Inverse Cotangent Calculator

Inverse cotangent is the function that returns the angle whose cotangent equals a given number. Inverse cotangent is written as cot⁻¹(x), arccot(x) or acot(x), and it reverses the cotangent function: cot(45°) = 1, so cot⁻¹(1) = 45°.

Arccotangent is one of the 6 inverse trigonometric functions. It accepts every real number and returns a principal value between 0° and 180° (0 and π radians), with both end points excluded.

In a right triangle, the cotangent of an angle θ equals the adjacent side divided by the opposite side, the reciprocal of the tangent. The inverse cotangent takes that adjacent/opposite ratio and returns θ. A roof that runs 4 m for every 3 m of rise has a pitch of θ = cot⁻¹(4/3) = 36.87°.

• Input: any real number, such as a run/rise ratio.
• Output: an angle strictly between 0° and 180°.
• Link to arctan: cot⁻¹(x) = 90° − tan⁻¹(x) for every x.

The −1 in cot⁻¹(x) marks the inverse function. The reciprocal 1/cot(x) is the tangent tan(x).

To calculate inverse cotangent, take the arctangent of the reciprocal: cot⁻¹(x) = tan⁻¹(1/x) for x > 0, and add 180° for x < 0. Follow these 4 steps:

1. Find the cotangent value x. For a right triangle, x = adjacent ÷ opposite. Example: 12 ÷ 5 = 2.4.
2. Take the reciprocal: 1/2.4 = 0.4167.
3. Apply the arctangent: tan⁻¹(0.4167) = 22.62°. For a negative x, add 180° to the arctangent result.
4. Convert the unit if needed: 22.62° × π/180 = 0.3948 rad.

The formula cot⁻¹(x) = 90° − tan⁻¹(x) needs no case split: cot⁻¹(2.4) = 90° − 67.38° = 22.62°, and cot⁻¹(−1) = 90° − (−45°) = 135°.

At x = 0, the reciprocal is undefined, but cot⁻¹(0) = 90° (π/2): cot(90°) = cos 90° / sin 90° = 0.

Enter any real number, or the adjacent and opposite sides, and the calculator returns cot⁻¹(x) as you type. The steps:

1. Pick an input method: Decimal / x for a cotangent value, or Adj / Opp for 2 sides of a right triangle.
2. Type the value. The field accepts decimals (2.4), fractions (12/5), roots (√3), π and expressions.
3. Choose the result unit: degrees, radians, gradians, turns, arcminutes, arcseconds, milliradians, microradians or multiples of π.
4. Choose the rounding, from whole numbers to 15 decimal places.
5. Read the angle in all 9 units, the exact value, the triangle, the unit circle, the graph and the worked steps.

The calculator uses the 0° to 180° convention, the same one as Excel's ACOT. Results can be copied, shared as a link, printed or kept in the history list.

RC phase mode finds the phase angle of a series RC circuit, φ = arccot(ωRC) with ω = 2πf. Enter R in ohms, C in microfarads and f in hertz. 1 kΩ, 0.1 μF and 1 kHz give ωRC = 0.6283 and φ = 57.86°; at the cut-off frequency, 1591.5 Hz, φ = 45°.

The inverse cotangent formula is θ = cot⁻¹(adjacent / opposite). θ is the angle, the adjacent side touches θ, and the opposite side faces θ.

Related forms of the formula:

• Reciprocal form: cot⁻¹ x = tan⁻¹(1/x) for x > 0, and tan⁻¹(1/x) + 180° for x < 0.
• Complement rule: cot⁻¹ x + tan⁻¹ x = 90° for every real x.
• Identity: cot(cot⁻¹ x) = x for every x, and cot⁻¹(cot θ) = θ for 0° < θ < 180°.
• Derivative: d/dx cot⁻¹ x = −1/(1 + x²).
• Integral: ∫ cot⁻¹ x dx = x·cot⁻¹ x + ½ ln(1 + x²) + C.

Example: a wheelchair ramp runs 12 m for 1 m of rise. cot θ = 12, so θ = cot⁻¹(12) = 4.76°.

Inverse cotangent in degrees returns an angle between 0° and 180°. cot⁻¹(1) = 45°, cot⁻¹(2) = 26.57° and cot⁻¹(−1) = 135°.

Positive inputs give angles below 90°, negative inputs give angles above 90°, and cot⁻¹(0) = 90°. Large positive numbers approach 0°: cot⁻¹(100) = 0.57°.

To convert radians to degrees, multiply by 180/π: cot⁻¹(2) = 0.4636 rad × 57.2958 = 26.57°, or 26° 33′ 54.2″ in degrees, minutes and seconds.

