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The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent or semi-tangent. [2] Leonhard Euler used it to evaluate the integral ∫ d x / ( a + b cos x ) {\textstyle \int dx/(a+b\cos x)} in his 1768 integral calculus textbook , [ 3 ] and Adrien-Marie Legendre described ...
In trigonometry, the law of tangents or tangent rule [1] is a statement about the relationship between the tangents of two angles of a triangle and the lengths of the opposing sides. In Figure 1, a , b , and c are the lengths of the three sides of the triangle, and α , β , and γ are the angles opposite those three respective sides.
The internal and external tangent lines are useful in solving the belt problem, which is to calculate the length of a belt or rope needed to fit snugly over two pulleys. If the belt is considered to be a mathematical line of negligible thickness, and if both pulleys are assumed to lie in exactly the same plane, the problem devolves to summing ...
The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of calculus in the 17th
By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes , who stated it in 1643. Frederick Soddy 's 1936 poem The Kiss Precise summarizes the theorem in terms of the bends (signed inverse radii) of the four circles:
Visual calculus, invented by Mamikon Mnatsakanian (known as Mamikon), is an approach to solving a variety of integral calculus problems. [1] Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculation. Mamikon collaborated with Tom Apostol on the 2013 book New Horizons in Geometry describing the ...
In that case, Apollonius' problem degenerates to the CCP limiting case, which is the problem of finding a solution circle tangent to the two remaining given circles that passes through the point P. Inversion in a circle centered on P transforms the two given circles into new circles, and the solution circle into a line.
The curve was first proposed and studied by René Descartes in 1638. [1] Its claim to fame lies in an incident in the development of calculus.Descartes challenged Pierre de Fermat to find the tangent line to the curve at an arbitrary point since Fermat had recently discovered a method for finding tangent lines.
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