Unified Formulation
Define a generalized sine function, depending also on a real parameter :
The law of sines in constant curvature reads as
By substituting, and, one obtains respectively the euclidian, spherical, and hyperbolic cases of the law of sines described above.
Let indicate the circumference of a circle of radius in a space of constant curvature . Then . Therefore the law of sines can also be expressed as:
This formulation was discovered by János Bolyai.
Read more about this topic: Law Of Sines
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“In necessary things, unity; in disputed things, liberty; in all things, charity.”
—Variously Ascribed.
The formulation was used as a motto by the English Nonconformist clergyman Richard Baxter (1615-1691)