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Half-Side Formula

In spherical geometry, there is a special branch concerned with the relations and problems of trigonometric functions of spherical triangles that are made by intersecting great circles on the surface of a sphere. This special branch is known as the "spherical trigonometry". It plays a vital role in various calculations related to geodesy, navigation, and astronomy. Nowadays, the use of spherical geometry is losing its existence because many advanced technologies have been introduced that can be easily used instead of complicated spherical geometry. But the half-side formula is still being in use. Let us go ahead and learn about this formula and its applications.

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Definition

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Half side formulas play an important role in spherical trigonometry. These formulas deal with the relationships between the side lengths and angles of a spherical triangle. The spherical triangles are said to be the triangles constructed on the surface of a sphere, therefore, the sides of a spherical triangle are curved and hence the rules and formulas of a spherical triangle are quite different than that of a plane triangle.

Formulas

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The half side formulas used in spherical trigonometry are given below.

Let us suppose that a spherical triangle has angles $\alpha$, $\beta$ and $\gamma$. Eventually, the sides opposite to them are denoted by a, b and c respectively. Then,

Half of side "a"
$tan\ ($$\frac{a}{2}$$) = R\ cos (S - \alpha)$
Half of side "b"
$tan\ ($$\frac{b}{2}$$) = R\ cos (S - \beta)$
Half of side "c"
$tan\ ($$\frac{c}{2}$$) = R\ cos (S - \gamma)$
Where

R is calculated by the following formula :

R = $\sqrt{\frac{-cos \ S}{cos(S - \alpha) \ cos(S - \beta) \ cos(S - \gamma)}}$

and

S is defined as half of the sum of angles of spherical triangle = $\frac{\alpha\ +\ \beta\ +\ \gamma}{2}$

Spherical Triangle

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A spherical triangle is defined as a triangle made up of three arcs that intersect pairwise forming three vertices on the sphere surface as shown in the diagram below.

Spherical Triangle

Assume that a spherical triangle is formed on a sphere whose radius is R and center is assumed to be at origin and vertices are A, B, and C.

Half-Side Formula

Then,  we have the following formulas :

$a . b$ = $R^{2}\ cos($$\frac{c}{R}$$)$

$b . c$ = $R^{2}\ cos($$\frac{a}{R}$$)$

$a . c$ = $R^{2}\ cos($$\frac{b}{R}$$)$

The formulas that express relations between sides and angles of a spherical triangle are :
 
cos a = cos b  cos c + cos A sin b  sin c

cos b = cos c  cos a + cos B sin c  sin a

cos c = cos a  cos b + cos C sin a  sin b
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