives {{mvar|aCbAcB}}. Next replace the parts that are not adjacent to {{mvar|C}} (that is {{mvar|A}}, {{mvar|c}}, and {{mvar|B}}) by their complements and then delete the angle {{mvar|C}} 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 ''middle'' part) will be adjacent to two parts and opposite the other two parts. The ten Napier's Rules are given by
*sine of the middle part = the product of the tangents of the adjacent parts
*sine of the middle part = the product of the cosines of the opposite parts
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 {{mvar|a}} we have:
<math display=block>\begin{align}
  \sin a &= \tan(\tfrac{\pi}{2} - B)\,\tan b  \\[2pt]
  &= \cos(\tfrac{\pi}{2} - c)\, \cos(\tfrac{\pi}{2} - A) \\[2pt]
  &= \cot B\,\tan b \\[4pt]
  &= \sin c\,\sin A. 
\end{align}</math>
The full set of rules for the right spherical triangle is (Todhunter,<ref name=todhunter/> Art.62)
<math display="block">\begin{alignat}{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{alignat}</math>

===Napier's rules for quadrantal triangles===
[[File:Spherical trigonometry Napier quadrantal 01.svg|center|thumb|300px|A quadrantal spherical triangle together with Napier's circle for use in his mnemonics]]

A quadrantal spherical triangle is defined  to be a spherical triangle in which one of the sides subtends an angle of {{pi}}/2 radians at the centre of the sphere: on the unit sphere the side has length {{pi}}/2. In the case that the side  {{mvar|c}} has length {{pi}}/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 {{math|△''A'B'C' ''}} with sides {{mvar|a', b', c'}} such that {{math|1=''A' ''= {{pi}} − ''a''}}, {{math|1=''a' ''= {{pi}} − ''A''}} etc. The results are:
<math display="block">\begin{alignat}{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{alignat}</math>

===Five-part rules===
Substituting the second cosine rule  into the first and simplifying gives:
<math display=block>\begin{align}
  \cos a &= (\cos a \,\cos c + \sin a \,\sin c \,\cos B) \cos c + \sin b \,\sin c \,\cos A \\[4pt]
  \cos a \,\sin^2 c &= \sin a \,\cos c  \,\sin c \,\cos B + \sin b \,\sin c \,\cos A
\end{align}</math>
Cancelling the factor of {{math|sin ''c''}} gives
<math display=block>\cos a \sin c = \sin a \,\cos c \,\cos B + \sin b \,\cos A</math>

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

===Cagnoli's Equation===
Multiplying the first cosine rule  by {{math|cos ''A''}}  gives
<math display=block>\cos a \cos A = \cos b \,\cos c \,\cos A + \sin b \,\sin c - \sin b \,\sin c \,\sin^2 A.</math>
Similarly multiplying the first supplementary cosine rule  by {{math|cos ''a''}}  yields
<math display=block>\cos a \cos A = -\cos B \,\cos C \,\cos a + \sin B \,\sin C - \sin B \,\sin C \,\sin^2 a.</math>
Subtracting the two and noting that it follows from the sine rules that <math>  \sin b \,\sin c \,\sin^2 A = \sin B \,\sin C \,\sin^2 a </math>  produces Cagnoli's equation
<math display=block>\sin b \,\sin c + \cos b \,\cos c \,\cos A = \sin B \,\sin C - \cos B \,\cos C \,\cos a</math>
which is a relation between the six parts of the spherical triangle.<ref>{{cite book
|last = Chauvenet
|first = William
|title = A Treatise on Plane and Spherical Trigonometry
|publisher = J. B. Lippincott & Co.
|location = Philadelphia
|year = 1867
|page = 165
|url = https://books.google.com/books?id=d8E8AAAAYAAJ
|access-date = 2021-07-11
|archive-date = 2021-07-11
|archive-url = https://web.archive.org/web/20210711194635/https://books.google.com/books?id=d8E8AAAAYAAJ
|url-status = live
}}</ref>

