Everything about Spherical Trigonometry totally explained
Spherical trigonometry is a part of
spherical geometry that deals with
polygons (especially
triangles) on the
sphere and explains how to find relations between the involved
angles. This is of great importance for calculations in
astronomy and earth-surface and orbital and space
navigation.
Al-Jayyani (989-1079), an
Arabic mathematician in
Islamic Spain, wrote the first treatise on spherical trigonometry,
circa 1060, entitled
The book of unknown arcs of a sphere, which "contains formulae for
right-handed 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.
Lines on a sphere
On the surface of a sphere, the closest analogue to straight
lines are
great circles, for example circles whose center coincide with the center of the sphere. For example,
meridians and the
equator are great circles on the
Earth, while non-equatorial lines of latitude are not great circles. As with a line segment in a
plane, an
arc of a great circle (subtending less than 180°) on a sphere is the shortest path lying on the sphere between its two
endpoints. Great circles are special cases of the concept of a
geodesic.
An area on the sphere which is bounded by arcs of
great circles is called a
spherical polygon. Note that, unlike the case on a plane, spherical "
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.
The
sides of these polygons are most conveniently specified not by their length but by the angle under which its endpoints appear when looked at from the sphere's center. Note that this
arc angle, measured in
radians, when multiplied by the sphere's
radius equals the arc length.
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 a spherical triangle is always larger than the 180° found in every planar triangle. The amount by which the sum of the angles exceeds 180° is called the
spherical excess E: E = α + β + γ − 180° (where α, β and γ refers to the angle of each corner). By
Girard's theorem, this surplus determines the surface area of any spherical triangle. To determine this, the spherical excess must be expressed in radians; the surface area A is then given in terms of the sphere's radius R by the expression:
» A = R
2 · E.
From this formula, which is an application of the
Gauss-Bonnet theorem, it follows that there are no 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.
To solve a geometric problem on the sphere, one dissects the relevant figure into
right spherical triangles (for example: one of the triangle's corner angles is 90°) because one can then use Napier's pentagon:
Napier's pentagon (also known as Napier's circle) is a
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 (for example: 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° (for example 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
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 »
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 angle limit.
They also satisfy an analogue of the
law of sines »
A more thorough list of identities is available
here
Further Information
Get more info on 'Spherical Trigonometry'.
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