What is spherical trigonometry and on what theorems is this based?

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

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Spherical trigonometry , unlike  the plane  trigonometry, deals with  relation between the sides and angles of  spherical triangles.

The spherical triangle is a triangle discribed by the arcs of great circles on the sphere. The great circles on a sphere are the circles whose centre is same as the centre of the sphere.

Let ABC be the spherical triangle with vertices A,B and C, formed by  the great cirle's arc AB, arc BC  and arc CA. Let arc AB=c, arc BC=a and arc AC = b.

Then the relation between the sides, a, b and c and angles A, B and C are given by sin rule and cosine rules:

Cosine rule:

Cos a = cos b*cos c +sin b*sin c cos A, which gives the side a interms of the other two sides and the included angle.

Sine rule: As in plane trigonometry, in spherical trigonometry also we a similar relationship. The sine rule could be derived from the cosine rule also.

sine a / sine A  = sin b / sine B = sin c/ sine C.

Spherical trigonometry is used in astronomy. The angular apparent positions of the planets and stars  with respect each other, with respect to horizon and ecliptic could be dealt with using spherical trigonometry.

 

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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Spherical trigonometry mathematics is the discipline which deals with solving triangles formed on the surface of a sphere from   arcs of large circles.

Spherical trigonometry has a big  importance, theoretical and practical, and it's applicable on a larger scale in astronomy, superior geodesy, in cartography, in crystallography, in the mining geometry, theory of instruments and other sciences, when, to study the relative position in space of points, lines and planes, it  is used an helpful sphere.
It's called spherical surface, or area, the geometrical locus of points in space, equal far off from a fixed point O - the center of this surface.

The space bounded by the surface of a sphere is called also sphere.

Spherical surface can be defined as the surface produced by rotating a semicircle around it's diameter.
Segment of straight line joining the center of the sphere with any point on its surface is called the sphere radius R and the segment of straight line , joining two points on the spherical surface passing through its center and is called diameter, obviously the same sphere radii are equal between them and the diameter is equal with two rays.

In the spherical geometry are used the following theorems:
Theorem 1: Section of a sphere with a certain plane, is a circle.


Theorem 2: great circles divide the sphere and its surface into two equal parts.

Theorem 3: Through two given points on the surface of a sphere, if they are not placed at the ends of the same diameter, it can be traced a large circle and only one.

Theorem 4: The intersection plane of two large circles   is one of their diameter and divides them into two equal parts.


Theorem 5: The shortest distance between two points on the sphere, on its surface, is an large arc smaller than 180 °.

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