What this calculator does
This tool builds a table of the three basic hyperbolic functions — hyperbolic sine (sinh), hyperbolic cosine (cosh) and hyperbolic tangent (tanh) — for every value of x in a range you choose. You set a start value, an end value and a step increment, and the calculator iterates across the interval, computing each function at each point. It is handy for studying function shapes, checking homework, or generating data points to plot.
How to use it
Enter the Start value and End value of x, then a positive Increment (step). Optionally pick how many significant digits to display. The increment must be greater than zero, and the end value should be at least the start value. The number of rows is approximately \( \lfloor (\text{end} - \text{start}) / \text{step} \rfloor + 1 \); very small steps over wide ranges are capped at 2000 rows to keep the table manageable.
The formulas explained
The hyperbolic functions are defined from the exponential function: $$\sinh x = \frac{e^{x}-e^{-x}}{2},\quad \cosh x = \frac{e^{x}+e^{-x}}{2}$$ Their ratio gives $$\tanh x = \frac{\sinh x}{\cosh x} = \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$$ Because \(\cosh x\) is always at least 1, tanh never divides by zero, and tanh always lies strictly between \(-1\) and \(1\). sinh and tanh are odd functions while cosh is even, so \(\sinh 0 = 0\), \(\cosh 0 = 1\) and \(\tanh 0 = 0\).
Worked example
With start = \(-2\), end = 2, step = 1 you get five rows for x = \(-2, -1, 0, 1, 2\). At \(x = 2\): \(e^{2} = 7.389056\) and \(e^{-2} = 0.135335\), so $$\sinh 2 = \frac{7.389056 - 0.135335}{2} = 3.626860$$ $$\cosh 2 = \frac{7.389056 + 0.135335}{2} = 3.762196$$ $$\tanh 2 = \frac{3.626860}{3.762196} = 0.964028$$ By symmetry \(x = -2\) gives the negatives of sinh and tanh and the same cosh.
FAQ
Why must the step be positive? A zero or negative step would never advance toward the end value, producing an infinite loop or no progress, so it is rejected.
What happens for very large x? For \(|x|\) beyond roughly 710, sinh and cosh overflow the range of a double and show as Infinity; the tool flags this with a warning.
Does x have units? No. The argument x is a dimensionless real number and is used directly with no scaling.
