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The Gudermannian function gives a direct relationship between the circular functions and the hyperbolic functions that does not involve complex numbers. The graph of the function a cosh( x / a ) is the catenary , the curve formed by a uniform flexible chain, hanging freely between two fixed points under uniform gravity.
Similar expressions can be written for tanh x, coth x, sech x, and csch x. Geometrically, this change of variables is a one-dimensional stereographic projection of the hyperbolic line onto the real interval, analogous to the Poincaré disk model of the hyperbolic plane.
This principal value of the square root function is denoted in what follows. Similarly, the principal value of the logarithm, denoted in what follows, is defined as the value for which the imaginary part has the smallest absolute value. It is defined everywhere except for non-positive real values of the variable, for which two different values ...
A formula for computing the trigonometric identities for the one-third angle exists, but it requires finding the zeroes of the cubic equation 4x 3 − 3x + d = 0, where is the value of the cosine function at the one-third angle and d is the known value of the cosine function at the full angle.
The Gudermannian function relates the area of a circular sector to the area of a hyperbolic sector, via a common stereographic projection.If twice the area of the blue hyperbolic sector is ψ, then twice the area of the red circular sector is ϕ = gd ψ.
The hyperbolic secant distribution shares many properties with the standard normal distribution: it is symmetric with unit variance and zero mean, median and mode, and its probability density function is proportional to its characteristic function.
The sides of this rhombus have length 1. The angle between the horizontal line and the shown diagonal is 1 / 2 (a + b).This is a geometric way to prove the particular tangent half-angle formula that says tan 1 / 2 (a + b) = (sin a + sin b) / (cos a + cos b).
The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen ...