Osborn's rule

  • Thread starter PFuser1232
  • Start date
  • #1
479
20
Osborn's rule:
"The prescription that a trigonometric identity can be converted to an analogous identity for hyperbolic functions by expanding, exchanging trigonometric functions with their hyperbolic counterparts, and then flipping the sign of each term involving the product of two hyperbolic sines."
I understand how to apply Osborn's rule to identities involving a product of two hyperbolic sines, but I'm not entirely sure what happens when there is an "odd/even product" of hyperbolic sines. For instance, does ##\sin^4{x}## become ##-\sinh^4{x}##? What about ##\sin^3{x}##?
 

Answers and Replies

  • #3
fzero
Science Advisor
Homework Helper
Gold Member
3,119
289
It's more important to understand the reasoning behind Osborn's rule rather than memorize the rule itself. What you should memorize is the Euler identity

$$e^{ix} = \cos x + i \sin x.$$

By taking the complex conjugate, we can solve for

$$ \cos x = \frac{e^{ix} + e^{-ix} }{2},~~~\sin x = \frac{e^{ix} - e^{-ix}}{2i}.$$

These provide a way to relate the trig functions to the hyperbolic ones and we find that

$$ \cos ix = \cosh x,~~~ \sin i x = i \sinh x. ~~(*)$$

Given these relations we can compute

$$ \sin^4 ix = \sinh^4 x,~~~\sin^3 ix = -i \sinh^3 x.$$

Osborne's rule, whatever the particular statement should be, is what follows from applying the relations (*) to the various trig identities. Whether it is better for you to memorize the rule, or remember the above logic and quickly derive the hyperbolic identities from the trig identities is something you should decide for yourself after some exercise in converting identities.
 
  • Like
Likes PFuser1232

Related Threads on Osborn's rule

  • Last Post
Replies
2
Views
2K
Replies
1
Views
9K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
4
Views
3K
  • Last Post
3
Replies
65
Views
8K
  • Last Post
Replies
3
Views
4K
  • Last Post
Replies
6
Views
8K
  • Last Post
Replies
5
Views
742
  • Last Post
Replies
2
Views
6K
  • Last Post
3
Replies
59
Views
9K
Top