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Anyway, on to my question. I started watching Susskinds special relativity lectures the other day. At one point he starts deriving the lorentz transformations using hyperbolic functions, which I know to be standard. What I do not understand is this.

\begin{align}

x' &= x*cosh(\theta) - t*sinh(\theta)

\end{align}

\begin{align}

t' &= - x*sinh(\theta) + t*cosh(\theta)

\end{align}

In the above coordinate transformation from the x-t frame to the x'-t' frame, I do not understand where his negative signs come from. I understand coordinate transformations using cosine and sine functions just fine, using triangles to determine where the signs of each component comes from. But, I have been unable to do the same thing with the hyperbolic functions. I cannot see where he gets the signs from, and I do not want to simply take the transformations on faith alone that they are true. I would prefer to understand where they come from. So, can anybody explain geometrically how to come to this transformation? Thanks in advance.