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If the law of cosines is used to solve for c, the necessity of inverting the cosine magnifies rounding errors when c is small. In this case, the alternative formulation of the law of haversines is preferable. [3] A variation on the law of cosines, the second spherical law of cosines, [4] (also called the cosine rule for angles [1]) states:
Spherical triangle solved by the law of cosines. As in Euclidean geometry, one can use the law of cosines to determine the angles A , B , C from the knowledge of the sides a , b , c . In contrast to Euclidean geometry, the reverse is also possible in both non-Euclidean models: the angles A , B , C determine the sides a , b , c .
The spherical cosine formulae were originally proved by elementary geometry and the planar cosine rule (Todhunter, [1] Art.37). He also gives a derivation using simple coordinate geometry and the planar cosine rule (Art.60). The approach outlined here uses simpler vector methods. (These methods are also discussed at Spherical law of cosines.)
Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3) , the group of rotations in three dimensions, and thus play a central ...
The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-". [32] With these functions, one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. [33]
On computer systems with low floating point precision, the spherical law of cosines formula can have large rounding errors if the distance is small (if the two points are a kilometer apart on the surface of the Earth, the cosine of the central angle is near 0.99999999).
Consider the projective (spherical) triangle at the point ; the vertices of this projective triangle are the three lines that join with the other three vertices of the tetrahedron. The edges will have spherical lengths α i , j , α i , k , α i , l {\displaystyle \alpha _{i,j},\alpha _{i,k},\alpha _{i,l}} and the respective opposite spherical ...
Spherical geometry or spherics (from Ancient Greek σφαιρικά) is the geometry of the two-dimensional surface of a sphere [a] or the n-dimensional surface of higher dimensional spheres.