Inverse Hyperbolic Sine Calculator

Inverse hyperbolic sine is the function that returns the number y whose hyperbolic sine equals x. It is written sinh⁻¹(x), arsinh(x), arcsinh(x) or asinh(x), and it reverses sinh: sinh(0.8814) = 1, so sinh⁻¹(1) = 0.8814.

The hyperbolic sine is sinh(y) = (eʸ − e⁻ʸ)/2. Solving that equation for y gives a closed form with a natural logarithm:

• Input: any real number, positive, negative or 0.
• Output: any real number, with the same sign as x.
• Special values: sinh⁻¹(0) = 0, sinh⁻¹(1) = ln(1 + √2) = 0.8814, sinh⁻¹(3/4) = ln 2 = 0.6931.

The output is not an angle in degrees. On the unit hyperbola x² − y² = 1, the point (cosh t, sinh t) sweeps a region of area t/2 from the vertex (1, 0). sinh⁻¹ reads the height sinh t and returns t, twice that area. The name “arsinh” stands for “area sine”.

The −1 in sinh⁻¹(x) marks the inverse function, not an exponent. The reciprocal 1/sinh(x) is the hyperbolic cosecant csch(x).

To calculate inverse hyperbolic sine, square x, add 1, take the square root, add x, then take the natural log. Follow these 4 steps for x = 2:

1. Square x and add 1: 2² + 1 = 5.
2. Take the square root: √5 = 2.236068.
3. Add x: 2 + 2.236068 = 4.236068.
4. Take the natural log: ln(4.236068) = 1.443635.

Check the answer with sinh: sinh(1.443635) = (e^1.443635 − e^−1.443635)/2 = (4.236068 − 0.236068)/2 = 2, which matches x.

Fractions with a Pythagorean pattern give exact logs. For x = 3/4, √((3/4)² + 1) = 5/4, so sinh⁻¹(3/4) = ln(3/4 + 5/4) = ln 2 = 0.693147. For x = 12/5, the root is 13/5 and sinh⁻¹(12/5) = ln 5 = 1.609438.

For a negative x, compute the positive value and flip the sign: sinh⁻¹(−2) = −1.443635.

Type any real number and the calculator returns sinh⁻¹(x) as you type, with the log formula, the exact value and the steps. The steps:

1. Type x in the field. It accepts decimals (0.5), fractions (3/4), roots (√2), π, e and expressions such as 2e − 1.
2. Choose the rounding, from whole numbers to 15 decimal places.
3. Read the result, the exact log form, the 6 hyperbolic functions of the result and eʸ.
4. Switch the diagram between the unit hyperbola, the curve sinh(y) = x and the graph of sinh⁻¹.
5. Open the steps to see the log formula worked out for your value.

Every real number has a sinh⁻¹, so the calculator never shows a domain error for a valid number. Results can be copied, shared as a link, printed or kept in the history list.

Hanging cable mode uses the catenary. Enter the catenary parameter a (horizontal tension ÷ weight per metre) and a cable length s measured from the lowest point. The horizontal distance is a·arsinh(s/a): a = 50 m and s = 30 m give 50 × arsinh(0.6) = 28.44 m.

The inverse hyperbolic sine formula is sinh⁻¹(x) = ln(x + √(x² + 1)). It holds for every real x, and the log argument x + √(x² + 1) is always positive.

Where the formula comes from: set x = sinh y = (eʸ − e⁻ʸ)/2 and let u = eʸ. Then u² − 2xu − 1 = 0, so u = x + √(x² + 1) (the positive root). Taking the log gives y = ln(x + √(x² + 1)).

Related forms:

• Identity: sinh(sinh⁻¹ x) = x and sinh⁻¹(sinh y) = y for all real numbers.
• Odd function: sinh⁻¹(−x) = −sinh⁻¹(x).
• Reciprocal link: sinh⁻¹(x) = csch⁻¹(1/x) for x ≠ 0.
• Through other inverses: sinh⁻¹(x) = tanh⁻¹(x / √(x² + 1)) and, for x ≥ 0, sinh⁻¹(x) = cosh⁻¹(√(x² + 1)).
• Sum rule: sinh⁻¹ a + sinh⁻¹ b = sinh⁻¹(a√(1 + b²) + b√(1 + a²)).

The domain of inverse hyperbolic sine is all real numbers, and the range is all real numbers. sinh⁻¹(x) is defined for every x, from −∞ to ∞.

sinh y runs from −∞ to ∞ and rises the whole way, so it takes each value exactly once. No restriction is needed to invert it, unlike sin x, cosh x or sech x.

