How to derive the formulas for hyperbolic sine?

The formula for hyperbolic sine is derived from the exponential function.

To derive the formula for hyperbolic sine, we start with the exponential function:

e^x = 1 + x + (x^2)/2! + (x^3)/3! + ...

We can rewrite this as:

e^x = 1/2 (e^x + e^-x) + 1/2 (e^x - e^-x)

Now, let y = e^x - e^-x. Solving for e^x and e^-x, we get:

e^x = (y + sqrt(y^2 + 4))/2
e^-x = (sqrt(y^2 + 4) - y)/2

Substituting these expressions into the equation for y, we get:

y = e^x - e^-x = (y + sqrt(y^2 + 4))/2 - (sqrt(y^2 + 4) - y)/2

Simplifying, we get:

y^2 = (e^x - e^-x)^2 = (y + sqrt(y^2 + 4))(sqrt(y^2 + 4) - y)

Expanding and simplifying, we get:

y^2 = 4 sinh^2(x)

Therefore, the formula for hyperbolic sine is:

sinh(x) = (e^x - e^-x)/2