Formula and Method for Hyperbolic Functions
The hyperbolic functions are built directly from the exponential function eˣ rather than from angles on a circle. The two basic definitions are sinh(x) = (eˣ − e⁻ˣ) / 2 (hyperbolic sine) and cosh(x) = (eˣ + e⁻ˣ) / 2 (hyperbolic cosine). Every other hyperbolic function is built from these two: tanh(x) = sinh(x)/cosh(x), coth(x) = cosh(x)/sinh(x), sech(x) = 1/cosh(x), and csch(x) = 1/sinh(x). Their name comes from the fact that the point (cosh(t), sinh(t)) traces the right branch of the unit hyperbola x² − y² = 1, the same way (cos t, sin t) traces the unit circle.
How the calculation works
Enter any real number x — it is a plain number, not an angle, so there is no degrees/radians choice. The calculator raises e to the power x and to the power −x, then combines those two values using the definitions above to get sinh(x) and cosh(x). tanh(x), coth(x), sech(x), and csch(x) then follow as simple ratios or reciprocals of sinh(x) and cosh(x). Because cosh(x) is never zero, sech(x) is always defined; because sinh(x) equals zero only at x = 0, coth(x) and csch(x) are undefined only at that single point.
Common mistakes
- Treating x as an angle: unlike sin and cos, hyperbolic functions do not take degrees or radians — x is just a real number (for example, a distance, a rate × time, or a rapidity).
- Confusing sinh/cosh with sin/cos: sinh(x) can be negative or any real value and grows without bound, while sin(x) always stays between −1 and 1; cosh(x) is always ≥ 1, never less than 1 like cos(x) can be.
- Forgetting the identity sign: the hyperbolic identity is cosh²(x) − sinh²(x) = 1 (a minus sign), not cosh²(x) + sinh²(x) = 1 like the circular identity sin²θ + cos²θ = 1.
- Dividing by zero: coth(x) and csch(x) blow up as x approaches 0 from either side — check whether your value of x is exactly 0 before trusting those two results.
Real-world applications
- The catenary curve formed by a hanging cable or chain under gravity is y = a·cosh(x/a), used in bridge and power-line design.
- Special relativity uses tanh to relate velocity to rapidity, since velocities combine additively in rapidity but not directly.
- Hyperbolic functions solve certain differential equations (e.g. those describing heat flow along a fin or the shape of a soap film) that circular trig functions cannot.
- tanh is a common activation function in neural networks because it smoothly maps any real input to the range (−1, 1).