Spherical trigonometry

Definitions

Great circle: Great and small circles on the sphere
A circle on the sphere that has the same centre as the sphere (an intersection of the sphere and any plane through the sphere centre). On the globe, all meridians (longitude lines) and the Equator is a great circle. Great circles are the geodesics of the sphere, i. e. the shortest way between two points.
Small circle: Any other circle on the sphere is a small circle. On the globe, all parallel circles (latitude lines) except the Equator are small circles.
Spherical triangle: Spherical triangle
A spherical shape delimited by three great circle arcs. On a unit sphere the length of the arcs numerically equals to the respective central angle in radians, therefore, length of the sides of a spherical triangle is often denoted by angles.

Cosine rules

Using the denotations on the figure, the following identities apply to all spherical triangles:

cos a = cos b cos c + sin b sin c cos α
cos b = cos a cos c + sin a sin c cos β
cos c = cos a cos b + sin a sin b cos γ

Derivations can be found e.g. here.

Applications of cosine rules

Calculating spherical distance

A spherical triangle for calculating distance of two points

Let's find the spherical distance between points P₁ (φ₁, λ₁) and P₂ (φ₂, λ₂)! Considering the spherical triangle of P₁, P₁ and the North Pole (see figure) and using the cosine rule for the side δ, we get the following formula:

cos δ = cos(90°-φ₁) cos(90°-φ₂) + sin(90°-φ₁) sin(90°-φ₂) cos(λ₂-λ₁)

As cos(90°-x) = sin x and sin(90°-x) = cos x, we can use a simplified form:

cos δ = sin φ₁ sin φ₂ + cos φ₁ cos φ₂ cos(λ₂-λ₁),

where δ is the spherical distance of the two points on a unit sphere.

Calculating metacoordinates

A spherical triangle for calculating metacoordinates

Let's define a meta coordinate system with meta pole at the point NP* (φ₀, λ₀), having the geodesic towards the true North Pole (NP) as prime meridian. The (φ*, λ*) metacoordinates of any P (φ, λ) point can be calculated by considering the spherical triangle P NP NP* (see figure):

sin φ* = sin φ sin φ₀ + cos φ cos φ₀ cos(λ₀-λ),

sin φ = sin φ* sin φ₀ + cos φ* cos φ₀ cos λ*, from which cos λ* can be expressed:

cos λ* = ( sin φ - sin φ* sin φ₀ ) / ( cos φ* cos φ₀ )