Sinh - Value, Derivative, and Arsinh

A sinh calculator returns the value of the hyperbolic sine for any real input x, plus the matching cosh derivative, the tanh ratio, and the inverse arsinh. The hyperbolic sine is the odd part of the exponential function, so the page sits between a clean scientific calculator and a purpose-built tool: it is the exact answer to the question 'what is sinh of x?' without you having to remember the formula or evaluate the exponentials by hand.

• • Coursework in calculus and differential equations: Check your analytic work for sinh, cosh, and tanh against a reference value when an exercise walks through the (e^x - e^-x) / 2 definition.
• • Physics and engineering preliminaries: Plug x in when a problem references a Lorentz boost parameter, a catenary constant, or a hyperbolic-angle parameter.
• • Catenary and suspension curve checks: Compute sinh(x) to feed into the y = a cosh(x / a) form of a hanging cable, and use inverse mode to recover a parameter from a target height and span.
• • Forward and inverse mode for arsinh: Use the inverse option to solve sinh(x) = y for x = ln(y + sqrt(y^2 + 1)).

The hyperbolic sine belongs to a small family of six functions - sinh, cosh, tanh, coth, sech, csch - that share identities with the ordinary sine and cosine, except the unit circle is replaced by the unit hyperbola x^2 - y^2 = 1. Treating sinh as the primary function and cosh, tanh, and arsinh as supporting outputs lets this single page stand in for almost any forward or inverse use a reader needs.

Because the value, derivative, and inverse all return together, the page is useful for a quick check, for stepping through a problem in lecture, and for sanity-testing code that calls the same exponentials.

When the value feeds into a hanging-cable problem and you need the matching cosh-based curve shape, the Catenary Curve Calculator page handles the geometry from the same sinh output.

The page runs the closed-form definition of sinh on the value of x you type and returns the supporting hyperbolic values from the same pair of exponentials. There is no lookup table or approximation; the result is what you would get from a scientific calculator with an e^x key.

• x: Real argument of the hyperbolic sine. The form accepts any real number; very large |x| will overflow IEEE 754 double precision and the page will show Infinity rather than a finite value.
• e^x: First exponential term, computed with the host's built-in exp function for full double-precision.
• e^-x: Second exponential term, computed from -x in the same way. The page keeps both terms visible so the subtraction is transparent.
• sinh(x): Half the difference of the two exponentials. Always odd, always defined for real x.
• cosh(x): Half the sum of the two exponentials, returned as the derivative of sinh and as a standalone value for any follow-up calculation.

The same pair of exponentials drives every supporting output. cosh is the sum, so the page returns d/dx sinh(x) = cosh(x) for free, and tanh is the ratio. In inverse mode the page applies arsinh(x) = ln(x + sqrt(x^2 + 1)) to the input you typed.

According to Wolfram MathWorld, the hyperbolic sine is defined as sinh(z) = (e^z - e^(-z)) / 2 and its derivative with respect to z is cosh(z).

According to Wikipedia (Inverse hyperbolic functions), the inverse hyperbolic sine is given in closed form by arsinh(x) = ln(x + sqrt(x^2 + 1)) for every real x, with derivative 1 / sqrt(x^2 + 1).

For a problem that uses the inverse trig family on the same input, the Arctan Calculator page covers arctan, arctan of a slope, and the matching radian-to-degree conversion in one place.

Four short ideas are enough to make sense of the value, the derivative, and the inverse that the page returns.

These four ideas are the level of detail a textbook or lecture problem is likely to ask about. For the identity cosh^2 x - sinh^2 x = 1, the page lets you read both numbers at the same time and confirm the identity in the displayed digits.

To compare the inverse hyperbolic identity with the analogous identity from the unit circle, the Arccos Calculator page returns arccos values that pair with arsinh on the same input.

Five short steps cover the most common case, the inverse case, and the troubleshooting cases where the input is wrong or out of range.

1. 1 Enter x in the real field: Type any real number, positive or negative. The default of x = 1 returns the standard 1.17520119... value so you can sanity-check the page on the first load.
2. 2 Pick a mode: Forward mode is the default and returns sinh(x). Switch to inverse mode when you already have a target sinh value and want to recover the x that produced it.
3. 3 Read the primary value: The black result tile at the top of the panel shows sinh(x) in forward mode, or x = arsinh(input) in inverse mode. The number updates as you type.
4. 4 Check the supporting outputs: cosh(x) (which doubles as the derivative), tanh(x), and the two exponentials e^x and e^-x are returned in the rows underneath, ready to verify the formula or chain into the next problem.
5. 5 Reset or pivot to a related page: Hit Reset to go back to x = 1. If the next step is a catenary, an angle conversion, or a coordinate transform, the related pages at the bottom of the article handle that.

Try x = 2 in forward mode. The page returns sinh(2) = 3.6268604078, cosh(2) = 3.7621956911, tanh(2) = 0.9640275801, and the two exponentials. Now switch to inverse mode, leave 2 in the box, and the page returns x = 1.4436354752 - that is arsinh(2).

If your next step is to convert the hyperbolic angle into a circular angle, the Arcsin Calculator page returns the arcsin of the unit-hyperbola y-coordinate in degrees or radians.

These benefits come up in homework, in research, and in any workflow where the answer has to be reproducible.

• • Skip the exponentials by hand: Two exponentials, a subtraction, and a division are error-prone on paper. The page keeps full double-precision, so the result is what a scientific calculator would show.
• • Read the derivative for free: Because d/dx sinh(x) = cosh(x), the page shows cosh next to the primary value, covering a follow-up question in a calculus problem.
• • Cover the whole hyperbolic set in one pass: sinh, cosh, tanh, and arsinh come back together, useful when the same input feeds two or three later lines of a derivation.
• • Switch straight into inverse mode: When the problem gives you a target sinh value, the inverse mode returns the x that produces it using the closed form arsinh(x) = ln(x + sqrt(x^2 + 1)).
• • Stay close to the source definition: The e^x and e^-x components are visible in the results panel, so the user can confirm the subtraction in the formula is honest.

The page is a useful check, not a replacement for understanding the formula. Use it to confirm a homework answer or to sanity-check the value you are about to paste into a longer program.

When the value sits inside a larger e^x expression and you want to confirm the underlying exponent before you add more terms, the Exponent Calculator page runs the e^x component by itself.

The closed form is identical in every case, but a handful of factors change how the result should be read.

• • The page is the real-input case. For a complex argument, sinh has its own definitions and branch cuts, and the closed form here is not the right tool.
• • The result is dimensionless. If the input x is supposed to carry units (meters, seconds, radians), the user has to carry them outside the page.
• • Inverse mode is arsinh, not 1 / sinh. The multiplicative inverse 1 / sinh(x) is csch(x) and belongs on a different page.

According to Wikipedia (Hyperbolic functions), the hyperbolic sine is the odd part of the exponential function with the closed form sinh(x) = (e^x - e^(-x)) / 2, and it satisfies the identity cosh^2(x) - sinh^2(x) = 1.

If the input x carries a unit such as meters and you need the same hyperbolic value expressed in a different length unit, the Distance Converter page converts the unit without touching the arithmetic.