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Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are great circles. Spherical trigonometry is of great importance for calculations in astronomy, geodesy, and ...
In spherical trigonometry, the law of cosines (also called the cosine rule for sides [1]) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Spherical triangle solved by the law of cosines. Given a unit sphere, a "spherical triangle" on the surface of the sphere ...
Spherical trigonometry was studied by early Greek mathematicians such as Theodosius of Bithynia, a Greek astronomer and mathematician who wrote Spherics, a book on the geometry of the sphere, [2] and Menelaus of Alexandria, who wrote a book on spherical trigonometry called Sphaerica and developed Menelaus' theorem. [3] [4]
Once the radius is fixed, the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. The plane passing through the origin and perpendicular to the polar axis (where the polar angle is a right angle ) is called the reference plane (sometimes fundamental plane ).
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The haversine formula determines the great-circle distance between two points on a sphere given their longitudes and latitudes.Important in navigation, it is a special case of a more general formula in spherical trigonometry, the law of haversines, that relates the sides and angles of spherical triangles.
For centuries, spherical trigonometry has been used for locating solar, lunar, and stellar positions, [56] predicting eclipses, and describing the orbits of the planets. [ 57 ] In modern times, the technique of triangulation is used in astronomy to measure the distance to nearby stars, [ 58 ] as well as in satellite navigation systems .
Infection rates dropped and stabilised throughout 2022 and 2023, leading to the end of COVID-19's classification as a severe transmissible disease in June 2023. [22] Although the pandemic has heavily disrupted the country's economy, [23] Vietnam's GDP growth rate has remained one of the highest in Asia-Pacific, at 2.91% in 2020. Due to the more ...