Search results
Results from the WOW.Com Content Network
Since the function cosh x is even, only even exponents for x occur in its Taylor series. The sum of the sinh and cosh series is the infinite series expression of the exponential function. The following series are followed by a description of a subset of their domain of convergence, where the series is convergent and its sum equals the function.
This geometric argument relies on definitions of arc length and area, which act as assumptions, so it is rather a condition imposed in construction of trigonometric functions than a provable property. [2] For the sine function, we can handle other values. If θ > π /2, then θ > 1. But sin θ ≤ 1 (because of the Pythagorean identity), so sin ...
Then exchange all the cosine and sine terms to cosh and sinh terms. However, for all products or implied products of two sine terms replace it with the negative product of two sinh terms. This is because − i sin ( i x ) {\displaystyle -i\sin(ix)} is equivalent to sinh ( x ) {\displaystyle \sinh(x)} , so when multiplied to together the ...
The fact that the triple-angle formula for sine and cosine only involves powers of a single function allows one to relate the geometric problem of a compass and straightedge construction of angle trisection to the algebraic problem of solving a cubic equation, which allows one to prove that trisection is in general impossible using the given tools.
For arcoth, the argument of the logarithm is in (−∞, 0], if and only if z belongs to the real interval [−1, 1]. Therefore, these formulas define convenient principal values, for which the branch cuts are (−∞, −1] and [1, ∞) for the inverse hyperbolic tangent, and [−1, 1] for the inverse hyperbolic cotangent.
Circles about the points (0,0), (0,1), (0,2) and (0,3) of radius 3.5 in the Lobachevsky hyperbolic coordinates. Construct a Cartesian-like coordinate system as follows. Choose a line (the x -axis) in the hyperbolic plane (with a standardized curvature of -1) and label the points on it by their distance from an origin ( x =0) point on the x ...
Since cosh x + sinh x = e x, an analog to de Moivre's formula also applies to the hyperbolic trigonometry. For all integers n, ( + ) = + . If n is a rational number (but not necessarily an integer), then cosh nx + sinh nx will be one of the values of (cosh x + sinh x) n. [4]
3.1 Integrals of hyperbolic tangent, cotangent, secant, cosecant functions. 3.2 Integrals involving hyperbolic sine and cosine functions.