The relationship between hyperbolic and circular functions

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Homework Help Overview

The discussion revolves around the relationship between hyperbolic functions and circular functions, particularly focusing on their mathematical identities and parametric representations. Participants are exploring the underlying reasons for the similarities and differences between these two types of functions.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • The original poster seeks sources or explanations for the analogy between hyperbolic and circular functions. Some participants provide mathematical representations of the functions and question the implications of substituting hyperbolic equivalents into trigonometric identities.

Discussion Status

The discussion is active, with participants sharing insights and references. There is an acknowledgment of a specific rule related to the identities, but no consensus has been reached on a comprehensive explanation.

Contextual Notes

Participants are navigating the complexity of the topic, with some expressing frustration over the clarity of available resources. The discussion includes references to specific mathematical rules and identities without resolving the broader conceptual questions.

nobahar
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Hello!
A book on calculus was introducing hyperbolic functions and pointed out that the identities such as cosh x and sinh x, etc. for hyperbolic functions were analogous to cos x and sin x for circular functions. I tried finding some internet sources explaining why this is so, but they tend to be overly complicated or unsatisfactory. Could someone point me in the direction of any recommended sources; or, perhaps, offer a brief explanation (presumably the explanation is not so brief!…)
Thanks in advance.
 
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Circular functions are parametritizations of the equation
x^2+y^2=1
Hyperbolic functions are parametritizations of the equation
y^2-x^2=1
so that explains the anaology
futher if complex numbers are used we see hyperbolas and circles as equivalent in some sense
 
lurflurf said:
Circular functions are parametritizations of the equation
x^2+y^2=1
Hyperbolic functions are parametritizations of the equation
y^2-x^2=1
Thanks ever so much lurflurf.
Because for any trig identities, apparently you simply substitute in the hyperbolic equivalent, changing the sign for a product of two sins. Is this simply a consequence of the minus sign in the above equation, and that the others remain 'unaffected'?
 
Yes!
That is called Osborne's Rule.
see
http://mathworld.wolfram.com/OsbornesRule.html
 

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