Inverse Hyperbolic Cosecant Calculator

Inverse hyperbolic cosecant is the function that returns the number y whose hyperbolic cosecant equals x, for any x except 0. It is written csch⁻¹(x), arcsch(x), arccsch(x) or acsch(x): csch(0.4812) = 2, so csch⁻¹(2) = 0.4812.

The hyperbolic cosecant is csch(y) = 1/sinh(y) = 2/(eʸ − e⁻ʸ). It takes every value except 0, and the inverse is a natural log:

• Input: any real number except 0.
• Output: any real number except 0, with the same sign as x.
• Special values: csch⁻¹(1) = ln(1 + √2) = 0.8814, csch⁻¹(4/3) = ln 2 = 0.6931, csch⁻¹(3/4) = ln 3 = 1.0986.
• Shortcut: csch⁻¹(x) = sinh⁻¹(1/x).

Large inputs give small outputs, csch⁻¹(x) ≈ 1/x, and inputs near 0 give large outputs: csch⁻¹(0.01) = 5.2983.

On the unit hyperbola, the tangent line at (cosh t, sinh t) crosses the y-axis at −csch t. csch⁻¹ turns that crossing into t, twice the shaded area.

To calculate inverse hyperbolic cosecant, take the reciprocal 1/x, then apply sinh⁻¹: ln(1/x + √(1/x² + 1)). Follow these 4 steps for x = 2:

1. Check that x ≠ 0, then take the reciprocal: 1/2 = 0.5.
2. Square it and add 1: 0.25 + 1 = 1.25.
3. Take the square root and add the reciprocal: 0.5 + √1.25 = 0.5 + 1.118034 = 1.618034.
4. Take the natural log: ln(1.618034) = 0.481212.

The log argument 1.618034 is the golden ratio φ = (1 + √5)/2, so csch⁻¹(2) = ln φ exactly.

Exact cases: csch⁻¹(4/3) = ln((3 + 5)/4) = ln 2, csch⁻¹(3/4) = ln((4 + 5)/3) = ln 3 and csch⁻¹(5/12) = ln((12 + 13)/5) = ln 5.

Type any number except 0 and the calculator returns csch⁻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 (4/3), roots (√2), π, e and expressions.
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 csch(y) = x and the graph of csch⁻¹.
5. Open the steps to see the log formula worked out for your value.

The calculator rejects 0 and states the reason. Results can be copied, shared as a link, printed or kept in the history list.

Mercator map mode places a latitude on a Mercator map: y = R·arcsch(cot φ), with R the radius of the globe the map is drawn from. London at 51.5° on a 10 cm globe sits 10.52 cm above the equator; the scale there is 1.61 times the scale at the equator.

The inverse hyperbolic cosecant formula is csch⁻¹(x) = ln(1/x + √(1/x² + 1)) for x ≠ 0.

Where the formula comes from: csch y = x means sinh y = 1/x. Apply the sinh⁻¹ formula to 1/x: ln(1/x + √(1/x² + 1)).

Related forms:

• Reciprocal link: csch⁻¹(x) = sinh⁻¹(1/x).
• Single fraction: csch⁻¹(x) = ln((1 + √(1 + x²))/x) for x > 0.
• Identity: csch(csch⁻¹ x) = x for x ≠ 0, and csch⁻¹(csch y) = y for y ≠ 0.
• Odd function: csch⁻¹(−x) = −csch⁻¹(x).
• Large x: csch⁻¹(x) ≈ 1/x − 1/(6x³).

The domain of inverse hyperbolic cosecant is all real numbers except 0, and the range is all real numbers except 0.

csch y = 1/sinh y is never 0, and it is undefined at y = 0. So csch⁻¹(0) has no value, and csch⁻¹ never returns 0.

• Domain: (−∞, 0) ∪ (0, ∞)
• Range: (−∞, 0) ∪ (0, ∞)
• Vertical asymptote: x = 0
• Horizontal asymptote: y = 0, as x goes to ±∞
• Monotonic: decreasing on each half
• Symmetry: odd, csch⁻¹(−x) = −csch⁻¹(x)

csch⁻¹ is the only inverse hyperbolic function whose domain and range are both “all reals except 0”.

The graph of y = csch⁻¹(x) has 2 branches that fall away from the y-axis and flatten toward the x-axis.

• Right branch: from +∞ near x = 0 down through (1, 0.8814) and (2, 0.4812) toward 0.
• Left branch: the right branch rotated 180° about the origin, through (−1, −0.8814).
• Vertical asymptote x = 0 and horizontal asymptote y = 0.
• Mirror image of y = csch x across the line y = x; csch has the same 2 asymptotes.

For large |x| the curve follows y ≈ 1/x, and near 0 it follows ±ln(2/|x|).

The derivative of csch⁻¹(x) is −1/(|x|√(1 + x²)) for x ≠ 0. The slope is negative on both branches.

Proof: csch⁻¹ x = sinh⁻¹(1/x). By the chain rule, the derivative is (−1/x²) · 1/√(1/x² + 1) = −1/(x²·√((1 + x²)/x²)) = −1/(|x|√(1 + x²)).

• Example: at x = 1, the slope is −1/√2 = −0.7071.
• Integral it solves: ∫ dx/(x√(1 + x²)) = −csch⁻¹|x| + C.
• Integral of csch⁻¹: ∫ csch⁻¹ x dx = x·csch⁻¹ x + sinh⁻¹ x + C for x > 0.

