Finding the Root of a Transcendental Equation with Sinh

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In summary, the equation 2a sinh(25/a) = 51 is difficult to solve for a due to the presence of the overall multiplicative factor of a on the left-hand side. One possible approach is to multiply both sides by e^(25/a) and solve for a quadratic equation in terms of e^(25/a), but the presence of the a makes this method ineffective. Therefore, the equation must be solved numerically or in terms of the Lambert W function.
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Shukie
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I have the following equation:

[tex]2a \sinh(\frac{25}{a}) = 51[/tex]

How do I solve this for a?

I tried changing it to:

[tex]a(e^{\frac{25}{a}} - e^{\frac{-25}{a}}) = 51[/tex], but that didn't get me any further. Anyone?
 
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Well, if it weren't for that overall multiplicative factor of [itex]a[/itex] on the LHS, you could multiply both sides by e^(25/a) to get a quadratic equation in terms of e^(25/a). But, that [itex]a[/itex] ruins everything, leaving you with a transcendental equation that I'm afraid you'll have to solve numerically (or perhaps in terms of the lambert W function), I believe.
 

FAQ: Finding the Root of a Transcendental Equation with Sinh

What is the definition of sinh?

The sinh function, short for hyperbolic sine, is a mathematical function that maps the input values to their corresponding hyperbolic sine values. It is defined as sinh(x) = (e^x - e^-x)/2, where e is the base of natural logarithm.

How do you solve a function with sinh?

To solve a function with sinh, you will need to use the inverse function of sinh, which is sinh^-1. You can solve for x by using the formula x = sinh^-1(y), where y is the input value of the sinh function. Alternatively, you can use a calculator or mathematical software to calculate the inverse function.

What is the domain and range of sinh?

The domain of sinh is all real numbers, while the range is from -∞ to +∞. This means that the input values can be any real number, but the output values will always be within a certain range.

How is sinh related to other trigonometric functions?

Sinh is related to other trigonometric functions, such as sine and cosine, through the unit hyperbola. The unit hyperbola is the set of all points (x,y) on the Cartesian plane that satisfy the equation x^2 - y^2 = 1. The coordinates of the point on the unit hyperbola can be represented as (sinh(t),cosh(t)), where t is a real number. This relationship shows that sinh and cosh are analogous to sine and cosine in the unit circle.

What are the applications of sinh in science and engineering?

Sinh has various applications in science and engineering, including in electrical engineering, quantum mechanics, and signal processing. It is also used in the calculation of magnetic fields, heat transfer, and fluid mechanics. In addition, sinh is used in the study of black holes and gravitational waves in astrophysics.

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