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.

May 25, 2013 · e^ix = cos(x) + isin(x) e^-ix = cos(x) - isin(x) Add to get 2cos(x), subtract to get 2isin(x). I presume you can do the rest.

$\text{cosh}\ 2x = \text{cosh}^2x + \text{sinh}^2x= 2 \text{cosh}^2x - 1 = 1 + 2 \text{sinh}^2x$ $\text{tanh}\ 2x = \frac{2\text{tanh}\ x}{1 + \text{tanh}^2x}$ Half angle formulas

The two basic hyperbolic functions are "sinh" and "cosh": Hyperbolic Sine: sinh(x) = e x − e-x 2 (pronounced "shine") Hyperbolic Cosine: cosh(x) = e x + e-x 2 (pronounced "cosh") They use the natural exponential function e x. And are not the same as sin(x) and cos(x), but a little bit similar: sinh vs sin. cosh vs cos. Catenary

Feb 7, 2015 · cosh(x) = (e x + e-x)/2 and cos(x) = (e ix + e-ix)/2 which makes that identity very clear. You can justify that these definition match with our other understandings of trigonometric functions by considering the Taylor series.

Jun 24, 2024 · ∫ (cosh x).dx = sinh x + C. The table added below shows the integration of various hyperbolic functions.

Aug 31, 2018 · They’re related by Euler’s formula. Since $e^{ix}=\cos x+i \sin x$ we have $e^{-ix}=\cos x-i \sin x$. This reveals, $$\cosh (ix)=\cos x$$ $$\sinh (ix)=i \sin x$$

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Feb 6, 2015 · Hint: Use the definition of the hyperbolic functions sinh(x) and cosh(x) in terms of the exponential function. Then use x = iy and Euler's formula to establish the connection between the hyperbolic functions and the trig functions.

Feb 9, 2018 · cosh ⁡ x = cos ⁡ i ⁢ x, sinh ⁡ x =-i ⁢ sin ⁡ i ⁢ x. As for the Taylor expansion of the third important hyperbolic function tangens hyperbolica ( http://planetmath.org/HyperbolicFunctions ) tanh, it is obtained via division of the Taylor series ( http://planetmath.org/TaylorSeriesViaDivision ) (2) and (1); the begin of the quotient ...