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In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle.Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola.
3.1 Integrals of hyperbolic tangent, cotangent, secant, cosecant functions 3.2 Integrals involving hyperbolic sine and cosine functions 3.3 Integrals involving hyperbolic and trigonometric functions
Then the area of the hyperbolic sector is the area of the triangle minus the curved region past the vertex at (,): = = ( (+)), which simplifies to the area hyperbolic cosine = = (+). Solving for x {\displaystyle x} yields the exponential form of the hyperbolic cosine: x = cosh a = e a + e − a 2 . {\displaystyle x=\cosh a={\frac ...
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Ptolemy's theorem states that the sum of the products of the lengths of opposite sides is equal to the product of the lengths of the diagonals. When those side-lengths are expressed in terms of the sin and cos values shown in the figure above, this yields the angle sum trigonometric identity for sine: sin(α + β) = sin α cos β + cos α sin β.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
The hyperbolastic rate equation of type I, denoted H1, is given by = (()) (+ +),where is any real number and is the population size at .The parameter represents carrying capacity, and parameters and jointly represent growth rate.
The Gudermannian and its inverse were used to make trigonometric tables of circular functions also function as tables of hyperbolic functions. Given a hyperbolic angle , hyperbolic functions could be found by first looking up = in a Gudermannian table and then looking up the appropriate circular function of , or by directly locating in an ...