Spherical Trigonometry

Define a Spherical Triangle on the surface of a unit Sphere, centered at a point $O$, with vertices $A$, $B$, and $C$. Define Angles $a\equiv\angle BOC$, $b\equiv\angle COA$, and $c\equiv\angle AOB$. Let the Angle between Planes $AOB$ and $AOC$ be $A$, the Angle between Planes $BOC$ and $AOB$ be $B$, and the Angle between Planes $BOC$ and $AOC$ be $C$. Define the Vectors

$${\bf a} \equiv \overrightarrow{OA}$$ (1)

$${\bf b} \equiv \overrightarrow{OB}$$ (2)

$${\bf c} \equiv \overrightarrow{OC}.$$ (3)

Then

$$(\hat {\bf a}\times \hat {\bf b})\cdot (\hat {\bf a}\times \hat {\bf c}) = (\vert\hat {\bf a}\vert\, \vert\hat {\bf b}\vert\sin c)(\vert\hat {\bf a}\vert \,\vert\hat {\bf c}\vert\sin b)\cos A$$

$$= \sin b\sin c\cos A.$$ (4)

Equivalently,

$$(\hat {\bf a}\times \hat {\bf b})\cdot (\hat {\bf a}\times \hat {\bf c}) = \hat {\bf a}\cdot [\hat {\bf b}\times (\hat {\bf a}\times \hat {\bf c})]$$

$$= \hat {\bf a}\cdot [\hat {\bf a}(\hat {\bf b}\cdot \hat {\bf c}) -\hat {\bf c}(\hat {\bf a}\cdot \hat {\bf b})]$$

$$= (\hat {\bf b}\cdot \hat {\bf c}) -(\hat {\bf a}\cdot \hat {\bf c})(\hat {\bf a}\cdot \hat {\bf b})$$

$$= \cos a-\cos c\cos b.$$ (5)

Since these two expressions are equal, we obtain the identity

$$\cos a = \cos b\cos c+\sin b\sin c\cos A$$ (6)

The identity

$$\sin A = {\vert(\hat {\bf a}\times \hat {\bf b})\times (\hat {\bf a}\times... ...}]+\hat {\bf b}[\hat {\bf a},\hat {\bf a},\hat {\bf c}]\vert\over \sin b\sin c}$$

$$= {[\hat {\bf a},\hat {\bf b},\hat {\bf c}]\over \sin b\sin c},$$ (7)

where $[{\bf a}, {\bf b}, {\bf c}]$ is the Scalar Triple Product, gives a spherical analog of the Law of Sines,

$${\sin A\over \sin a} = {\sin B\over \sin b} = {\sin C\over \sin c} = {6\mathop{\rm Vol}(OABC)\over \sin a\sin b\sin c},$$ (8)

where $\mathop{\rm Vol}(OABC)$ is the Volume of the Tetrahedron. From (7) and (8), it follows that

$$\sin a\cos B = \cos b\sin c-\sin b\cos c\cos A$$ (9)

$$\cos a\cos C = \sin a\cot b-\sin C\cot B.$$ (10)

These are the fundamental equalities of spherical trigonometry.

There are also spherical analogs of the Law of Cosines for the sides of a spherical triangle,

$$\cos a = \cos b\cos c+\sin b\sin c\cos A$$ (11)

$$\cos b = \cos c\cos a+\sin c\sin a\cos B$$ (12)

$$\cos c = \cos a\cos b+\sin a\sin b\cos C,$$ (13)

and the angles of a spherical triangle,

$$\cos A = -\cos B\cos C+\sin B\sin C\cos a$$ (14)

$$\cos B = -\cos C\cos A+\sin C\sin A\cos b$$ (15)

$$\cos C = -\cos A\cos B+\sin A\sin B\cos c$$ (16)

(Beyer 1987), as well as the Law of Tangents

$${\tan[{\textstyle{1\over 2}}(a-b)]\over\tan[{\textstyle{1\ov... ...style{1\over 2}}(A-B)]\over\tan[{\textstyle{1\over 2}}(A+B)]}.$$ (17)

