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Osborn's rule

  1. Jun 19, 2015 #1
    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}##?
  2. jcsd
  3. Jun 19, 2015 #2


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  4. Jun 19, 2015 #3


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    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.
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