Arccos Calculator - Inverse Cosine Function
Calculate the inverse cosine (arccos) of any value in [-1, 1] and get the result in degrees or radians instantly.
Enter a value between -1 and 1, choose the output unit, and set the decimal precision to compute arccos.
Arccos Calculator - Inverse Cosine Function
Calculate the inverse cosine (arccos) of any value in [-1, 1] and get the result in degrees or radians instantly.
Valid domain: -1 ≤ x ≤ 1
About the arccos calculator
The arccosine function, written arccos(x) or cos⁻¹(x), is the inverse of the cosine function. Given a value x between -1 and 1, arccos(x) returns the angle whose cosine is x. The output is always in the principal-value range: 0° to 180° in degrees, or 0 to π in radians. This range is chosen by mathematical convention to make the function single-valued and continuous.
The arccosine arises naturally in geometry whenever you know the ratio of two sides of a right triangle and need the angle. If the adjacent side has length a and the hypotenuse has length c, then the angle between them is arccos(a/c). For example, in a right triangle with adjacent side 3 and hypotenuse 5, the angle is arccos(0.6) ≈ 53.13°.
The domain restriction x ∈ [-1, 1] comes directly from the fact that the cosine of any real angle is always between -1 and 1. Values outside this range do not correspond to any real angle, so arccos is undefined there. In the complex number system, arccos can be extended to all complex numbers, but for real-world applications the domain [-1, 1] covers every practical case.
Key reference values of arccos are worth memorizing. arccos(1) = 0° (the cosine of 0° is 1). arccos(0) = 90° (the cosine of a right angle is 0). arccos(-1) = 180° (the cosine of a straight angle is -1). arccos(0.5) = 60°, and arccos(√2/2) = 45° — both arising from the standard 30-60-90 and 45-45-90 triangles.
In physics and engineering, arccos appears in calculations involving dot products, work and energy (the work done by a force is F × d × cos(θ), so θ = arccos(W / (F × d))), optics (Snell's law inversion), and computer graphics (finding the angle between two vectors). In navigation, the great-circle distance formula uses arccos to find the angular separation between two points on a sphere given their coordinates.
The decimal precision option in this calculator is useful for matching the precision required by a specific context. Scientific work might require 8 or more decimal places, while everyday engineering might only need 2. The underlying computation is performed in full IEEE-754 double precision (about 15-17 significant decimal digits), so reducing displayed precision simply rounds the result for readability.
Arccos examples
Four reference values that cover the main points on the arccosine curve.
| Input | arccos(x) | Explanation |
|---|---|---|
| arccos(1) | 0° = 0 rad | The cosine of 0° is 1, so arccos(1) = 0°. This is the starting point of the arccos range. |
| arccos(0) | 90° ≈ 1.5708 rad | The cosine of 90° is 0, so arccos(0) = 90°. This represents a perfect right angle. |
| arccos(0.5) | 60° ≈ 1.0472 rad | Cosine of 60° equals 0.5. Arises from the 30-60-90 triangle where the adjacent side is half the hypotenuse. |
| arccos(-1) | 180° ≈ 3.1416 rad | The cosine of 180° is -1. arccos(-1) = π radians, the maximum output of the function. |
How to use the arccos calculator
- Type the value you want to find the inverse cosine of in the Input Value field. The value must be between -1 and 1.
- Choose the Output Unit: click Degrees to get the answer in degrees, or Radians to get it in radians.
- Set the Decimal Precision to control how many decimal places are shown in the result (0 to 10).
- Click Calculate Arccos. The result appears below, showing arccos(x) = result in your chosen unit.
- Click Reset to clear the fields and start a fresh calculation.
Arccos calculator FAQ
What is the domain of arccos?
The domain of arccos is [-1, 1]. Any input outside this interval is undefined for real numbers because no real angle has a cosine greater than 1 or less than -1. If you enter a value outside this range the calculator shows a domain error.
What is the range (output) of arccos?
The principal value of arccos returns angles in [0°, 180°] when expressed in degrees, or [0, π] when expressed in radians. This restricted range makes arccos a proper function with exactly one output for each input.
What is the difference between arccos and cos⁻¹?
They are the same function written in different notations. arccos(x) and cos⁻¹(x) both mean the inverse cosine. The superscript -1 notation can cause confusion because it looks like a reciprocal (1/cos), but in the context of inverse trigonometric functions it specifically means the angle whose cosine is x.
How do I find arccos of a value greater than 1?
You cannot — arccos is undefined for real inputs outside [-1, 1]. If you have a value slightly above 1 due to floating-point rounding (e.g., 1.0000001), round it to 1 before using the calculator. In complex analysis arccos can be extended beyond this domain, but that requires a different formula.
How is arccos related to arcsin?
For any x in [-1, 1], arccos(x) + arcsin(x) = 90° (or π/2 radians). This complementary relationship comes from the co-function identity: the sine of an angle equals the cosine of its complement. So if you know arcsin(x), you can find arccos(x) by subtracting from 90°.
Can I use this calculator to find angles in triangles?
Yes. In a right triangle, if you know the adjacent side and the hypotenuse, the angle is arccos(adjacent/hypotenuse). For non-right triangles, the law of cosines gives c² = a² + b² − 2ab·cos(C), so C = arccos((a² + b² − c²) / (2ab)). Enter the cosine ratio directly into this calculator to get the angle.