Cos-1 Calculator

Find the angle whose cosine equals a given value. Enter x between -1 and 1 to get θ = cos⁻¹(x) instantly in degrees, radians, and gradians.

Quick Facts

Domain & range
x must satisfy −1 ≤ x ≤ 1; the result θ is always between 0° and 180°
cos⁻¹(x) is the principal inverse of cosine, restricted to this range so every valid x maps to exactly one angle.
Useful identity
cos⁻¹(−x) = 180° − cos⁻¹(x)
This symmetry follows from cosine being reflected about 90°.

Your Results

Calculated
Angle θ (preferred unit)
-
arccos(x) in your chosen unit
Angle θ (other unit)
-
Same angle, converted
Angle in gradians
-
θ × 10/9 (400 grad = full turn)
Angle classification
-
Acute, right, straight, or obtuse

Ready

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

Understanding the Inverse Cosine (cos⁻¹)

The inverse cosine, written cos⁻¹(x) or arccos(x), answers the question "which angle has a cosine of x?" It undoes the cosine function: if cos(θ) = x, then θ = cos⁻¹(x). Because cosine only ever produces values between −1 and 1, the input x must also fall in that range — there is no real angle whose cosine is, say, 1.5.

The formula

For any x with −1 ≤ x ≤ 1, the angle is computed as:

  • θ = cos⁻¹(x), returned natively in radians (as with JavaScript's Math.acos), then converted to degrees with θ° = θ_rad × 180/π, or to gradians with θ_grad = θ° × 10/9.

The result is always the principal value — the one angle between 0° and 180° (0 and π radians) whose cosine equals x. Cosine repeats every 360°, so infinitely many angles technically share a cosine value (cos(60°) = cos(−60°) = cos(300°) = 0.5), but cos⁻¹ always reports the single angle in that 0°–180° window.

Reading a few reference points

  • x = 1 → θ = 0° (zero angle)
  • x = 0.5 → θ = 60°
  • x = 0 → θ = 90° (a right angle)
  • x = −0.5 → θ = 120°
  • x = −1 → θ = 180° (a straight angle)

Right-triangle and vector use

In a right triangle, cos(θ) = adjacent / hypotenuse, so if you know both side lengths, dividing them gives x and cos⁻¹(x) recovers the angle between that side and the hypotenuse. The same idea extends to vectors: the angle between two vectors u and v is θ = cos⁻¹((u·v) / (|u||v|)), a formula used in physics, computer graphics, and cosine-similarity comparisons.

Degrees, radians, and gradians

Radians are the native unit for cos⁻¹ in mathematics and most programming languages. Degrees are more familiar for everyday geometry, and gradians (400 per full turn) turn up in surveying and some European engineering contexts. This calculator converts automatically between all three so you don't have to.

Frequently Asked Questions

What does cos⁻¹(x) mean?
cos⁻¹(x), also written arccos(x), is the inverse cosine function. It returns the angle θ, between 0° and 180°, whose cosine equals x. If cos(θ) = x, then θ = cos⁻¹(x).
Why must x be between −1 and 1?
Cosine of any real angle always produces a value between −1 and 1 inclusive. Since cos⁻¹ undoes cosine, it can only accept inputs in that same range — there is no real angle whose cosine falls outside [−1, 1].
Is cos⁻¹(x) the same as 1/cos(x)?
No. cos⁻¹(x), or arccos, is the inverse function that returns an angle. 1/cos(x) is the reciprocal of cosine, called secant, sec(x), and returns a ratio, not an angle. The "−1" in cos⁻¹ is inverse-function notation, not an exponent.
How do I convert the answer from radians to degrees?
Multiply the radian value by 180/π (about 57.2958). This calculator does that conversion automatically and shows both units, along with the equivalent in gradians.

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