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Spherical trigonometry

Spherical trigonometry is a branch of spherical geometry which deals with polygons (especially triangles) on the sphere and the relationships between the sides and the angles. This is of great importance for calculations in astronomy and earth-surface, orbital and space navigation.

History

Spherical triangles were studied by early Greek mathematics|Greek mathematicians such as Menelaus of Alexandria, who wrote a book on spherical triangles called ''Sphaerica'' and developed Menelaus' theorem. E. S. Kennedy, however, points out that while it was possible in ancient mathematics to compute the magnitudes of a spherical figure, in principle, by use of the table of Chord (geometry)|chords and Menelaus' theorem, the application of the theorem to spherical problems was very difficult in practice. (cf. , in ) Spherical trigonometry was founded as an independent discipline by medieval Mathematics in medieval Islam|Islamic mathematicians. In order to observe holy days on the Islamic calendar in which timings were determined by phases of the moon, the astronomers initially used Menalaus' method to calculate the place of the moon and stars, though this method proved to be clumsy and difficult. It involved setting up two intersecting right triangles; by applying Menelaus' theorem it was possible to solve one of the six sides, but only if the other five sides were known. To tell the time from the sun's altitude, for instance, repeated applications of Menelaus' theorem were required. For medieval Astronomy in medieval Islam|Islamic astronomers, there was an obvious challenge to find a simpler trigonometric method. In the early 9th century, Muhammad ibn Mūsā al-Khwārizmī was an early pioneer in spherical trigonometry and wrote a treatise on the subject. In the 10th century, Abū al-Wafā' al-Būzjānī established the angle addition formulas, e.g., sin(''a'' + ''b''), and discovered the Law of sines|sine formula for spherical trigonometry:Jacques Sesiano, "Islamic mathematics", p. 157, in
-

\frac{\sin \alpha}{\sin a} = \frac{\sin \beta}{\sin b} = \frac{\sin \gamma}{\sin c}.

Here, ''a'', ''b'', and ''c'' are the angles at the centre of the sphere subtended by the three sides of the triangle, and ''α'', ''β'', and ''γ'' are the angles between the sides, where angle ''α'' is opposite the side which subtends angle ''a'', etc. Al-Jayyani (989-1079), an Islamic mathematics|Arabic mathematician in Al-Andalus|Islamic Iberian Peninsula, wrote what some consider the first treatise on spherical trigonometry, ''circa'' 1060, entitled ''The book of unknown arcs of a sphere'', in which spherical trigonometry was brought into its modern form. Al-Jayyani's book "contains formulae for Special right triangles|right-angle triangles, the general law of sines and the solution of a spherical triangle by means of the polar triangle." This treatise later had a "strong influence on European mathematics", and his "definition of ratios as numbers" and "method of solving a spherical triangle when all sides are unknown" are likely to have influenced Regiomontanus. In the 13th century, Iranian mathematician Nasīr al-Dīn al-Tūsī was the first to treat trigonometry as a mathematical discipline independent from astronomy, and he further developed spherical trigonometry, bringing it to its present form. He listed the six distinct cases of a right-angled triangle in spherical trigonometry. In his ''On the Sector Figure'', he also stated the law of sines for plane and spherical triangles, and discovered the law of tangents for spherical triangles.

Lines on a sphere

On the surface of a sphere, the closest analogue to straight line (mathematics)|lines are great circles, i.e. circles whose center coincide with the center of the sphere. For example, simplifying the shape of the Earth (the geoid) to a sphere, the meridian (geography)|meridians and the equator are great circles on its surface, while non-equatorial lines of latitude are small circles. As with a line segment in a plane (mathematics)|plane, an Arc (geometry)|arc of a great circle (subtending less than 180°) on a sphere is the shortest path lying on the sphere between its two Point (geometry)|endpoints. Great circles are special cases of the concept of a geodesic. An area on the sphere bounded by arcs of great circles is called a '''spherical polygon'''. Note that, unlike the case on a plane, spherical "digon|biangles" (two-sided analogs to triangle) are possible (such as a slice cut out of an orange). Such a polygon is also called a lune (mathematics)#spherical geometry|lune. The sides of these polygons are specified not by their lengths, but by the angles at the sphere's center subtended to the endpoints of the sides. Note that this ''arc angle'', measured in radians, when multiplied by the sphere's radius equals the arc length. (In the special case of polygons on the surface of a sphere of radius one, the arc length of any side equals its subtended angle.) Hence, a '''spherical triangle''' is specified as usual by its corner angles and its sides, but the sides are given not by their length, but by their arc angle. The sum of the vertex angles of spherical triangles is always larger than the sum of the angles of plane triangles, which is exactly 180°. The amount ''E'' by which the sum of the angles exceeds 180° is called '''spherical excess''':
-

