Cosh Calculator

The hyperbolic cosine looks like a close cousin of the ordinary cosine — cosh, sinh, tanh mirror cos, sin, tan in name and in many formulas — but it is built from something quite different: exponentials. cosh(x) is defined as (eˣ + e⁻ˣ) / 2, the average of a growing exponential and a decaying one. A cosh calculator evaluates that definition for you, and the calculator at the top of this page shows the working openly: it displays eˣ and e⁻ˣ, averages them for cosh, differences them for sinh, divides for tanh, and even checks the famous identity cosh² − sinh² = 1 on your numbers. One point deserves saying at the start, because it trips up many searchers: the input to cosh is not an angle. There are no degrees and no radians to convert. You enter a plain real number x, and out comes a plain real number.

Direct answer: For x = 1, cosh(1) = 1.543081, and for x = 2, cosh(2) = 3.762196.

Why does such a function exist at all? Because the curve y = cosh(x) is the catenary — the exact shape a uniform chain, rope, or cable takes when it hangs between two supports under its own weight. It also appears wherever exponentials grow in both directions at once: the solutions of many differential equations in physics and engineering are combinations of cosh and sinh rather than of cos and sin, because the underlying behaviour is growth and decay, not oscillation. If you have met the ordinary trig functions through circles, meet the hyperbolic ones through the hyperbola x² − y² = 1, which they parametrise exactly as cos and sin parametrise the circle — hence the name hyperbolic.

What cosh, sinh, and tanh Are

The definitions are short enough to hold in one glance. cosh(x) = (eˣ + e⁻ˣ) / 2. sinh(x) = (eˣ − e⁻ˣ) / 2, the same pair subtracted instead of added. tanh(x) = sinh(x) / cosh(x), which simplifies to (eˣ − e⁻ˣ) / (eˣ + e⁻ˣ). Everything about these functions flows from those definitions, and the calculator's step display is simply the definitions being evaluated: compute the two exponentials once, and cosh, sinh, and tanh are three small combinations of them.

Some values anchor the picture. At x = 0: e⁰ = 1 and e⁰ = 1 again, so cosh(0) = (1 + 1) / 2 = 1 exactly, sinh(0) = 0, and tanh(0) = 0. At x = 1, the live test: e¹ ≈ 2.718282 and e⁻¹ ≈ 0.367879, so cosh(1) = (2.718282 + 0.367879) / 2 = 1.543081, sinh(1) = (2.718282 − 0.367879) / 2 = 1.175201, and tanh(1) = 1.175201 / 1.543081 ≈ 0.761594. cosh is an even function — cosh(−x) = cosh(x), because swapping x for −x just swaps the two exponentials and their average is unchanged — while sinh and tanh are odd functions that flip sign with x. And cosh is always at least 1, with its minimum at x = 0, rising without bound on both sides: that U-shape, symmetric about the vertical axis, is the catenary silhouette.

The identity cosh²(x) − sinh²(x) = 1 holds for every x, mirroring cos² + sin² = 1 with a sign change that reflects the hyperbola-versus-circle geometry. Check it on the live test values at full precision: cosh(1)² − sinh(1)² is exactly 1 — the calculator performs this check on your own inputs and prints 1.000000, so a correct evaluation announces itself; if the identity ever failed to come out at 1, the arithmetic would be wrong. (If you square the six-decimal printouts yourself you may land a unit in the last place off 1, which is rounding in the displayed digits, not in the computation.) Other useful relations: cosh(2x) = 2cosh²(x) − 1, sinh(2x) = 2 sinh(x) cosh(x), and tanh approaches 1 as x grows (tanh(5) ≈ 0.999909) without ever passing it, which is why tanh is used as a smooth squashing function in statistics and machine learning, where its S-curve appears as an activation function.