Common values in degrees:

• cot⁻¹(√3) = 30°
• cot⁻¹(1) = 45°
• cot⁻¹(1/√3) = 60°
• cot⁻¹(0) = 90°
• cot⁻¹(−1/√3) = 120°
• cot⁻¹(−1) = 135°
• cot⁻¹(−√3) = 150°

Inverse cotangent in radians returns an angle between 0 and π (0 and 3.1416). cot⁻¹(1) = π/4 = 0.7854 rad and cot⁻¹(0) = π/2 = 1.5708 rad.

On the unit circle, cot θ = cos θ / sin θ, the ratio of the x- and y-coordinates. The cotangent line y = 1 makes the value visible: the ray at angle θ meets that line at x = cot θ.

To convert degrees to radians, multiply by π/180: 135° × π/180 = 3π/4.

• cot⁻¹(√3) = π/6 = 0.5236 rad
• cot⁻¹(1) = π/4 = 0.7854 rad
• cot⁻¹(1/√3) = π/3 = 1.0472 rad
• cot⁻¹(−1) = 3π/4 = 2.3562 rad
• cot⁻¹(−√3) = 5π/6 = 2.618 rad

The domain of inverse cotangent is all real numbers, and the range is 0° < θ < 180° (0 < θ < π).

Cotangent takes every real value between 0° and 180°, so every number has an arccotangent. The end points 0° and 180° are excluded: cot 0° and cot 180° are undefined.

• Domain: (−∞, ∞)
• Range: (0, π)
• Monotonic: decreasing over the whole domain
• Symmetry: cot⁻¹(−x) = π − cot⁻¹(x)
• Horizontal asymptotes: y = 0 as x → ∞ and y = π as x → −∞

Some software uses a second convention, (−90°, 90°], where cot⁻¹(x) = tan⁻¹(1/x) for every x ≠ 0. The 2 conventions agree for x > 0 and differ by 180° for x < 0: cot⁻¹(−1) is 135° in the first and −45° in the second.

Scientific calculators have no cot⁻¹ key, so enter tan⁻¹(1 ÷ x) and add 180° when x is negative. On a Casio fx-991, the keys for cot⁻¹(2) are SHIFT, tan, 1 ÷ 2, ), = and the screen shows 26.56505118.

1. Set the angle mode: D for degrees, R for radians.
2. Press SHIFT tan (Casio) or 2nd TAN (TI) for tan⁻¹.
3. Type 1 ÷ x, close the bracket and press =.
4. For a negative x, add 180 (or π in radian mode).

A shortcut without the case split: type 90 − tan⁻¹(x). For x = −2: 90 − (−63.43) = 153.43°.

For x = 0, the reciprocal 1 ÷ 0 gives Math ERROR; the answer is 90°.

On the Android (Google) Calculator, open the scientific keys, tap INV, tap tan⁻¹, type 1 ÷ x and tap =. cot⁻¹(2) = tan⁻¹(1 ÷ 2) returns 26.56505118 in DEG mode.

1. Tap the expand arrow to show the scientific keys.
2. Check the mode label: DEG for degrees.
3. Tap INV, then tap tan⁻¹.
4. Type 1 ÷ 2 and tap =.

For a negative x, add 180 to the result: cot⁻¹(−2) = −26.565 + 180 = 153.435°.

On the iPhone Calculator, switch to Scientific, compute 1 ÷ x, then tap 2nd and tan⁻¹. For cot⁻¹(2), type 1 ÷ 2 =, tap 2nd, tap tan⁻¹: the display shows 26.56505118.

1. Turn the iPhone sideways, or choose Scientific from the calculator icon menu.
2. Check that Rad is not shown on the display (degrees).
3. Type 1 ÷ 2 and tap =. The display shows 0.5.
4. Tap 2nd, then tap tan⁻¹.

The 1/x key in Scientific mode replaces the division: type 2, tap 1/x, then 2nd and tan⁻¹.

Windows Calculator has a cot⁻¹ key: choose Scientific, open Trigonometry, select 2nd, then select cot⁻¹. The walkthrough below uses 1 ÷ x with tan⁻¹, the method that works on every calculator.

1. Open the menu (☰) and choose Scientific.
2. Check the angle button: DEG.
3. Type 1 ÷ 2 and press =.
4. Open Trigonometry, select 2nd, then select tan⁻¹.

With the cot⁻¹ key: type 2, open Trigonometry, select 2nd and select cot⁻¹. The result is 26.56505118 for 2 and 153.43494882 for −2.

Excel and Google Sheets have ACOT; most programming languages do not, so use π/2 − atan(x). Results come out in radians.