E = \alpha + \beta + \gamma - \pi,\

where α, β and γ denote the angles. '''Girard's theorem,''' named after the 16th century French mathematician Albert Girard (earlier discovered but not published by the English mathematician Thomas Harriot), states that this surplus determines the surface area of any spherical triangle:
-

A = R^2 \cdot E,

where ''R'' is the radius of the sphere. The analogous result holds for hyperbolic triangles, with "excess" replaced by "defect"; these are both special cases of the Gauss-Bonnet theorem. It follows from here that there are no non-trivial similar triangles (triangles with equal angles but different side lengths and area) on a sphere. In the special case of a sphere of radius 1, the area simply equals the excess angle: ''A = E''. One can also use Girard's formula to obtain the discrete Gauss-Bonnet theorem. To solve a geometric problem on the sphere, one dissects the relevant figure into ''right spherical triangles'' (i.e.: one of the triangle's corner angles is 90°) because one can then use Napier's pentagon: John Napier|Napier's pentagon (also known as Napier's circle) is a mnemonic|mnemonic aid that helps to find all relations between the angles in a right spherical triangle. Write the six angles of the triangle (three vertex angles, three arc angles) in the form of a circle, sticking to the order as they appear in the triangle (i.e.: start with a corner angle, write the arc angle of an attached side next to it, proceed with the next corner angle, etc. and close the circle). Then cross out the 90° corner angle and replace the arc angles adjacent to it by their complement to 90° (i.e. replace, say, ''a'' by 90° − ''a''). The five numbers that you now have on your paper form Napier's Pentagon (or Napier's Circle). For them, it holds that the cosine of each angle is equal to:
- the product of the cotangents of the angles written next to it
- the product of the sines of the two angles written opposed to it As an example, starting with the angle

\overline{b}

, we can obtain the formula:
-

\cos(\overline{b}) = \cot(\overline{a}) \cot(A) = \sin(B) \sin(c).

See also the Haversine formula, which relates the lengths of sides and angles in spherical triangles in a numerically stable form for navigation.

Identities

Spherical triangles satisfy a spherical law of cosines
-

\cos c= \cos a \cos b + \sin a \sin b \cos C. \!

The identity may be derived by considering the triangles formed by the tangent lines to the spherical triangle subtending angle ''C'' and using the plane law of cosines. Moreover, it reduces to the plane law in the small area limit. They also satisfy an analogue of the law of sines
-

\frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C}.

A more thorough list of identities is available here And finally, they satisfy the half side formulae.

References

- Isaac Todhunter: ''Spherical Trigonometry: For the Use of Colleges and Schools''. Macmillan & Co. 1863 (complete online version (Google Books))

External links

- Wolfram's mathworld: Spherical Trigonometry a more thorough list of identities, with some derivation
- Wolfram's mathworld: Spherical Triangle nice applet
- Intro to Spherical Trig. Includes discussion of The Napier circle and Napier's rules
- Spherical Trigonometry — for the use of colleges and schools by I. Todhunter, M.A., F.R.S. Historical Math Monograph posted by Cornell University Library.
- A Visual Proof of Girard's Theorem by Okay Arik, the Wolfram Demonstrations Project. Category:Spherical trigonometry

Related Images

- Spherical triangle
- Napier's Circle shows the relations of parts of a right spherical triangle
- Spherical triangle solved by the law of cosines.

Sources: StartLearningNow, Wikipedia | Usage license: GNU FDL

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