Let

$$s \equiv {\textstyle{1\over 2}}(a+b+c)$$ (18)

$$S \equiv {\textstyle{1\over 2}}(A+B+C),$$ (19)

then the half-angle formulas are

$$\tan({\textstyle{1\over 2}}A) = {k\over\sin(s-a)}$$ (20)

$$\tan({\textstyle{1\over 2}}B) = {k\over\sin(s-b)}$$ (21)

$$\tan({\textstyle{1\over 2}}C) = {k\over\sin(s-c)},$$ (22)

where

$$k^2={\sin(s-a)\sin(s-b)\sin(s-c)\over\sin s}=\tan^2 r,$$ (23)

and the half-side formulas are

$$\tan({\textstyle{1\over 2}}a) = K\cos(S-A)$$ (24)

$$\tan({\textstyle{1\over 2}}b) = K\cos(S-B)$$ (25)

$$\tan({\textstyle{1\over 2}}c) = K\cos(S-C),$$ (26)

where

$$K^2=-{\cos S\over\cos(S-A)\cos(S-B)\cos(S-C)}=\tan^2 r,$$ (27)

where $r$ is the Radius of the Sphere on which the spherical triangle lies.

Additional formulas include the Haversine formulas

$$\mathop{\rm hav}\nolimits a = \mathop{\rm hav}\nolimits (b-c)+\sin b\sin c\sin(s-c)$$ (28)

$$\mathop{\rm hav}\nolimits A = {\sin(s-b)\sin(s-c)\over\sin b\sin c}$$ (29)

$$= {\mathop{\rm hav}\nolimits a-\mathop{\rm hav}\nolimits (b-c)\over\sin b\sin c}$$ (30)

$$= \mathop{\rm hav}\nolimits [\pi-(B+C)]+\sin B\sin C\mathop{\rm hav}\nolimits a,$$ (31)

Gauss's Formulas

$${\sin[{\textstyle{1\over 2}}(a-b)]\over\sin({\textstyle{1\over 2}}c)} = {\sin[{\textstyle{1\over 2}}(A-B)]\over\cos({\textstyle{1\over 2}}C)}$$ (32)

$${\sin[{\textstyle{1\over 2}}(a+b)]\over\sin({\textstyle{1\over 2}}c)} = {\cos[{\textstyle{1\over 2}}(A-B)]\over\sin({\textstyle{1\over 2}}C)}$$ (33)

$${\cos[{\textstyle{1\over 2}}(a-b)]\over\cos({\textstyle{1\over 2}}c)} = {\sin[{\textstyle{1\over 2}}(A+B)]\over\cos({\textstyle{1\over 2}}C)}$$ (34)

$${\cos[{\textstyle{1\over 2}}(a+b)]\over\cos({\textstyle{1\over 2}}c)} = {\cos[{\textstyle{1\over 2}}(A+B)]\over\sin({\textstyle{1\over 2}}C)},$$ (35)

and Napier's Analogies

$${\sin[{\textstyle{1\over 2}}(A-B)]\over\sin[{\textstyle{1\over 2}}(A+B)]} = {\tan[{\textstyle{1\over 2}}(a-b)]\over\tan({\textstyle{1\over 2}}c)}$$ (36)

$${\cos[{\textstyle{1\over 2}}(A-B)]\over\cos[{\textstyle{1\over 2}}(A+B)]} = {\tan[{\textstyle{1\over 2}}(a+b)]\over\tan({\textstyle{1\over 2}}c)}$$ (37)

$${\sin[{\textstyle{1\over 2}}(a-b)]\over\sin[{\textstyle{1\over 2}}(a+b)]} = {\tan[{\textstyle{1\over 2}}(A-B)]\over\cot({\textstyle{1\over 2}}C)}$$ (38)

$${\cos[{\textstyle{1\over 2}}(a-b)]\over\cos[{\textstyle{1\over 2}}(a+b)]} = {\tan[{\textstyle{1\over 2}}(A+B)]\over\cot({\textstyle{1\over 2}}C)}$$ (39)

(Beyer 1987).

See also Angular Defect, Descartes Total Angular Defect, Gauss's Formulas, Girard's Spherical Excess Formula, Law of Cosines, Law of Sines, Law of Tangents, L'Huilier's Theorem, Napier's Analogies, Spherical Excess, Spherical Geometry, Spherical Polygon, Spherical Triangle

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 131 and 147-150, 1987.

Danby, J. M. Fundamentals of Celestial Mechanics, 2nd ed., rev. ed. Richmond, VA: Willmann-Bell, 1988.

Smart, W. M. Text-Book on Spherical Astronomy, 6th ed. Cambridge, England: Cambridge University Press, 1960.