• Excel and Google Sheets: =ACOT(2) returns 0.4636; =DEGREES(ACOT(2)) returns 26.565. ACOT uses the 0 to π range.
• Python: math.pi/2 - math.atan(2) returns 0.4636. math.atan2(1, x) gives the same 0 to π convention.
• JavaScript: Math.PI / 2 - Math.atan(2) returns 0.4636.
• MATLAB and Octave: acot(2) and acotd(2). For negative x, MATLAB returns a negative angle: acotd(−1) = −45, not 135.
• Wolfram Alpha: ArcCot[2]. Wolfram uses the (−π/2, π/2] convention: ArcCot[−1] = −π/4.

Check the convention before mixing tools. For x < 0, add π (180°) to a negative result to switch to the 0 to π range.

Without a calculator, use exact values for special angles, or the arctangent series of 1/x.

1. Special values: cot 30° = √3, cot 45° = 1 and cot 60° = 1/√3, so cot⁻¹(√3) = 30°, cot⁻¹(1) = 45° and cot⁻¹(1/√3) = 60°.
2. Series for x ≥ 1: cot⁻¹ x = 1/x − 1/(3x³) + 1/(5x⁵) − 1/(7x⁷) + …
3. Series for |x| ≤ 1: cot⁻¹ x = π/2 − (x − x³/3 + x⁵/5 − …).

Example: cot⁻¹(2) ≈ 0.5 − 0.041667 + 0.00625 − 0.001116 = 0.463467 rad = 26.555°. The true value is 0.463648 rad (26.565°), and each extra term halves the error.

Large inputs converge fastest: for x = 10, the first term 0.1 rad already lands within 0.0004 rad of cot⁻¹(10) = 0.09967.

The graph of y = cot⁻¹(x) is a decreasing curve that falls from y = π on the left to y = 0 on the right and crosses the y-axis at (0, π/2).

• Passes (−1, 135°), (0, 90°) and (1, 45°).
• Horizontal asymptotes y = π (x → −∞) and y = 0 (x → ∞); the curve never touches either line.
• Slope −1/(1 + x²): steepest at x = 0 with slope −1.
• Point symmetry about (0, π/2), the inflection point.
• The arctangent graph turned upside down and shifted up by π/2.

The curve is continuous with no gaps, the reason the 0 to π convention is the standard one in calculus.

The inverse cotangent table lists cot⁻¹(x) for common values of x, in degrees and radians. Click a row to load it into the calculator.

Exact values to know: cot⁻¹(√3) = π/6, cot⁻¹(1) = π/4, cot⁻¹(1/√3) = π/3, cot⁻¹(0) = π/2 and cot⁻¹(−1) = 3π/4.

The 15° and 75° rows use 2 + √3 and 2 − √3, the cotangents of π/12 and 5π/12.

The inverse cotangent of a negative number is an obtuse angle: cot⁻¹(−x) = 180° − cot⁻¹(x). cot⁻¹(−1) = 135°, cot⁻¹(−√3) = 150° and cot⁻¹(−2) = 153.43°.

The angle lands in quadrant II, between 90° and 180°, where cosine is negative and sine is positive, so the cotangent cos/sin is negative.

What is cot⁻¹(−1)?

cot⁻¹(−1) = 135° = 3π/4 rad. cot(135°) = cos 135° / sin 135° = (−√2/2) / (√2/2) = −1. Tools that use the (−90°, 90°] convention, such as MATLAB and Wolfram Alpha, return −45° for the same input.

arccot takes adjacent/opposite and returns 0° to 180°; arctan takes opposite/adjacent and returns −90° to 90°. The 2 always add up to 90°.

For x = 2: tan⁻¹(2) = 63.43° and cot⁻¹(2) = 26.57°, and 63.43° + 26.57° = 90°. Arccotangent pairs with arccosine in shape: both decrease and both live between 0° and 180°.

Inverse cotangent turns a horizontal distance divided by a height into an angle, so it fits any problem stated as run over rise.

• Sun elevation: a 2 m pole that casts a 3 m shadow has the sun at cot⁻¹(3/2) = 33.69° above the horizon.
• Roof pitch: a 12 in run per 5 in rise gives a pitch of cot⁻¹(12/5) = 22.62°.
• Camera lenses: a 36 mm sensor behind a 50 mm lens gives a horizontal field of view of 2·cot⁻¹(2·50/36) = 39.6°.
• Ramps and roads: a grade of 1 in 20 equals cot⁻¹(20) = 2.86°.
• AC circuits: the phase angle of a series RC circuit has size cot⁻¹(ωRC), with ω the angular frequency, R the resistance and C the capacitance.