• Domain: (−∞, ∞)
• Range: (−∞, ∞)
• Monotonic: increasing over the whole domain
• Symmetry: odd, sinh⁻¹(−x) = −sinh⁻¹(x)
• Zero: sinh⁻¹(0) = 0, the only root
• Growth: sinh⁻¹(x) ≈ ln(2x) for large x, so the curve rises like a logarithm

The calculator accepts any real input. Very large values, such as 10³⁰⁰, still return a finite result: sinh⁻¹(10³⁰⁰) ≈ 691.47.

The graph of y = sinh⁻¹(x) is an S-shaped increasing curve through the origin that grows like ln(2x) for large x.

• Passes through (0, 0), (1, 0.8814) and (−1, −0.8814).
• Slope 1 at the origin, the steepest point; the slope 1/√(x² + 1) falls toward 0 as |x| grows.
• Inflection point at the origin: the curve bends up for x < 0 and down for x > 0.
• No asymptotes: the curve rises without bound, but slowly. sinh⁻¹(100) = 5.3.
• Mirror image of y = sinh x across the line y = x.

Near 0, sinh⁻¹(x) ≈ x, so the graph hugs the line y = x between −0.5 and 0.5.

The derivative of sinh⁻¹(x) is 1/√(x² + 1). The slope is 1 at x = 0 and falls toward 0 as x moves away from 0 in either direction.

Proof: let y = sinh⁻¹ x, so x = sinh y. Differentiate: 1 = cosh y · dy/dx. Since cosh y = √(1 + sinh² y) = √(1 + x²), dy/dx = 1/√(1 + x²).

• Chain rule: d/dx sinh⁻¹(u) = u′/√(u² + 1). For sinh⁻¹(3x): 3/√(9x² + 1).
• Integral it solves: ∫ dx/√(x² + 1) = sinh⁻¹ x + C, and ∫ dx/√(x² + a²) = sinh⁻¹(x/a) + C.
• Integral of sinh⁻¹: ∫ sinh⁻¹ x dx = x·sinh⁻¹ x − √(x² + 1) + C.
• Second derivative: −x/(x² + 1)^(3/2), which is 0 at the inflection point x = 0.

On a scientific calculator, press hyp, then SHIFT (or 2nd), then sin, type the value and press =. On a Casio fx-82 or fx-991MS, the keys for sinh⁻¹(2) are hyp, SHIFT, sin, 2, ), = and the screen shows 1.443635475.

1. Press hyp. The sin key now means sinh.
2. Press SHIFT (Casio, Sharp) or 2nd (Texas Instruments) to reach the inverse, sinh⁻¹.
3. Type the value, close the bracket and press =.

The degree/radian setting does not change hyperbolic results. On Casio fx-991EX and fx-570EX models, hyperbolic functions sit in the OPTN menu. On a TI-84, choose asinh( from the CATALOG (2nd, 0).

A calculator without hyp keys can use the formula: ln(2 + √(2² + 1)) = ln(4.236068) = 1.443635.

The Android (Google) Calculator has no sinh⁻¹ key, so type the formula ln(x + √(x² + 1)). For x = 1, type ln(1 + √2) and tap =: the result is 0.881373587.

1. Tap the expand arrow, or turn the phone to landscape, to show the scientific keys.
2. Tap ln. The display shows ln(.
3. Type 1, tap +, tap √ and type 2. For x = 1, x² + 1 = 2.
4. Close the bracket and tap =.

For another x, type the full form, such as ln(2 + √(2^2 + 1)) for x = 2. The angle mode (DEG or RAD) does not affect the answer.

On the iPhone Calculator, switch to Scientific, type the value, tap 2nd, then tap sinh⁻¹. Type 2, tap 2nd and tap sinh⁻¹: the display shows 1.443635475.

1. Turn the iPhone sideways, or tap the calculator icon and choose Scientific (iOS 18 and later).
2. Type 2.
3. Tap 2nd. The sinh key changes to sinh⁻¹.
4. Tap sinh⁻¹. The result appears at once.

The iPhone applies sinh⁻¹ to the number on the display, so type the value first. The Rad label has no effect on hyperbolic functions.

In Windows Calculator, choose Scientific, type the value, open Trigonometry, select 2nd and hyp, then select sinh⁻¹. For 2, the result is 1.443635475.

1. Open the menu (☰) and choose Scientific.
2. Type 2.
3. Open the Trigonometry list and select 2nd, then hyp.
4. Select sinh⁻¹. The result is 1.443635475.

