Derive $\sin{ix}=i\sinh{x}$ from $(5)$. What is $\sin{i}$? $$\cos{x}=\frac{1}{2}\left(e^{ix}+e^{-ix}\right)\quad\text{and}\quad\sin{x}=\frac{1}{2i}\left(e^{ix}-e^{-ix}\right)\tag 5$$

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.

If x = sinh y, then y = sinh-1 a is called the inverse hyperbolic sine of x. Similarly we define the other inverse hyperbolic functions. The inverse hyperbolic functions are multiple-valued and as in the case of inverse trigonometric functions we restrict ourselves to principal values for which they can be considered as single-valued.

Given that $\tan(y) =\sinh(x)$ show that $\sin(y) = \pm \tanh(x) $. I know: $\tan\theta=\dfrac{\sin\theta}{\cos\theta}$ $\tanh\theta=\dfrac{\sinh\theta}{\cosh\theta}$ Also, $\tanh\theta = \dfr...

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

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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, $$\cosh (ix)=\cos x$$ $$\sinh (ix)=i \sin x$$

$ \Rightarrow \sinh (ix) = i\sin x $ So, the correct answer is “Option A”. Note : Always remember the standard hyperbolic formula for the trigonometric function and the exponential form of the trigonometric function.

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Nov 15, 2021 · In this short note we show how to deal with sines and cosine of imaginary arguments. In particular, we’ll show that cos (ix) = cosh x , sin (ix) = i sinh x , cosh (ix) = cos x , sinh (ix) = i sin x , where x is defined over the complex numbers. (The reader is assumed to know about complex exponentials and hyperbolic functions.) sinh (u) = .