Sinh x formula

The hyperbolic trigonometric functions extend the notion of the \[x = \cosh a = \dfrac\), is also referred to as a catenary. The shape of a dangling chain is a hyperbolic cosine. The St. Louis Gateway Arch—the shape of an upside-down hyperbolic cosine hyperbolic functions, also define the shape of the path a spaceship takes when it uses the "gravitational slingshot" effect to alter its course via a planet's gravitational pull propelling it away from that planet at high velocity. The spaceship traces out a hyperbola as it uses the "slingshot" effect. One of the key characteristics that motivates the hyperbolic trigonometric functions is the striking similarity to trigonometric functions, which can be seen from \[\begin = \cosh a \pm \sinh a,\] which is the equivalent of Euler's formula for hyperbolic functions. \( _\square\) Billy Tangent naively thought that the hyperbolic cosine function and the standard cosine function were the same. To make sure, he tried one real value and, sure enough, he got the same result. What value did he try? If you think there are multiple values for which this would work, enter 99999 as your answer. If you think there are no values for which this would work, enter 88888 as your answer. There are six hyperbolic trigonometric functions: • \(\sinh x = \dfrac\). Their graphic representations are shown here: Graphs of the six trigonometric hyperbolic functions The parametric equations for a unit circle are given by • \(x = \cos t\) • \(y = \sin t\)...

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It is possible if $n$ is odd, and it is impossible if $n$ is even. To see the first recall that $\sinh x=\dfrac$. Its possible for odd integer $n$: $$\sinh((2k+1)x) = \sum_ \sinh^r x $$ For example, $$ \sinh (9x) = 256 \sinh^9x + 576 \sinh^7 x + 432 \sinh^5 x + 120 \sinh^3 x + 9 \sinh x $$ For even integer $n$ it is also almost possible, but in the form $$ \sinh(2kx) = \cosh x \sum a_r\sinh^r x $$ and you can't express that $\cosh x$ as a polynomial in $\sinh x$. Not a complete solution. We have $$\sinh(nx)=\dfrac(n-k)\pi \right]$$ evaluated at $x=-iu$ to obtain the expression that the OP wanted (with an additional complex scaling factor). Note as already mentioned by other users this is only possible for odd $n$. Only then you will be able to rewrite the $\cos x$ terms by using $1-\sin^2 x$.

Learning Objectives • 6.9.1 Apply the formulas for derivatives and integrals of the hyperbolic functions. • 6.9.2 Apply the formulas for the derivatives of the inverse hyperbolic functions and their associated integrals. • 6.9.3 Describe the common applied conditions of a catenary curve. We were introduced to hyperbolic functions in Derivatives and Integrals of the Hyperbolic Functions Recall that the hyperbolic sine and hyperbolic cosine are defined as d d x ( sinh x ) = d d x ( e x − e − x 2 ) = 1 2 [ d d x ( e x ) − d d x ( e − x ) ] = 1 2 [ e x + e − x ] = cosh x . d d x ( sinh x ) = d d x ( e x − e − x 2 ) = 1 2 [ d d x ( e x ) − d d x ( e − x ) ] = 1 2 [ e x + e − x ] = cosh x . Similarly, ( d / d x ) cosh x = sinh x . ( d / d x ) cosh x = sinh x . We summarize the differentiation formulas for the hyperbolic functions in the following table. Table 6.2 Derivatives of the Hyperbolic Functions Let’s take a moment to compare the derivatives of the hyperbolic functions with the derivatives of the standard trigonometric functions. There are a lot of similarities, but differences as well. For example, the derivatives of the sine functions match: ( d / d x ) sin x = cos x ( d / d x ) sin x = cos x and ( d / d x ) sinh x = cosh x . ( d / d x ) sinh x = cosh x . The derivatives of the cosine functions, however, differ in sign: ( d / d x ) cos x = − sin x , ( d / d x ) cos x = − sin x , but ( d / d x ) cosh x = sinh x . ( d / d x ) cosh x = sinh x . As we continue our examinati...

Question : Find sinh(x+iy) formula Solution : Let the complex number as \[z=sinh\left(x+iy\right)\] We know that \[\sin\] Please Login or Register to View Answer or Ask a Question

Trigonometric functions are similar to Hyperbolic functions. Â Hyperbolic functions also can be seen in many linear differential equations, for example in the cubic equations, the calculation of angles and distances in hyperbolic geometry are done through this formula. This has importance in electromagnetic theory, heat transfer, and special relativity. The basic hyperbolic formulas are sinh, cosh, tanh. \[\large e^\] Solved Examples

