Lorentz transformation conclusion with rotation of axis

[TheDestroyer]

Hi Guys,

I've attached 2 pages from the book of landau "The Classical Theory of fields", I have a question about the lorentz transformations in pages 10,11

after reaching the step:

tanh(psy)= V/c

How did he split the latter into sinh(psy) and cosh(psy) and added the "gamma" constant which is 1/sqrt(1-V^2/c^2) ?

Can we add any constant we want? of course there is a reason

anyone can explain?

I know the Einsteins way of concluding these transformations but I want to understand the rotation of axes method

thanks in advance, please reply as soon as possible :)

 

[robphy]

Think about a [Minkowski-] right-triangle... and its associated Pythagorean theorem [the square-interval]:

write:
$$\cosh^2 \psi - \sinh^2\psi = 1$$
as:
$$\cosh^2 \psi(1 - \tanh^2\psi) = 1$$

 

[orubi]

Thanks