== Solution of triangles ==
{{main|Solution of triangles#Solving spherical triangles}}

===Oblique triangles===
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 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.
[[File:Spherical trigonometry triangle cases.svg|thumb|center|500px]]

*'''Case 1: three sides given (SSS).'''  The cosine rule may be used to give the angles {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} but, to avoid ambiguities, the half angle formulae are preferred.
*'''Case 2: two sides and an included angle given (SAS).''' The cosine rule gives {{mvar|a}} and then we are back to Case&nbsp;1.
*'''Case 3: two sides and an opposite angle given (SSA).''' The sine rule gives {{mvar|C}} and then we have Case&nbsp;7. There are either one or two solutions.
*'''Case 4: two angles and an included side given (ASA).''' The four-part cotangent formulae for sets ({{mvar|cBaC}}) and ({{mvar|BaCb}}) give {{mvar|c}} and {{mvar|b}}, then {{mvar|A}} follows from the sine rule.
*'''Case 5: two angles and an opposite side given (AAS).''' The sine rule gives {{mvar|b}} and then we have Case&nbsp;7 (rotated). There are either one or two solutions.
*'''Case 6: three angles given (AAA).''' The supplemental cosine rule may be used to give the sides {{mvar|a}}, {{mvar|b}}, and {{mvar|c}} but, to avoid ambiguities, the half-side formulae are preferred.
*'''Case 7: two angles and two opposite sides given (SSAA).''' Use Napier's analogies for {{mvar|a}} and {{mvar|A}}; or, use Case 3 (SSA) or case 5 (AAS).

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 half-angles will be less than {{pi}}/2 and therefore free from ambiguity. There is a full discussion in Todhunter. The article [[Solution of triangles#Solving spherical triangles]] presents variants on these methods with a slightly different notation.

There is a full discussion of the solution of oblique triangles in Todhunter.<ref name=todhunter/>{{rp|Chap. VI}} See also the discussion in Ross.<ref>Ross, Debra Anne. ''Master Math: Trigonometry'', Career Press, 2002.</ref> [[Nasir al-Din al-Tusi]] was the first to list the six distinct cases (2–7 in the diagram) of a right triangle in spherical trigonometry.<ref>{{MacTutor|id=Al-Tusi_Nasir|title=Nasir al-Din al-Tusi}} "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"</ref>

[[File:Spherical trigonometry solution construction.svg|thumb|100px]]

===Solution by right-angled triangles===
Another approach is to split the triangle into two right-angled triangles. For example, take the Case&nbsp;3 example where {{mvar|b}}, {{mvar|c}}, and {{mvar|B}} are given. Construct the great circle from {{mvar|A}} that is normal to the side {{mvar|BC}} at the point {{mvar|D}}. Use Napier's rules to solve the triangle {{math|△''ABD''}}: use {{mvar|c}} and {{mvar|B}} to find the sides {{mvar|AD}} and {{mvar|BD}} and the angle {{math|∠''BAD''}}. Then use Napier's rules to solve the triangle {{math|△''ACD''}}: that is use {{mvar|AD}} and {{mvar|b}} to find the side {{mvar|DC}} and the angles {{mvar|C}} and {{math|∠''DAC''}}. The angle {{mvar|A}} and side {{mvar|a}} follow by addition.

===Numerical considerations===
Not all of the rules obtained are numerically robust in extreme examples, for example when an angle approaches zero or&nbsp;{{pi}}. Problems and solutions may have to be examined carefully, particularly when writing code to solve an arbitrary triangle.

==Area and spherical excess{{anchor|Area|Excess}}==
{{see also|Solid angle|Geodesic polygon}}
[[File:Lexell's theorem.png|thumb|[[Lexell's theorem]]: the triangles of constant area on a fixed base {{mvar|AB}} have their free vertex {{mvar|C}} along a [[Spherical circle|small circle]] through the points antipodal to {{mvar|A}} and {{mvar|B}}.]]

