Hyperbolic Functions and Their Inverses
Build sinh and cosh from the exponential function, prove the identity that replaces the Pythagorean one, and write the inverses as logarithms.
What a learner can do afterwards
- Define sinh and cosh in terms of the exponential and sketch both graphs
- Prove the identity relating cosh squared and sinh squared and compare it with the circular case
- Express an inverse hyperbolic function as a logarithm and state its domain
1 · Read
Hyperbolic functions are exponentials in disguise. You define sinh x as the half difference (e to the x minus e to the minus x) over 2, and cosh x as the half sum (e to the x plus e to the minus x) over 2. The sinh graph is odd and passes through the origin, while the cosh graph is even, bottoms out at 1, and passes through (0, 1). A hanging chain follows a cosh curve, which engineers call the catenary.
The key identity is cosh squared x minus sinh squared x equals 1 for every real x. Expanding both definitions makes the cross terms cancel and leaves 1. It parametrises the unit hyperbola exactly as cosine and sine parametrise the unit circle. Its circular twin is cos squared x plus sin squared x equals 1, with a plus where the hyperbolic case has a minus.
Inverses of hyperbolic functions are logarithms in disguise. Solving y equals sinh x gives arsinh x equals ln(x plus root(x squared plus 1)), valid for every real x. Read domains off the graphs: arcosh needs x at least 1 since cosh bottoms out at 1, while artanh needs inputs strictly between minus 1 and 1. Differentiation respects the disguise, so the derivative of arsinh is 1 over root(x squared plus 1).
Values at zero settle every confusion: cosh 0 equals 1 and sinh 0 equals 0, so cosh 0 minus sinh 0 equals 1. Derivatives cycle with a sign twist: the derivative of sinh is cosh and the derivative of cosh is sinh. So for f(x) equals sinh x plus cosh x, f prime of 0 equals 1 plus 0, which is 1.
Half difference and half sum of exponentials give sinh and cosh, whose squares always differ by 1.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.