Arccotangent Calculator
Enter any number to find its arccotangent, the angle θ = cot⁻¹(x), in degrees or radians.
Updated
Arccotangent Calculator
Our Arccotangent Calculator finds the angle whose cotangent is a given number, arccot(x) or cot⁻¹(x). Type any number, a fraction such as 4/3 or a root such as √3 and the angle appears straight away in degrees or radians, to 10 decimal places. Common values give exact answers (arccot(√3) = 30° = π/6), and the calculator also checks the result and lists the other angles with the same cotangent. Every real number has an arccotangent, including 0 (arccot 0 = 90°).
Quick answer
The arccotangent of x is the angle whose cotangent is x, between 0° and 180°. For example, arccot(1) = 45° because cot 45° = 1. Enter any number above and the calculator shows the angle instantly.
- Formula: if cot θ = x, then θ = arccot(x) = 90° − arctan(x)
- Key values: arccot(√3) = 30°, arccot(1) = 45°, arccot(0) = 90°, arccot(−1) = 135°
- Domain: every real number
- Range: strictly between 0° and 180° (0 and π), never reaching either
What Is Arccotangent?
Arccotangent is the inverse of the cotangent function. Cotangent takes an angle and gives a ratio; arccotangent takes the ratio back to the angle. In a right triangle, if you know the side next to an angle and the side opposite it, arccotangent gives the angle:
θ = arccot(adjacent ÷ opp)
It is written arccot x, cot⁻¹ x, acot x or arcctg x. The ⁻¹ means "inverse function", not a power: cot⁻¹ x is not 1 ÷ cot x (that is the tangent).
Arccotangent Table of Common Values
| x | Degrees | Radians |
|---|---|---|
| 2 + √3 ≈ 3.7321 | 15° | π/12 |
| √3 ≈ 1.7321 | 30° | π/6 |
| 1 | 45° | π/4 |
| √3/3 ≈ 0.5774 | 60° | π/3 |
| 0 | 90° | π/2 |
| −√3/3 ≈ −0.5774 | 120° | 2π/3 |
| −1 | 135° | 3π/4 |
| −√3 ≈ −1.7321 | 150° | 5π/6 |
Other values have no simple exact angle, for example arccot(2) ≈ 26.5651° and arccot(0.25) ≈ 75.9638°. Large positive values give angles close to 0°, and large negative values give angles close to 180°.
The Arccotangent Graph
- Shape: a falling S-curve through (0, 90°), flattening out on both sides.
- Always decreasing: a bigger x always gives a smaller angle.
- Horizontal asymptotes: y = 0° and y = 180°, approached but never reached.
- Symmetry: arccot(−x) = 180° − arccot x, symmetric about the point (0, 90°).
- Mirror of arctan: it is the arctangent graph flipped and raised by 90°.
Two Conventions for Negative Values
For positive x everyone agrees. For negative x there are two common definitions:
| Range | arccot(−1) |
|---|---|
| 0° to 180° 90° − arctan x; this calculator and Excel ACOT | 135° |
| −90° to 90° arctan(1 ÷ x); some textbooks | −45° |
The 0° to 180° version is continuous, works at x = 0 and matches arccot x + arctan x = 90°. The two answers always differ by exactly 180°, so subtract 180° from a negative-input result here to get the other convention.
Why Arccotangent Gives One Angle
Cotangent repeats every 180°, so many angles share the same cotangent: cot 45° = cot 225° = 1. A function must give one answer, so arccotangent returns the principal value, the one between 0° and 180°. Every angle with cot θ = x is:
For cot θ = 1 between 0° and 360° that gives 45° and 225°. The calculator shows these in its "Other angles" row.
Arccotangent Rules and Identities
- Inverse: cot(arccot x) = x for every real x.
- With arctan: arccot x + arctan x = 90° (π/2).
- Positive values: arccot x = arctan(1 ÷ x) for x > 0.
- Negative values: arccot(−x) = 180° − arccot x.
- Zero: arccot 0 = 90°, because cot 90° = 0.
- Calculus: the derivative of arccot x is −1 ÷ (1 + x2).
Worked Examples
Angle in a triangle. A right triangle has an adjacent side of 4 and an opposite side of 3:
= 90° − arctan(4/3)
≈ 36.87°
Ramp angle. An accessible ramp runs 12 units for every 1 unit of rise, so cot θ = 12 and θ = arccot(12) ≈ 4.76°.
Negative values. arccot(−2) ≈ 153.4349° and arccot(−1/2) ≈ 116.5651°. A negative cotangent always gives an angle between 90° and 180°.
