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neginf
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Where do the definitions of hyperbolic functions in terms of exponentials come from ?
neginf said:Thank you for that.
I wonder if the definitions in terms of exponentials can be gotten from the geometric definition. That would seem more natural than adding and subtracting trig functions with imaginary arguments.
Hyperbolic functions are important in mathematics because they have a wide range of applications in various fields such as physics, engineering, and statistics. They are particularly useful in solving problems involving curves and surfaces.
Hyperbolic functions are defined in terms of exponential functions. Specifically, the hyperbolic sine (sinh) and cosine (cosh) functions can be expressed as combinations of the exponential function e^x and its inverse, ln(x).
The purpose of defining hyperbolic functions in terms of exponentials is to make calculations and equations involving hyperbolic functions easier. This is because exponential functions have well-established properties and can be manipulated algebraically, making it simpler to work with hyperbolic functions.
Yes, hyperbolic functions have many real-life applications. For example, they are used in modeling the shape of a hanging chain or cable, calculating the trajectory of a satellite orbit, and analyzing heat and diffusion processes.
Yes, just like any other mathematical function, hyperbolic functions can also be graphed. They have distinct curves that resemble the graphs of sine and cosine functions, but with different properties and behaviors.