Min Max Min Max Re Im The inverse hyperbolic sine (Beyer 1987, p.181; Zwillinger 1995, p.481), sometimes called the area hyperbolic sine (Harris and Stocker 1998, p.264) is the The variants or (Harris and Stocker 1998, p.263) are sometimes used to refer to explicit is sometimes used for the principal value, with being used for the multivalued function (Abramowitz and Stegun 1972, p.87). The notations (Jeffrey 2000, p.124) and (Gradshteyn and Ryzhik 2000, p.xxx) are sometimes also used. Note that in the notation , is the denotes an not the multiplicative inverse. Its is implemented in the z] and in the GNU C library as asinh( double x). The inverse hyperbolic sine is a and . This follows from the definition of as More things to try: • • • References Abramowitz, M. and Stegun, I.A. (Eds.). "Inverse Circular Functions."§4.4 in Beyer, W.H. GNU C Library. "Mathematics: Inverse Trigonometric Functions." Gradshteyn, I.S. and Ryzhik, I.M. Harris, J.W. and Stocker, H. Jeffrey, A. "Inverse Trigonometric and Hyperbolic Functions."§2.7 in Sloane, N.J.A. Sequences Spanier, J. and Oldham, K.B. "Inverse Trigonometric Functions." Ch.35 in Zwillinger, D. (Ed.). "Inverse Hyperbolic Functions."§6.8 in Referenced on Wolfram|Alpha Cite this as: MathWorld--A Wolfram Web Resource. Subject classifications • • • • • • • • • • • • • • • • • • • • Created, developed and nurtured by Eric Weisstein at Wolfram Research

Trigonometric functions are similar to Hyperbolic functions. Â Hyperbolic functions also can be seen in many linear differential equations, for example in the cubic equations, the calculation of angles and distances in hyperbolic geometry are done through this formula. This has importance in electromagnetic theory, heat transfer, and special relativity. The basic hyperbolic formulas are sinh, cosh, tanh. \[\large e^\] Solved Examples

The hyperbolic trigonometric functions extend the notion of the \[x = \cosh a = \dfrac\), is also referred to as a catenary. The shape of a dangling chain is a hyperbolic cosine. The St. Louis Gateway Arch—the shape of an upside-down hyperbolic cosine hyperbolic functions, also define the shape of the path a spaceship takes when it uses the "gravitational slingshot" effect to alter its course via a planet's gravitational pull propelling it away from that planet at high velocity. The spaceship traces out a hyperbola as it uses the "slingshot" effect. One of the key characteristics that motivates the hyperbolic trigonometric functions is the striking similarity to trigonometric functions, which can be seen from \[\begin = \cosh a \pm \sinh a,\] which is the equivalent of Euler's formula for hyperbolic functions. \( _\square\) Billy Tangent naively thought that the hyperbolic cosine function and the standard cosine function were the same. To make sure, he tried one real value and, sure enough, he got the same result. What value did he try? If you think there are multiple values for which this would work, enter 99999 as your answer. If you think there are no values for which this would work, enter 88888 as your answer. There are six hyperbolic trigonometric functions: • \(\sinh x = \dfrac\). Their graphic representations are shown here: Graphs of the six trigonometric hyperbolic functions The parametric equations for a unit circle are given by • \(x = \cos t\) • \(y = \sin t\)...

Learning Objectives • 6.9.1 Apply the formulas for derivatives and integrals of the hyperbolic functions. • 6.9.2 Apply the formulas for the derivatives of the inverse hyperbolic functions and their associated integrals. • 6.9.3 Describe the common applied conditions of a catenary curve. We were introduced to hyperbolic functions in Derivatives and Integrals of the Hyperbolic Functions Recall that the hyperbolic sine and hyperbolic cosine are defined as d d x ( sinh x ) = d d x ( e x − e − x 2 ) = 1 2 [ d d x ( e x ) − d d x ( e − x ) ] = 1 2 [ e x + e − x ] = cosh x . d d x ( sinh x ) = d d x ( e x − e − x 2 ) = 1 2 [ d d x ( e x ) − d d x ( e − x ) ] = 1 2 [ e x + e − x ] = cosh x . Similarly, ( d / d x ) cosh x = sinh x . ( d / d x ) cosh x = sinh x . We summarize the differentiation formulas for the hyperbolic functions in the following table. Table 6.2 Derivatives of the Hyperbolic Functions Let’s take a moment to compare the derivatives of the hyperbolic functions with the derivatives of the standard trigonometric functions. There are a lot of similarities, but differences as well. For example, the derivatives of the sine functions match: ( d / d x ) sin x = cos x ( d / d x ) sin x = cos x and ( d / d x ) sinh x = cosh x . ( d / d x ) sinh x = cosh x . The derivatives of the cosine functions, however, differ in sign: ( d / d x ) cos x = − sin x , ( d / d x ) cos x = − sin x , but ( d / d x ) cosh x = sinh x . ( d / d x ) cosh x = sinh x . As we continue our examinati...