How to Use the Calculator

Using the tool takes a few seconds. Follow these steps:

1. Enter a real number x. The live test uses 1. Decimals and negatives are welcome, and |x| can be up to 700.
2. Press "Calculate cosh(x)".
3. Read cosh(x) as the headline result, then the companion values sinh(x) and tanh(x), the identity check, and the step-by-step working that shows eˣ and e⁻ˣ and how they were combined.

There is deliberately no angle mode and no degrees/radians switch on this tool, because hyperbolic functions do not take angles. If a formula in front of you contains cosh of something, that something is a dimensionless number — often a ratio like distance over a characteristic length, or a product like rate times time — and you enter it exactly as written. If instead you are holding an angle and reaching for the ordinary cosine, you want a trigonometry tool, not this one; the Limits section below says more about that boundary.

Worked Examples

Example 1: cosh(1), the live test

Enter 1. The exponentials are e¹ = 2.718282 and e⁻¹ = 0.367879. Average them: (2.718282 + 0.367879) / 2 = 3.086161 / 2 = 1.543081, so cosh(1) = 1.543081. Difference them: sinh(1) = 1.175201. Divide: tanh(1) = 0.761594. Identity check: cosh² − sinh² = 1.000000, as it must be. This example is worth doing once by hand precisely because all three functions come from the same two exponentials — a single pair of values, combined three ways.

Example 2: cosh(0) and cosh(−1)

Enter 0: both exponentials are 1, cosh(0) = 1, sinh(0) = 0, tanh(0) = 0. Now enter −1: the exponentials swap, e⁻¹ = 0.367879 becomes the small one and e¹ = 2.718282 the large one, but cosh(−1) = 1.543081 — identical to cosh(1), confirming the even symmetry — while sinh(−1) = −1.175201 and tanh(−1) = −0.761594 flip sign. A hanging cable does not care which end you measure from, and the evenness of cosh is that indifference in algebraic form.

Example 3: cosh(2) and the double-value identities

Enter 2: e² ≈ 7.389056, e⁻² ≈ 0.135335, so cosh(2) = (7.389056 + 0.135335) / 2 = 3.762196 and sinh(2) = 3.626860, tanh(2) ≈ 0.964028. Test the doubling identity against Example 1: 2cosh²(1) − 1 = 2 × 2.381098 − 1 = 3.762196 — matching cosh(2) exactly as the identity promises, and 2 sinh(1) cosh(1) = 2 × 1.175201 × 1.543081 ≈ 3.626860 = sinh(2). These cross-checks are the practical way to trust a calculator: independent routes to the same number.

Example 4: A larger input, cosh(5)

Enter 5: e⁵ ≈ 148.413159, e⁻⁵ ≈ 0.006738. The decaying exponential is already almost nothing, so cosh(5) ≈ (148.413159 + 0.006738) / 2 ≈ 74.209949, essentially e⁵ / 2, and sinh(5) ≈ 74.203211 is barely smaller. tanh(5) ≈ 0.999909, a hair under 1. The lesson generalises: for large positive x, cosh and sinh both behave like eˣ / 2, and their difference, e⁻ˣ, fades away. It is also why the tool caps |x| at 700 — e⁷⁰⁰ is astronomically large but still representable, while e⁷¹⁰ overflows ordinary double-precision arithmetic entirely, and the tool says so rather than printing infinity dressed as an answer.

The Catenary: cosh in the Physical World

Hang a chain between two posts and photograph it. The curve in the photograph is not a parabola, though Galileo guessed it was. It is the catenary, y = a × cosh(x / a), where a is a constant set by the chain's weight and the tension pulling its ends. The Gateway Arch in St. Louis is an inverted catenary, chosen because that shape channels the structure's weight cleanly to the ground. Suspension bridge cables, power lines sagging between pylons, and a rope held loosely in two hands all draw cosh curves, scaled and shifted. In equations, cosh and sinh appear as the natural solutions whenever a system's rate of change is proportional to its current value in both directions — certain heat-flow, diffusion, and relativity calculations among them — because eˣ and e⁻ˣ are the atoms of growth and decay, and cosh and sinh are their even and odd combinations. Special relativity uses the hyperbolic cousins to parametrise velocity (rapidity), where tanh of the rapidity is velocity as a fraction of light speed — which is exactly why tanh's ceiling of 1 is a feature there, not a quirk.