The DEG, RAD and GRAD button has no effect on hyperbolic functions.

Software calls inverse hyperbolic sine ASINH or asinh, and every tool returns the same plain number. No degree conversion is needed.

• Excel and Google Sheets: =ASINH(2) returns 1.443635475.
• Python: math.asinh(2) returns 1.4436354751788103. NumPy uses np.arcsinh.
• JavaScript: Math.asinh(2) returns 1.4436354751788103.
• MATLAB and Octave: asinh(2).
• C, C++ and Java: asinh(2) in C99 and C++11. Java has no Math.asinh, so use Math.log(x + Math.sqrt(x*x + 1)).
• Wolfram Language: ArcSinh[2]; Wolfram Alpha lists the exact form log(2 + √5).

For large negative x, the plain formula ln(x + √(x² + 1)) loses digits: x + √(x² + 1) is a tiny difference of 2 big numbers. Built-in asinh functions avoid this by using −asinh(−x).

Without a calculator, use the Taylor series sinh⁻¹ x = x − x³/6 + 3x⁵/40 − 5x⁷/112 + … for |x| ≤ 1, or ln(2x) + 1/(4x²) − … for large x.

1. Small x (|x| ≤ 1): sinh⁻¹ x = Σ (−1)ⁿ (2n)! / (4ⁿ (n!)² (2n + 1)) · x^(2n+1).
2. Large x (|x| > 1): sinh⁻¹ x = ln(2x) + 1/(4x²) − 3/(32x⁴) + …
3. Exact cases: fractions p/q with p² + q² a perfect square give an exact log, such as sinh⁻¹(3/4) = ln 2 and sinh⁻¹(5/12) = ln(3/2).

Example: sinh⁻¹(0.5) ≈ 0.5 − 0.020833 + 0.002344 − 0.000349 = 0.481162. The true value is 0.481212.

The series has alternating signs, so each partial sum lands on the opposite side of the true value, and the error is smaller than the first term left out.

The inverse hyperbolic sine table lists sinh⁻¹(x) for common values of x, with exact log forms where they exist. Click a row to load it into the calculator.

Values to know: sinh⁻¹(1) = ln(1 + √2) = 0.8814, sinh⁻¹(3/4) = ln 2 = 0.6931 and sinh⁻¹(4/3) = ln 3 = 1.0986.

Negative rows are the positive rows with the sign flipped: sinh⁻¹(−2) = −1.4436.

The inverse hyperbolic sine of a negative number is negative: sinh⁻¹(−x) = −sinh⁻¹(x). sinh⁻¹(−1) = −0.8814 and sinh⁻¹(−2) = −1.4436.

sinh⁻¹ is an odd function, so its graph has point symmetry about the origin. The log formula gives the same result: ln(−1 + √2) = ln(0.414214) = −0.8814.

What is sinh⁻¹(−1)?

sinh⁻¹(−1) = ln(√2 − 1) = −0.881374. sinh(−0.881374) = −1, and ln(√2 − 1) = −ln(√2 + 1) since (√2 − 1)(√2 + 1) = 1.

arsinh inverts the hyperbolic sine and accepts every real number; arcsin inverts the circular sine and accepts only −1 to 1.

The 2 functions meet through complex numbers: arsinh(x) = −i·arcsin(ix). Near 0, both equal x minus a small cubic term: arsinh x ≈ x − x³/6 and arcsin x ≈ x + x³/6.

Inverse hyperbolic sine places latitude lines on Mercator maps, transforms skewed data that contains zeros, and measures length along hanging cables.

• Mercator maps: a latitude φ sits at height y = R·sinh⁻¹(tan φ) on the map. At 60° N, y = R·1.317, which is 1.317 Earth radii above the equator, and the map scale grows to 2×.
• Statistics and economics: the inverse hyperbolic sine transform sinh⁻¹(x) behaves like ln(2x) for large values but stays defined at 0 and for negative values, so analysts use it on income, trade and wealth data with zeros.
• Astronomy: the Sloan Digital Sky Survey reports faint objects in “asinh magnitudes”, which stay finite when the measured flux is 0 or negative.
• Cables and chains: along a hanging cable y = a·cosh(x/a), the arc length from the lowest point is s = a·sinh(x/a), so x = a·sinh⁻¹(s/a).
• Integrals: ∫ dx/√(x² + a²) = sinh⁻¹(x/a) + C appears in physics problems such as the field of a charged wire.