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B Trigonometric identities

  1. Oct 10, 2017 #1
    Consider the following set of equations:

    ##r = \cosh\rho \cos\tau + \sinh\rho \cos\varphi##

    ##rt = \cosh\rho \sin\tau##

    ##rl\phi = \sinh\rho \sin\varphi##

    Is there some way to combine the equations to get rid of ##\varphi## and ##\tau## and express ##\rho## in terms of ##r, t, \phi##?

    I tried the following:

    ##r^{2} = (\cosh\rho \cos\tau + \sinh\rho \cos\varphi)^{2}##

    ##r^{2}(t-l\phi)^{2} = (\cosh\rho \sin\tau - \sinh\rho \sin\varphi)^{2}##

    so that we have

    ##r^{2} + r^{2}(t-l\phi)^{2} = \cosh^{2}\rho + \sinh^{2}\rho + 2\cos(\tau+\varphi)\sinh\rho\cosh\rho.##

    The above line is not exactly what I want, because of the factor ##\cos(\tau+\varphi)##!

    Is there some neat way to get rid of ##\varphi## and ##\tau## and express ##\rho## in terms of ##r, t, \phi##?
     
  2. jcsd
  3. Oct 10, 2017 #2

    jedishrfu

    Staff: Mentor

    Is this a homework problem? What course are you taking?

    We need some context here. Where did this problem come from and what do you need it for?
     
  4. Oct 10, 2017 #3

    Svein

    User Avatar
    Science Advisor

    There are connections between the hyperbolic and trigonometric functions:. For example:
    [tex]\sinh(x)=-i \sin(ix) [/tex][tex]\cosh(x)=\cos(ix) [/tex]
    See https://en.wikipedia.org/wiki/Hyperbolic_function
     
  5. Oct 10, 2017 #4

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    You can express ##\varphi## and ##\tau## as function of the other variables using the second and third equation. Not nice, but possible.
     
  6. Oct 14, 2017 #5
    This is not a homework problem. This is part of my research work.

    The equations are a modified form of (1.17) on page 13 of https://esc.fnwi.uva.nl/thesis/centraal/files/f37733672.pdf.

    I need it to make progress in my work.
     
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