The cosine rule is used in trigonometry to find aspects of non-right triangles. It is an extension of the Pythagorean theorem and can determine angles and side lengths. It was created by al-Kashi, but may have been devised by Euclid. The rule is useful in triangulation and only works in Euclidean geometry, but different versions can be used for non-Euclidean geometry.
The cosine rule is a commonly used formula in trigonometry to determine some aspect of a non-right triangle when other key parts of that triangle are known or can be otherwise determined. It is an effective extension of the Pythagorean theorem, which generally only works with right triangles and states that the square of the triangle’s hypotenuse equals the squares of the other two sides when added together (c2=a2+b2). The cosine rule is an extension of this mathematical principle which makes it effective for non-right triangles and states that with respect to a certain angle, the square of the side of the triangle opposite that angle equals the squares of the other two sides added together , minus twice both sides multiplied together by the cosine of that angle (c2=a2+b2-2ab cosC where C is the angle opposite side c).
Although many modern mathematical sources credit a Muslim mathematician named al-Kashi for the creation of the cosine rule, there is also some evidence to indicate that the ancient Greek mathematician Euclid had devised a similar principle. Much of modern algebra and trigonometry derives from the efforts of Muslims during the European Middle Ages, and it was around the 15th century that al-Kashi codified the formula in a way that is still understood today. In France the rule is even called Le théorème d’Al-Kashi or “al-Kashi’s theorem”.
In general, the cosine rule is used in triangulation and a number of other practical applications of trigonometry. It is particularly useful in systems where the lengths of all three sides are known or can be stated and the measure of the angles within the triangle must be determined. The cosine rule can also be used to determine the length of a side of a triangle if the lengths of the other two sides and the angle opposite that side are known.
Since the cosine rule deals with triangles made up of three straight sides and their angles, it generally only works in the context of Euclidean geometry. Different versions of the cosine rule can be used for non-Euclidean geometry such as spherical geometry and hyperbolic geometry. In these systems, a triangle consists of three points in curved space and the lines, usually curved lines, that connect them. The hyperbolic law of cosines and the spherical law of cosines work much like the Euclidean cosine rule in that they can allow someone to determine the three angles of a triangle as long as they know the three sides. Unlike the Euclidean cosine rules, however, these non-Euclidean laws can also allow someone to determine the dimensions of the three sides of a triangle if they know the three angles.
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