How this instrument works
sinh, cosh, and tanh are three separate functions that happen to share one set of raw materials: eˣ and e⁻ˣ. sinh(x) is half their difference, cosh(x) is half their sum, and tanh(x) is simply sinh(x) divided by cosh(x) — no fresh exponential required. Because the first two already contain everything the third needs, a sheet built to report just one of them wastes the other two answers sitting right there in the arithmetic.
The shapes differ sharply once you look past the shared ingredients. cosh(x) never drops below 1 and climbs without bound in both directions, tracing the curve a hanging chain settles into under its own weight. sinh(x) crosses zero at the origin and grows just as unboundedly, but signed — negative for negative x. tanh(x), their ratio, behaves like neither parent: squeezed between −1 and 1 for every real input, it is the smooth S-curve that gives neural networks a bounded gate and gives special relativity a clean way to add velocities through rapidity.
Push x far from zero and a limit appears: e⁻ˣ shrinks toward nothing, so sinh(x) and cosh(x) both collapse onto the same curve, eˣ⁄2, and their ratio tanh(x) creeps toward 1 without ever touching it. At x = 0 the opposite extreme holds exactly — e⁰ and e⁻⁰ are both 1, so sinh(0) = 0, cosh(0) = 1, and tanh(0) = 0, the same reference values ordinary sine, cosine, and tangent carry at zero degrees, despite arriving from exponentials instead of a circle.
- Type a value for x — the field takes any real number, whether that's a whole figure, a decimal, or a negative.
- Read sinh(x) and cosh(x) first; both are computed straight from eˣ and e⁻ˣ for the x you entered.
- Read tanh(x) directly underneath — it is sinh(x) divided by cosh(x), so you can check it by hand against the two figures above it.
- Change x and watch the three move together: sinh(x) and cosh(x) drift apart from cosh's floor of 1, while tanh(x) stays pinned between −1 and 1.
- To isolate the sign behaviour, try x and its negative back to back — cosh(x) repeats, while sinh(x) and tanh(x) flip sign.
Worked example — all three functions at x = 1
Feed x = 1 into the sheet and three figures land at once: sinh(x) = 1.1752011936438014, cosh(x) = 1.5430806348152437, and tanh(x) = 0.7615941559557649. Hand arithmetic backs up the first two directly: e¹ ≈ 2.718281828 and e⁻¹ ≈ 0.367879441, so sinh(1) = (2.718281828 − 0.367879441) ⁄ 2 = 1.175201194 and cosh(1) = (2.718281828 + 0.367879441) ⁄ 2 = 1.543080635, matching the display past the ninth decimal.
tanh(1) needs no fresh exponentials at all — it is the ratio of the two answers already on screen, 1.175201194 ⁄ 1.543080635 = 0.761594156. A relativity problem reading x = 1 as a rapidity would call that ratio the fraction of light speed the boosted frame is moving. The three figures also police each other: cosh²(1) − sinh²(1) works out to 1.543080635² − 1.175201194², which lands on 1.000000000, the identity that ties every triple this sheet produces back together.
Questions
Why does one sheet return sinh, cosh, and tanh together instead of just one?
Because all three are built from the same two ingredients, eˣ and e⁻ˣ, computing them separately would waste work: tanh(x) is nothing more than sinh(x) divided by cosh(x), so once those two exist the third is free. Applications rarely need just one either — special relativity's rapidity and a neural network's activation gate both lean on more than one of the three from a single input.
What exactly is tanh(x), and how does it differ from sinh and cosh?
tanh(x) is the ratio sinh(x) ⁄ cosh(x). Unlike its two unbounded parents, it never leaves the band between −1 and 1, creeping toward but never touching those limits as x grows large in either direction. That bounded, S-shaped squeeze is exactly why tanh is used as a gating function in neural networks, where an unbounded output would be unusable.
How do sinh, cosh, and tanh relate to ordinary sine, cosine, and tangent?
Swap the real input for an imaginary one and the hyperbolic trio turns back into the circular one: cosh(x) = cos(ix) and sinh(x) = −i·sin(ix). The circular functions trace the unit circle x² + y² = 1; the hyperbolic ones trace the hyperbola x² − y² = 1 instead, which is where the shared names and the very different real-line graphs both come from.
Why do sinh(x) and cosh(x) become almost equal for large x?
Because e⁻ˣ shrinks toward zero as x grows, leaving both functions dominated by the same surviving term, eˣ⁄2. At x = 5 this sheet gives sinh(5) ≈ 74.20321057778875 and cosh(5) ≈ 74.20994852478785 — agreeing to three decimal places — while tanh(5) ≈ 0.9999092042625951 sits a hair's breadth under its ceiling of 1.
Where do people most often slip up with sinh, cosh, and tanh?
Mishandling negative inputs. cosh is even, so cosh(−x) and cosh(x) are identical, but sinh and tanh are both odd, so sinh(−x) and tanh(−x) flip sign instead of repeating. Treating all three the same way under negation — or assuming tanh needs its own exponential formula rather than sinh(x) ⁄ cosh(x) — accounts for most hand-check errors against this sheet.
What are sinh(0), cosh(0), and tanh(0)?
Zero, one, and zero. At x = 0 both e⁰ and e⁻⁰ equal 1, so sinh(0) = (1 − 1) ⁄ 2 = 0, cosh(0) = (1 + 1) ⁄ 2 = 1, and tanh(0) = 0 ⁄ 1 = 0 — the same starting values ordinary sine, cosine, and tangent carry at zero, even though the hyperbolic trio is built from exponentials rather than a circle.