Consider an  {{mvar|N}}-sided spherical polygon and let {{mvar|A<sub>n</sub>}} denote the {{mvar|n}}-th interior angle. The area of such a polygon is given by (Todhunter,<ref name=todhunter/> Art.99)
<math display=block>{\text{Area of polygon} \atop \text{(on the unit sphere)}} \equiv E_N = \left(\sum_{n=1}^{N} A_{n}\right) - (N-2)\pi.</math>

{{anchor|Girard's theorem}}
For the case of a spherical triangle with angles {{mvar|A}}, {{mvar|B}}, and {{mvar|C}} this reduces to '''Girard's theorem'''
<math display=block> {\text{Area of triangle} \atop \text{(on the unit sphere)}} \equiv E = E_3 = A+B+C -\pi,</math>
where {{mvar|E}} is the amount by which the sum of the angles exceeds {{pi}} radians, called the '''spherical excess''' of the triangle. This theorem is named after its author, [[Albert Girard]].<ref>Another proof of Girard's theorem may be found at [http://math.rice.edu/~pcmi/sphere/gos4.html] {{Webarchive|url=https://web.archive.org/web/20121031004912/http://math.rice.edu/~pcmi/sphere/gos4.html|date=2012-10-31}}.</ref> An earlier proof was derived, but not published, by the English mathematician [[Thomas Harriot]]. On a sphere of radius {{mvar|R}} both of the above area expressions are multiplied by {{math|''R''<sup>2</sup>}}. The definition of the excess is independent of the radius of the sphere.

The converse result may be written as

<math display=block> A+B+C = \pi + \frac{4\pi \times \text{Area of triangle}}{\text{Area of the sphere}}.</math>

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{{pi}} (3{{pi}} for ''proper'' angles). For example,
an octant of a sphere is a spherical triangle with three right angles, so that the excess is {{pi}}/2. In practical applications it ''is'' often small: for example the triangles of geodetic survey typically have a spherical excess much less than 1' of arc.<ref name=clarke>This follows from [[Legendre's theorem on spherical triangles]] whenever the area of the triangle is small relative to the surface area of the entire Earth; see {{cite book
 |last        = Clarke
 |first       = Alexander Ross
 |author-link = Clarke
 |title       = Geodesy
 |year        = 1880
 |url         = https://archive.org/details/in.ernet.dli.2015.42772
 |publisher   = Clarendon Press
}}
(Chapters 2 and 9).</ref> On the Earth the excess of an equilateral triangle with sides 21.3&nbsp;km (and area 393&nbsp;km<sup>2</sup>) is approximately 1&nbsp;arc second.

There are many formulae for the excess. For example, Todhunter,<ref name=todhunter/> (Art.101—103) gives ten examples including that of [[Simon Antoine Jean L'Huilier|L'Huilier]]:
<math display=block>\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)}</math>
where <math>s = \tfrac{1}{2}(a+b+c)</math>. This formula is reminiscent of [[Heron's formula]] for planar triangles.

Because some triangles are badly characterized by
their edges (e.g., if <math display=inline>a = b \approx \frac12c</math>), it is often better to use
the formula for the excess in terms of two edges and their included angle
<math display=block>\tan\tfrac12 E = \frac
{\tan\frac12a\tan\frac12b\sin C}{1 + \tan\frac12a\tan\frac12b\cos C}.</math>

When triangle {{math|△''ABC''}} is a right triangle with right angle at {{mvar|C}}, then {{math|1=cos ''C'' = 0}} and {{math|1=sin ''C'' = 1}}, so this reduces to
<math display=block>\tan\tfrac12 E = \tan\tfrac12a\tan\tfrac12b.</math>

[[Angular defect|Angle deficit]] is defined similarly for [[hyperbolic geometry]].

