What does the 'h' in the hyperbola equation mean?

[Quarlep]

I am studying leonard susskind lectures and there I saw a hyperbola equation


I didnt understand that equation what it means coshw or sinhw or total equation(In 49 minute) what's the "h"in there I need help (you can look also 52 min)

 

[Mentallic]

The professor miswrote on the board, it should be

$$1=\cosh^2w-\sinh^2w$$

Where

$$\cosh{x} = \frac{e^x+e^{-x}}{2}$$

$$\sinh{x} = \frac{e^x-e^{-x}}{2}$$

They're represented in this way because they appear frequently and quite often have very similar properties to the usual cos and sin trig functions, such as

$$\cosh^2w-\sinh^2w=1$$
versus
$$\cos^2w+\sin^2w=1$$

 

[Quarlep]

whats the "h" means there I need to lear hyperbolic geometry to learn it I guess

 

[Mentallic]

The h stands for hyperbolic.
The sine function is denoted by sin, and evaluating the sine of $\pi$ is done by calculating $\sin(\pi)$.
Similarly, the hyperbolic sine function is denoted by sinh and evaluating this function at x=5 is done by $\sinh(5)$, except we can also represent the sinh function by
$$\sinh(x)=\frac{e^x-e^{-x}}{2}$$
hence
$$\sinh(5)=\frac{e^5-e^{-5}}{2}$$

 

[Quarlep]

I understand it thanks