Review of the Hyperbolic Functions

Minkowski space is called a hyperbolic geometry. Clearly the time coordinate is not treated the same as the space coordinates. We will find that when space and time coordinates are mixed in a transformation, that we use the hyperbolic functions $\sinh$ and $\cosh$ instead of $\sin$ and $\cos$. They are very symmetrically defined.

$$e^{i\phi} =\cos\phi+i\sin\phi e^\phi =\cosh\phi+\sinh\phi$$

$$e^{-i\phi} =\cos\phi-i\sin\phi e^{-\phi} =\cosh\phi-\sinh\phi$$

$$\cos\phi ={e^{i\phi}+e^{-i\phi}\over 2} \cosh\phi ={e^{\phi}+e^{-\phi}\over 2}$$

$$\sin\phi ={e^{i\phi}-e^{-i\phi}\over 2i} \sinh\phi ={e^{\phi}-e^{-\phi}\over 2}$$

$$\tan\phi ={e^{i\phi}-e^{-i\phi}\over e^{i\phi}+e^{-i\phi}} \tanh\phi ={e^{\phi}-e^{-\phi}\over e^{\phi}+e^{-\phi}}$$