How this instrument works
Hyperbolic sine takes the same two ingredients as hyperbolic cosine — eˣ and its mirror image e⁻ˣ — but subtracts instead of adding: sinh(x) = (eˣ − e⁻ˣ) ⁄ 2. That single sign flip changes everything about the shape. Where the sum of a growing exponential and its shrinking twin can never fall below a floor, their difference has no floor and no ceiling: sinh climbs without bound as the input grows, drops without bound as it falls, and crosses zero exactly once, right at the origin.
The function is odd — sinh(−x) = −sinh(x) — so its graph mirrors through the origin rather than through the vertical axis the way cosh's graph does. Close to zero it behaves almost exactly like a straight line, since the first term of its series expansion is simply the input itself; the curve only bends away from that line once the cubic and higher terms take over. That same curve is also, quite literally, a slope: differentiate the catenary equation y = a·cosh(x ⁄ a) and hyperbolic cosine turns into hyperbolic sine, dy⁄dx = sinh(x ⁄ a) — sinh doesn't just resemble part of a hanging cable, it is the steepness at every point along it.
Because sinh is a strictly increasing map from all real numbers onto all real numbers, its inverse arsinh(x) = ln(x + √(x² + 1)) is defined for every real input, with none of the domain restriction the inverse of cosh needs. In special relativity, if φ is a particle's rapidity, its momentum term works out proportional to sinh(φ) while the matching energy term is proportional to cosh(φ) — and the identity cosh²φ − sinh²φ = 1 is exactly what forces the relativistic relation E² − (pc)² = (mc²)² to hold for any rapidity at all. Despite the name, sinh has nothing to do with circular angles: sin(θ) oscillates forever between −1 and 1, while sinh(x) never repeats and never stays put.
- Enter any real number in the x field — sinh handles positive figures, negative figures, and zero alike, with no domain restriction.
- sinh(x) reports the result: setting x to 1 returns 1.1752011936438014, the exact value of (e¹ − e⁻¹) ⁄ 2.
- Flip the sign of x and watch sinh(x) flip too — enter −1 and the field returns −1.1752011936438014, confirming the odd symmetry.
- Set x to 0 to see the one fixed point every hyperbolic sine passes through: sinh(x) reports exactly 0.
- Move x further from zero in either direction to watch sinh(x) grow past any range an ordinary sine could ever reach.
Worked example — evaluating sinh at x = 1
Set x to 1 and the sheet returns value = 1.1752011936438014. Checking it by hand confirms the figure: e¹ ≈ 2.718281828459045, e⁻¹ ≈ 0.367879441171442, and their difference divided by two is 2.350402387287603 ⁄ 2 = 1.175201193643801 — the two results agree to the full sixteen digits shown, exactly what an instrument built to double-check its own arithmetic should do.
Enter −1 in place of 1 and sinh(x) reports −1.1752011936438014, the exact negative of the first answer — the odd symmetry flips the sign of the difference eˣ − e⁻ˣ without changing its size. Set x to 0 instead and the field returns exactly 0, since e⁰ and e⁻⁰ are both 1 and their difference vanishes; every other input pushes the value away from that single resting point in one direction or the other, with no ceiling stopping it as x grows.
The same function also measures slope. For the catenary y = a·cosh(x ⁄ a) with a = 2, the cable's steepness at x = 1 is dy⁄dx = sinh(1 ⁄ 2) = sinh(0.5) = 0.5210953054937474 — a rise of a little over half a unit of height for every unit moved sideways at that point, a distinct question from the height itself, which cosh answers.
Questions
What does sinh(x) mean, and why the minus sign?
sinh(x) is the hyperbolic sine, (eˣ − e⁻ˣ) ⁄ 2 — half the gap between a growing exponential and its shrinking mirror image. The minus sign is what separates it from cosh, which adds the same two terms instead; subtracting rather than adding is why sinh passes through zero and has no floor, while cosh never dips below 1.
Is sinh just a hyperbolic version of ordinary sine, so it oscillates the same way?
No — the two behave almost oppositely despite sharing a name. Ordinary sine oscillates forever between −1 and 1; sinh never repeats and has no bound in either direction, growing toward infinity as x increases and toward negative infinity as x decreases. The formal link runs through imaginary numbers: sinh(x) = −i·sin(ix), so hyperbolic sine is what ordinary sine becomes off the real axis.
What's the most common mistake people make computing sinh by hand?
Swapping the sign and computing cosh by accident — (eˣ + e⁻ˣ) ⁄ 2 instead of (eˣ − e⁻ˣ) ⁄ 2. The two formulas look almost identical on the page, but they produce very different curves: cosh(0) = 1 while sinh(0) = 0, and cosh never goes negative while sinh does for every negative input.
How does sinh relate to the slope of a catenary curve?
Differentiate the catenary equation y = a·cosh(x ⁄ a) with respect to x and hyperbolic cosine turns into hyperbolic sine: dy⁄dx = sinh(x ⁄ a). So while cosh(x ⁄ a) gives the cable's height at a point, sinh(x ⁄ a) gives its steepness there — the two functions describe the same hanging chain from different angles.
Does sinh have an inverse, and what does it look like?
Yes — arsinh(x) = ln(x + √(x² + 1)), and unlike the inverse of cosh, it is defined for every real number with no domain restriction, because sinh itself passes through every real value exactly once. arsinh also appears as the inverse Gudermannian function used in deriving the Mercator map projection.
Where does sinh show up in physics?
In special relativity, if φ is a particle's rapidity, its momentum term is proportional to sinh(φ) while its energy term is proportional to cosh(φ); because cosh²φ − sinh²φ = 1 always holds, those two terms automatically satisfy the relativistic relation E² − (pc)² = (mc²)² for any rapidity at all.