How this instrument works
Hyperbolic cosine is built from the same exponential that describes uncontrolled growth and decay, not from a circle. cosh(x) = (eˣ + e⁻ˣ)⁄2 averages a runaway-growing term against its shrinking mirror image, and because one half of that pair is always climbing back up whenever the other falls, the result never dips below 1 and never turns around to repeat itself the way ordinary cosine does.
Set x to 0 and both exponential terms equal 1, so cosh(0) = 1 — the single lowest point the function ever reaches. Move away from zero in either direction and the value only rises: eˣ and e⁻ˣ multiply to exactly 1 for every input, so their arithmetic average can never fall beneath their geometric mean of 1, an inequality that pins the floor of the whole function in place and makes it symmetric, since swapping the sign just swaps which exponential is doing the growing.
Swap the real axis for the imaginary one and cosh turns back into ordinary cosine: cosh(x) equals cos(ix), which is why the two names rhyme without the two graphs resembling each other at all on the real line. Far from zero the shrinking term e⁻ˣ becomes irrelevant and cosh collapses toward the simpler eˣ⁄2 — a function with no ceiling, unlike the −1-to-1 band that bounds its circular namesake forever.
- Type any real number into the x field — cosh accepts positive numbers, negative numbers, and zero without restriction.
- Read the answer in cosh(x): for x = 1 it shows 1.5430806348152437, matched against the exact value of (e¹ + e⁻¹)⁄2.
- Try the negative of that same number, −1 — cosh(x) returns the identical figure, confirming that the function is even.
- Push x further from zero in either direction to watch the result climb; there is no ceiling and no repeat, unlike ordinary cosine.
Worked example — evaluating cosh at x = 1
Set x to 1 and the sheet returns value = 1.5430806348152437. Working it by hand confirms the figure: e¹ ≈ 2.718281828, e⁻¹ ≈ 0.367879441, and their sum divided by two is 3.086161270 ⁄ 2 = 1.543080635 — the two results agree past the eighth decimal, exactly what an instrument built to double-check its own arithmetic should do.
Enter −1 instead and the field reports the same 1.5430806348152437, because swapping eˣ and e⁻ˣ inside the average changes nothing — cosh is even, its graph a mirror image across the y-axis, the same shape a hanging chain traces between two supports of equal height. Set x to 0 and cosh returns exactly 1, the lowest point that curve ever reaches, since e⁰ and e⁻⁰ are both 1.
Questions
What does cosh(x) mean, exactly?
cosh(x) is shorthand for the hyperbolic cosine, defined as (eˣ + e⁻ˣ) ⁄ 2 — the average of eˣ and its reciprocal. Where ordinary cosine comes from a circle, cosh comes from a hyperbola: the point (cosh t, sinh t) traces the curve x² − y² = 1, the hyperbolic twin of the circle traced by (cos t, sin t).
Is cosh just a hyperbolic version of cosine, so it behaves the same way?
No — despite the similar name, the two functions behave almost oppositely. Ordinary cosine oscillates forever between −1 and 1; cosh never repeats and never turns back down, climbing without bound as x moves away from zero in either direction. The formal link is complex: cosh(x) = cos(ix), so hyperbolic cosine is what plain cosine becomes along the imaginary axis.
Why can cosh(x) never be less than 1?
Because eˣ and e⁻ˣ are both positive and multiply to exactly 1, their arithmetic mean can never fall below their geometric mean, which is √(eˣ·e⁻ˣ) = 1. So (eˣ + e⁻ˣ)⁄2 ≥ 1 for every real x, with equality only at x = 0, the single point where cosh reaches its minimum.
How does cosh relate to sinh?
They satisfy cosh²(x) − sinh²(x) = 1 for every real number, the hyperbolic twin of the circular identity cos²θ + sin²θ = 1 — the minus sign is what turns a circle into a hyperbola. sinh is defined as (eˣ − e⁻ˣ)⁄2, so subtracting the squares of the two definitions cancels every eˣe⁻ˣ cross-term, leaving exactly 1.
Where does cosh(x) show up outside pure mathematics?
Its best-known role is the catenary: a chain or cable hanging freely under its own weight settles into a curve shaped by cosh, not a parabola, however similar the two shapes look near the bottom of the sag. Physicists also lean on cosh to write Lorentz factors and rapidity in special relativity, since it naturally stays at or above 1 the way a boost factor must.
What happens to cosh(x) for large values of x?
The shrinking term e⁻ˣ becomes negligible and cosh approaches eˣ⁄2 — growth that is essentially exponential once the input sits a few units from zero. The same holds going negative, using eˣ in place of e⁻ˣ: cosh(5) and cosh(−5) both round to about 74.21, close to e⁵⁄2 ≈ 74.21.