=== From latitude and longitude ===

The spherical excess of a spherical quadrangle bounded by the equator, the two meridians of longitudes <math>\lambda_1</math> and  <math>\lambda_2,</math> and the great-circle arc between two points with longitude and latitude <math>(\lambda_1, \varphi_1)</math> and <math>(\lambda_2, \varphi_2)</math> is
<math display="block">
\tan\tfrac12 E_4
= \frac {\sin\tfrac12(\varphi_2 + \varphi_1)}{\cos\tfrac12(\varphi_2 - \varphi_1)}
\tan\tfrac12(\lambda_2 - \lambda_1).
</math>

This result is obtained from one of Napier's analogies. In the limit where <math>\varphi_1, \varphi_2, \lambda_2 - \lambda_1</math> are all small, this reduces to the familiar trapezoidal area, <math display=inline>E_4 \approx \frac12 (\varphi_2 + \varphi_1) (\lambda_2 - \lambda_1)</math>.

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,<ref>{{cite conference |last1=Chamberlain |first1=Robert G. |last2=Duquette |first2=William H. |title=Some algorithms for polygons on a sphere. |date=17 April 2007 |url=https://trs.jpl.nasa.gov/handle/2014/41271 |conference=Association of American Geographers Annual Meeting |publisher=NASA JPL |access-date=7 August 2020 |archive-date=22 July 2020 |archive-url=https://web.archive.org/web/20200722072320/https://trs.jpl.nasa.gov/handle/2014/41271 |url-status=live }}</ref> by a [[line integral]] with [[Green's theorem]],<ref>{{cite web |title=Surface area of polygon on sphere or ellipsoid – MATLAB areaint |url=https://www.mathworks.com/help/map/ref/areaint.html |website=www.mathworks.com |access-date=2021-05-01 |archive-date=2021-05-01 |archive-url=https://web.archive.org/web/20210501073522/https://www.mathworks.com/help/map/ref/areaint.html |url-status=live }}</ref> or via an [[equal-area projection]] as commonly done in GIS. The other algorithms can still be used with the side lengths calculated using a [[great-circle distance]] formula.

==See also==
*[[Air navigation]]
*[[Celestial navigation]]
*[[Ellipsoidal trigonometry]]
*[[Great-circle distance]] or spherical distance
*[[Lenart sphere]]
*[[Schwarz triangle]]
*[[Spherical geometry]]
*[[Spherical polyhedron]]
*[[Triangulation (surveying)]]

==References==
{{Reflist}}

==External links==
* {{MathWorld |title=Spherical Trigonometry |id=SphericalTrigonometry}} a more thorough list of identities, with some derivation
* {{MathWorld |title=Spherical Triangle |id=SphericalTriangle}} a more thorough list of identities, with some derivation
* [http://gnomonique.fr/trisph/index_en.htm TriSph] A free software to solve the spherical triangles, configurable to different practical applications and configured for gnomonic
* [https://www.researchgate.net/publication/228849546_Revisiting_Spherical_Trigonometry_with_Orthogonal_Projectors "Revisiting Spherical Trigonometry with Orthogonal Projectors"] by Sudipto Banerjee. The paper derives the spherical law of cosines and law of sines using elementary linear algebra and projection matrices.
* {{WolframDemonstrations|title=A Visual Proof of Girard's Theorem|urlname=AVisualProofOfGirardsTheorem}} by Okay Arik
* [https://www.wdl.org/en/item/2856/ "The Book of Instruction on Deviant Planes and Simple Planes"], a manuscript in Arabic that dates back to 1740 and talks about spherical trigonometry, with diagrams
* [https://trs.jpl.nasa.gov/handle/2014/41271 Some Algorithms for Polygons on a Sphere] 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.
* [http://www.in-dubio-pro-geo.de/?file=plasph/stri0&english=1 Online computation of spherical triangles]
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[[Category:Spherical trigonometry| ]]