Inverse Cosine Calculator

Enter a value between -1 and 1 to find its inverse cosine, arccos(x) — the angle whose cosine equals that value — shown in degrees, radians, gradians, and turns.

Quick Facts

Domain
-1 ≤ x ≤ 1
Cosine of a real angle never falls outside this range, so arccos is undefined elsewhere.
Principal range
0° to 180° (0 to π rad)
arccos always returns the unique angle in this half-circle.
Complementary identity
arccos(x) = 90° − arcsin(x)
Because sin(90° − θ) = cos(θ).

Your Results

Calculated
Angle (degrees)
-
arccos(x) in degrees
Angle (radians)
-
arccos(x) in radians
Angle (gradians)
-
400 gradians per full circle
Angle (turns)
-
Fraction of a full revolution

Ready

Enter a value between -1 and 1, then press Calculate.

Formula and Method for Inverse Cosine

The inverse cosine, written arccos(x) or cos⁻¹(x), is the angle whose cosine equals x. It undoes the cosine function: if cos(θ) = x, then θ = arccos(x). Because cosine is not one-to-one over all real numbers (many angles share the same cosine value), arccos is defined to return only the principal value — the unique angle between 0° and 180° (0 and π radians) — for any x in its domain, -1 ≤ x ≤ 1.

How the calculation works

Enter a value x between -1 and 1. The calculator computes θ = arccos(x) in radians using the standard inverse cosine function, then converts that single radian value into the other common angle units: degrees (θ × 180/π), gradians (θ × 200/π, since a full circle is 400 gradians), and turns (θ ÷ 2π, the fraction of one full revolution). All four numbers describe the same angle — they only differ by the unit used to express it.

Key values to know

  • arccos(1) = 0° — the angle with cosine 1 is 0.
  • arccos(0) = 90° (π/2 rad) — cosine is zero exactly at a right angle.
  • arccos(-1) = 180° (π rad) — cosine is -1 at a straight angle.
  • arccos(0.5) = 60° (π/3 rad) and arccos(-0.5) = 120° (2π/3 rad) — common triangle values.

Common mistakes

  • Entering a value outside [-1, 1]: cosine of a real angle is always between -1 and 1, so any x outside that interval has no real solution.
  • Mixing up degrees and radians: arccos(0.5) equals 60°, which is π/3 radians (about 1.047), not 60 radians — always check which unit your downstream formula expects.
  • Forgetting the range restriction: arccos only ever returns a value from 0° to 180°; if you need an angle in a different quadrant, you must adjust the result manually based on the geometry of your problem.

Frequently Asked Questions

What are the domain and range of arccos(x)?
The domain of arccos(x) is -1 ≤ x ≤ 1, because the cosine of a real angle never falls outside that interval. The principal range of the result is 0° to 180° (0 to π radians), so arccos always returns the unique angle in that half-circle whose cosine equals x.
How is arccos related to arcsin?
They are complementary: arccos(x) = 90° − arcsin(x), or in radians, arccos(x) = π/2 − arcsin(x). This follows because sin(90° − θ) = cos(θ).
Why does the result look different in degrees versus radians?
Degrees and radians are two different units for the same angle. Multiply radians by 180/π to get degrees, or multiply degrees by π/180 to get radians. This calculator shows both, plus gradians and turns, so you can use whichever unit your work requires.
What happens if I enter a value outside -1 to 1?
There is no real answer. Because cosine values are always between -1 and 1, any x outside that range has no real angle whose cosine equals it, and the calculator will flag the input as invalid.

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