Most scientific calculators have no csch⁻¹ key, so use csch⁻¹(x) = sinh⁻¹(1/x): press hyp, SHIFT, sin, type 1 ÷ x and press =. On a Casio fx-82 or fx-991MS, csch⁻¹(2) is hyp, SHIFT, sin, 1, ÷, 2, ), = and the screen shows 0.4812118251.

1. Press hyp, then SHIFT (or 2nd), then sin for sinh⁻¹.
2. Type 1 ÷ x, the reciprocal of the value.
3. Close the bracket and press =.

On a TI-84, use asinh(1/x) from the CATALOG. Math ERROR means x = 0.

The Android (Google) Calculator has no csch⁻¹ key, so type the log formula. For x = 1, 1/x = 1 and 1/x² + 1 = 2, so 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.
4. Close the bracket and tap =.

For another x, type ln(1 ÷ x + √(1 ÷ x^2 + 1)) with your value.

On the iPhone Calculator, type the value, tap 1/x, tap 2nd, then tap sinh⁻¹. csch⁻¹(x) = sinh⁻¹(1/x), so for 2 the display shows 0.4812118251.

1. Turn the iPhone sideways, or tap the calculator icon and choose Scientific (iOS 18 and later).
2. Type 2.
3. Tap 1/x. The display shows 0.5.
4. Tap 2nd, then tap sinh⁻¹.

In Windows Calculator, choose Scientific, type the value, open Trigonometry, select 2nd and hyp, then select csch⁻¹. For 2, the result is 0.4812118251.

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

Windows Calculator has a direct csch⁻¹ key, so no reciprocal step is needed.

Excel, Python and JavaScript have no acsch, so use asinh(1/x); MATLAB and Wolfram have it built in.

• Excel and Google Sheets: =ASINH(1/2) returns 0.481211825.
• Python: math.asinh(1/2) returns 0.48121182505960347. NumPy: np.arcsinh(1/x).
• JavaScript: Math.asinh(1/2).
• MATLAB and Octave: acsch(2).
• C and C++: asinh(1.0/x).
• Wolfram Language: ArcCsch[2].

At x = 0, 1/x fails first: Excel returns #DIV/0!, Python raises ZeroDivisionError, and JavaScript gives Math.asinh(Infinity) = Infinity.

Without a calculator, use csch⁻¹ x = 1/x − 1/(6x³) + 3/(40x⁵) − … for |x| ≥ 1, or ln(2/x) + x²/4 − … for small positive x.

1. |x| ≥ 1: csch⁻¹ x = Σ (−1)ⁿ (2n)! / (4ⁿ (n!)² (2n + 1)) · x^(−(2n+1)), the sinh⁻¹ series in 1/x.
2. Small x > 0: csch⁻¹ x = ln(2/x) + x²/4 − 3x⁴/32 + …
3. Exact cases: fractions such as 4/3, 3/4, 5/12 and 12/5 give plain logs.

Example: csch⁻¹(2) ≈ 0.5 − 0.020833 + 0.002344 − 0.000349 = 0.481162. The true value is 0.481212.

The signs alternate, so the error is smaller than the first term left out.

The inverse hyperbolic cosecant table lists csch⁻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: csch⁻¹(1) = ln(1 + √2) = 0.8814, csch⁻¹(2) = ln((1 + √5)/2) = 0.4812, csch⁻¹(4/3) = ln 2 and csch⁻¹(3/4) = ln 3.

The inverse hyperbolic cosecant of a negative number is negative: csch⁻¹(−x) = −csch⁻¹(x). csch⁻¹(−1) = −0.8814 and csch⁻¹(−2) = −0.4812.

csch⁻¹ is an odd function, so the left branch of the graph is the right branch rotated 180° about the origin. Every negative number has a real csch⁻¹.

What is csch⁻¹(−1)?

csch⁻¹(−1) = ln(√2 − 1) = −0.881374. 1/x = −1, so the log argument is −1 + √2 = 0.414214.

arcsch accepts every x except 0 and returns any nonzero number; arccsc accepts only |x| ≥ 1 and returns an angle from −90° to 90°.

For large x both shrink like 1/x: arcsch(10) = 0.0998 and arccsc(10) = 0.1002 rad.

Inverse hyperbolic cosecant gives the electric and gravitational potential of a straight rod, and it solves integrals of the form ∫ dx/(x√(1 + x²)).

• Charged rod: at distance d from the middle of a rod of length 2ℓ carrying charge density λ, the potential is V = 2kλ·csch⁻¹(d/ℓ). For λ = 1 nC/m, ℓ = 10 cm and d = 5 cm, V = 2 × 8.99 × csch⁻¹(0.5) = 25.95 V.
• Far field check: for d much larger than ℓ, csch⁻¹(d/ℓ) ≈ ℓ/d, and V becomes kQ/d, the potential of a point charge Q = 2ℓλ.
• Gravity: a thin rod of mass density μ has gravitational potential −2Gμ·csch⁻¹(d/ℓ) at the same point.
• Integrals: ∫ dx/(x√(a² + x²)) = −(1/a)·csch⁻¹(|x|/a) + C appears in orbit and lens problems.