Arccotangent in Excel and Code
| Tool | arccot(2) in degrees |
|---|---|
| Excel / Google Sheets | =DEGREES(ACOT(2)) |
| Python | math.degrees(math.pi / 2 - math.atan(2)) |
| JavaScript | (Math.PI / 2 - Math.atan(2)) * 180 / Math.PI |
| Scientific calculator | DEG mode, 90 − tan⁻¹(2) |
Python and JavaScript have no arccot function, so use 90° − arctan x. Writing atan(1 / x) instead fails at x = 0 and gives the −90° to 90° convention for negative x.
Common Mistakes
- Reading cot⁻¹ as 1 ÷ cot. cot⁻¹ x is an angle; 1 ÷ cot θ is just tan θ.
- Mixing conventions. arccot(−1) is 135° here but −45° in some books.
- Calling arccot 0 undefined. It is 90°; only cot 0° itself is undefined.
- Expecting 0° or 180°. No finite number has an arccotangent of 0° or 180°.
- Wrong angle mode. arccot(1) is 45 in degrees but 0.7854 in radians.
Key Terms
- Inverse
- A function that undoes another; arccotangent undoes cotangent.
- Cotangent
- cos θ ÷ sin θ, or adjacent ÷ opposite.
- Range
- The possible outputs; here, between 0° and 180°.
- Asymptote
- A line the graph approaches but never reaches; y = 0° and y = 180° here.
- Arctan
- The inverse of tangent; arctan x + arccot x = 90°.
- Tangent
- 1 ÷ cot θ, often confused with cot⁻¹.
How to Use the Arccotangent Calculator
- Enter the value. Type the cotangent value x, which can be any number including 0, or press one of the buttons (√3, 1, √3/3 and −1). You can type a decimal such as 0.25, a fraction such as 4/3, or a root such as √3 or 1/√3.
- Choose degrees or radians. Pick Degrees (°) or Radians (rad) for the answer. The table below the result always shows both.
- Read and copy the result. The angle θ = arccot(x) appears as you type, between 0° and 180°, to 10 decimal places, with the exact angle for common values (for example arccot(√3) = 30° = π/6 rad). The table also checks the answer and gives the other angles with the same cotangent. Press Copy to copy the result, or Clear or Esc to start again.
Example: enter −1 and the result is 135°, or 3π/4 rad. The table shows that cot(135°) = −1 and that 315° and −45° have the same cotangent.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is arccotangent?
Arccotangent is the inverse of cotangent. arccot(x), also written cot−1(x), is the angle between 0° and 180° whose cotangent is x. For example arccot(1) = 45°, because cot 45° = 1.
What is arccot √3, arccot 1, arccot 0 and arccot(−1)?
arccot(√3) = 30° (π/6), arccot(1) = 45° (π/4), arccot(√3/3) = 60° (π/3), arccot(0) = 90° (π/2) and arccot(−1) = 135° (3π/4).
What is arccot(0)?
arccot(0) = 90°, or π/2 radians, because cot 90° = cos 90° ÷ sin 90° = 0 ÷ 1 = 0. It is defined, even though the shortcut arctan(1 ÷ x) cannot be used at x = 0.
What are the domain and range of arccotangent?
The domain is every real number. The range is strictly between 0° and 180° (0 and π); large positive x gives angles near 0° and large negative x gives angles near 180°.
Is cot⁻¹(x) the same as 1/cot(x)?
No. cot−1(x) is the inverse function and returns an angle: cot−1(1) = 45°. 1 ÷ cot(x) is simply tan(x).
Is arccot(−1) equal to 135° or −45°?
It depends on the convention. With the range 0° to 180°, used here and by Excel ACOT, arccot(−1) = 135°. Some textbooks define arccot x = arctan(1 ÷ x) with a range of −90° to 90°, which gives −45°. Both angles have a cotangent of −1.
How do I find all angles with a given cotangent?
Cotangent repeats every 180°, so use θ = arccot(x) + 180° × k for any whole number k. For cot θ = 1 between 0° and 360° that gives 45° and 225°.
What is arccot(2) in degrees?
arccot(2) ≈ 26.5651°, or about 0.4636 radians. It has no simple exact form. It equals arctan(0.5), because arccot x = arctan(1 ÷ x) for positive x.
How do you calculate arccotangent in Excel or on a calculator?
In Excel use =DEGREES(ACOT(2)) for degrees or =ACOT(2) for radians. Calculators have no arccot key, so in DEG mode work out 90 − tan−1(x). In code use Math.PI / 2 − Math.atan(x) in JavaScript or math.pi / 2 − math.atan(x) in Python, both in radians.
Is the Arccotangent Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Arccotangent Calculator to find the angle for any cotangent value in degrees or radians instantly.