Hyperbolic and Circular Functions, Side by Side

Because the names rhyme, students reasonably expect cosh to behave like cos with a small twist. The comparison is worth making explicit, because the differences are exactly where errors creep in. The circular functions parametrise a circle: cos and sin take an angle, repeat every 360 degrees, and stay between −1 and 1. The hyperbolic functions parametrise a hyperbola: cosh and sinh take a plain number, never repeat, and grow like eˣ / 2 in one direction. The formulas rhyme — cosh² − sinh² = 1 against cos² + sin² = 1, cosh(2x) = 2cosh²(x) − 1 against cos(2θ) = 2cos²(θ) − 1 — but note the sign flip in the identity, a pattern called Osborne's rule: translate a circular identity to a hyperbolic one by changing the sign of every product of two sinh terms. There is also a bridge through complex numbers: cosh(ix) = cos(x) and sinh(ix) = i sin(x), which is why the families are related at all, and why the angle of complex number calculator elsewhere on this site and this cosh tool are distant cousins. For everyday use, though, the practical rule is simpler than the theory: if your input is an angle from a triangle or a rotation, you want the circular family; if your input is a number from an exponential, growth, decay, or hanging-cable setting, you want this hyperbolic one, and entering it here needs no conversion of any kind.

Common Mistakes

The first common mistake is feeding cosh an angle. If a problem says 45 degrees, cosh(45) is a gigantic number (about half of e⁴⁵), not anything to do with the angle — the person almost certainly wants cos(45°). Hyperbolic inputs are plain numbers. The second mistake is the reverse: using cos where a catenary or growth problem needs cosh, a swap that drawing packages with both buttons side by side make easy. The third is misremembering the identity's sign: it is cosh² − sinh² = 1, with the minus, not the plus of the circular identity. The fourth is expecting tanh to exceed 1 for big inputs; it approaches 1 asymptotically and never arrives. The fifth is forgetting that cosh is even — cosh(−3) equals cosh(3) ≈ 10.067662 — and reporting a negative cosh, which cannot happen: cosh is never below 1 for real inputs. That last fact is a free error detector, and the tool's outputs respect it by construction.

Limits: What This Tool Does and Does Not Do

This tool computes cosh(x), sinh(x), and tanh(x) for a single real number x with |x| up to 700, directly from the exponential definitions, and shows the two exponentials, the combinations, and the identity check as its working. It does not compute the inverse functions (arcosh, arsinh, artanh), it does not handle complex inputs — cosh of a complex number is a different, two-part answer — and it does not work with angles, degrees, or radians in any form, because its functions do not take angles. It also does not plot the catenary; it returns the numbers a plot would be drawn from. Within its scope, values are given to six decimal places for ordinary inputs and in exponential notation for very large ones, and every displayed step is computed from the input you gave, not recalled from a table.

A word on accuracy: the exponentials are computed by the browser's built-in mathematics library, which is accurate to about fifteen significant decimal digits, far beyond the six places displayed. The displayed cosh is the rounded version of a much more precise internal value, and the identity check is performed on the internal values before rounding, which is why it prints exactly 1.000000 even when squaring the rounded printouts by hand lands a whisker away. Six decimal places is also the right display precision for the function's typical uses — catenary constants, relativity factors, activation values — where inputs themselves are rarely known beyond four or five significant figures. If you need more digits than the display shows, that is a job for a computer algebra system rather than a web calculator, and no answer on this page pretends otherwise.

Privacy: Your Numbers Stay in Your Browser

Every number you enter into this cosh calculator stays in your browser. Nothing is uploaded to a server, nothing is stored, and there is no account to create. The calculation runs entirely on your own device. Closing the page removes your inputs, and no one else sees them. This browser-only design is the same principle behind the other free tools on Your Smart Kit: a simple tool that does one job well, keeps your data private, and gives an answer you can check.

