Spherical trigonometry is a subdivision of spherical geometry which studies the relationship between trigonometric functions of the sides and angles of the spherical triangles which is a part of the great circle of a sphere.

The cross-section of a sphere is a hemisphere and the plane passing through the centre is a circle, called the great circle of the sphere. A spherical triangle is a triangle formed by arcs of great circles of a sphere. The spherical triangle doesn’t belong to Euclidean space.

Menelaus of Alexandria is a renowned Greek mathematician and astronomer whose major work is Spheric which later led to his famous work Menelaus’ Theorem. This theorem helps to solve the problems related to spherical triangles or distances across a sphere for the practical study of astronomy such as the trajectory of planets. The theory involves using a spherical quadrilateral to calculate the shortest curved distances between two points on a sphere.

Spherical triangles

Take the sides of arcs of great circles as a,b,c and these are measured by their angles subtended at center O of the sphere.

A, B, C are the angles opposite sides a, b, c respectively.

Area of spherical triangle

Girard’s theorem will help you solve for the area of a spherical triangle, which states that the area of a spherical triangle can be found by the spherical excess where the interior angles of the triangle are and the radius of the sphere is 1.

The area of a spherical triangle on the surface of the sphere of radius R is given by the formula ,

A=R2*E

Where E is the spherical excess in degrees( The amount by which the sum of three angles of a triangle on a sphere exceeds 180 degrees)

spherical excess E

E=A+B+C-π

Sum of interior angles of spherical triangle

The sum of the interior angles of a spherical triangle is greater than 180° and less than 540°.

180°﹤(A+B+C)﹤ 540°.

In spherical trigonometry, earth is assumed to be a perfect sphere. One minute (0° 1′) of arc from the center of the earth has a distance equivalent to one (1) nautical mile (6080 feet) on the arc of a great circle on the surface of the earth.

1 minute of arc = 1 nautical mile

1 nautical mile = 6080 feet

1 statute mile = 5280 feet

1 knot = 1 nautical mile per hour

Applications

Navigation

A location or position on the Earth is defined by its Latitude and Longitude. Latitude measures from the Equator to the North or to the South to the position and Longitude measures from the Prime Meridian of Greenwich to the East or to the West to the position.

Astronomy

The position of astronomical objects on the Celestial Sphere is determined by Declination which is measured from the Celestial Equator to the North or to the South towards the celestial position and by Greenwich Hour Angle (GHA), which is always measured Westward from the Celestial Prime Meridian to the position of the celestial object.

Spherical trigonometry is a subdivision of spherical geometry which studies the relationship between trigonometric functions of the sides and angles of the spherical triangles which is a part of the great circle of a sphere.

Spherical triangle is a triangle bounded by arc of great circles of a sphere. Note that for spherical triangles, sides a, b, and c are usually in angular units.