The relationship between hyperbolic and circular functions

[nobahar]

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.

 

[lurflurf]

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

 

[nobahar]

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'?

 

[lurflurf]

Yes!
That is called Osborne's Rule.
see
http://mathworld.wolfram.com/OsbornesRule.html