Frequently Asked Questions

What is cosh?

cosh(x), the hyperbolic cosine, is (eˣ + e⁻ˣ) / 2. It is the even part of the exponential function, its graph is the catenary curve of a hanging chain, and cosh(0) = 1 is its minimum value.

What is cosh(1)?

1.543081 (to six decimal places). With e¹ = 2.718282 and e⁻¹ = 0.367879, cosh(1) = (2.718282 + 0.367879) / 2. The same input gives sinh(1) = 1.175201 and tanh(1) = 0.761594.

Is cosh the same as cos?

No. cos is the circular cosine, which takes an angle and oscillates between −1 and 1. cosh is the hyperbolic cosine, which takes a plain real number, is never below 1, and grows exponentially. They share names, identities with sign changes, and little else.

Do I enter degrees or radians?

Neither. Hyperbolic functions do not take angles at all. Enter the real number x exactly as it appears in your formula.

Why does the calculator stop at |x| = 700?

Because cosh is computed from eˣ, and eˣ overflows standard double-precision arithmetic shortly past 709. The cap of 700 keeps every displayed value finite and honest, and the tool states the limit instead of returning infinity.

Can cosh be negative?

No, not for real x. cosh(x) ≥ 1 always, with equality only at x = 0. sinh and tanh can be negative — exactly when x is negative.

Related Tools on Your Smart Kit

If you found the cosh calculator useful, you may also like the other free calculators on Your Smart Kit, including the angle of complex number calculator for argument and modulus in the complex plane, the exponent calculator for powers and exponentials themselves, and the log calculator for undoing them. Each runs in your browser, free, with no sign-up, and each is built on the same principle: a simple tool that does one job well, keeps your data private, and gives an answer you can check. Try the cosh calculator at the top of this page with x = 1 first, watch the two exponentials combine, and then explore how quickly cosh grows from there — by x = 10 it already exceeds 11,000, and that steep climb is exactly the exponential character the definitions promise.

How do I check cosh(2) = 3.762196 by hand?

Use the definition with the two exponentials. For x = 2, e^2 = 7.389056 and e^-2 = 0.135335, so cosh(2) = (7.389056 + 0.135335) / 2 = 3.762196. The same check for x = 1 gives (2.718282 + 0.367879) / 2 = 1.543081. If your hand check differs in the sixth decimal, the cause is usually rounding of the exponentials, not the formula.

Why can cosh never fall below 1?

Because cosh is the average of a number and its reciprocal in exponential form. For any real x, e^x and e^-x are both positive and multiply to 1, so their average is at least 1, with equality only when both equal 1, at x = 0. That is why the chart bottoms out at 1 and rises on both sides, and why a negative cosh result for real x signals an input error.

When should I use cosh instead of cos?

Use cosh for shapes and growth described by exponentials, such as a hanging chain or a catenary curve, and use cos for angles, waves and rotations. A problem that gives degrees or a triangle angle calls for cos; a problem that gives a real number x with no angle, or a curve defined by e^x and e^-x, calls for cosh. Entering 45 into this tool therefore computes cosh(45), not the cosine of 45 degrees.

Quick-reference table for the live tests

Input x	e^x	e^-x	Tool result
1	2.718282	0.367879	cosh(1) = 1.543081
2	7.389056	0.135335	cosh(2) = 3.762196

Worked example with the live test figures

Find cosh(2) and its companion values. Step 1: e^2 = 7.389056 and e^-2 = 0.135335. Step 2: cosh(2) = (7.389056 + 0.135335) / 2 = 3.762196. Step 3: sinh(2) = (7.389056 - 0.135335) / 2 = 3.626860, and tanh(2) = 3.626860 / 3.762196 = 0.964028, matching the tool's companion values. For a different function on a single input, see the Angle of Complex Number Calculator and the Euler Phi Function Calculator.