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The two basic hyperbolic functions are "sinh" and "cosh": sinh (x) = ex − e-x 2. cosh (x) = ex + e-x 2. They use the natural exponential function ex. And are not the same as sin (x) and cos (x), but a little bit similar: One of the interesting uses of Hyperbolic Functions is the curve made by suspended cables or chains.
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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.
Hyperbolic Functions - sinh, cosh, tanh, coth, sech, csch Definition of hyperbolic functions. Hyperbolic sine of x $\text{sinh}\ x = \frac{e^{x} - e^{-x}}{2}$ Hyperbolic cosine of x $\text{cosh}\ x = \frac{e^x + e^{-x}}{2}$ Hyperbolic tangent of x $\text{tanh}\ x = \frac{e^x - e^{-x}}{e^x + e^{-x}}$ Hyperbolic cotangent of x
8 Δεκ 2016 · The following formula holds: $$\cos(z)=\cosh(iz),$$ where $\cos$ is the cosine and $\cosh$ is the hyperbolic cosine. Proof From the definition of $\cosh$ and the definition of $\cos$, $$\cosh(iz)=\dfrac{e^{iz}+e^{-iz}}{2}=\cos(z),$$ as was to be shown.
20 Απρ 2024 · Cos (cosine) is basic trigonometric function used to determine the ratio between the adjacent side and hypotenuse of right triangle, while cosh (hyperbolic cosine) relates to the sum of the exponential functions e^x and e^-x, used in hyperbolic geometry.
The hyperbolic trigonometric functions extend the notion of the parametric equations for a unit circle \((x = \cos t\) and \(y = \sin t)\) to the parametric equations for a hyperbola, which yield the following two fundamental hyperbolic equations: \[x = \cosh a = \dfrac{e^a + e^{-a}}{2},\quad y = \sinh a = \dfrac{e^a - e^{-a}}{2}.\]
If $(x,y)$ is a point on the right half of the hyperbola, and if we let $x=\cosh t$, then $\ds y=\pm\sqrt{x^2-1}=\pm\sqrt{\cosh^2 t-1}=\pm\sinh t$. So for some suitable $t$, $\cosh t$ and $\sinh t$ are the coordinates of a typical point on the hyperbola. In fact, it turns out that $t$ is twice the area shown in the first graph of figure 4.11.2.
