# What is the inverse cosine of x?

Finds arccos x, the angle from 0° to 180° whose cosine is x, in degrees and radians, exact as a multiple of π for the special values.

- Page: https://www.acalculator.org/math/inverse-cosine-calculator
- JSON spec: https://www.acalculator.org/math/inverse-cosine-calculator.json
- Version: c662f292890f

## Default answer

Example with the default inputs (Value x 0.5): arccos(0.5) = 60°.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| x | Value x | A number from −1 to 1: the cosine of the angle. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| angle | arccos x | The principal angle whose cos is x, in degrees. |
| radians | In radians | The same angle in radians. |
| exact | Exact angle | The angle as a multiple of π, when x is the value at a multiple of 15°. |

## Method

arccos x is the angle θ from 0° to 180° (0 to π) with cos θ = x. When x is within 10⁻¹⁴ of the cosine of a multiple of 15° in that range, the angle is that multiple exactly; otherwise it is computed in 64-bit floating point.

## Assumptions

- x is from −1 to 1; the cosine of a real angle is never outside that range.
- The answer is the principal value. Other angles with the same cosine are −θ and those angles plus whole turns.

## Worked examples

1. x = 0.5 gives angle = 1.047198, radians = 1.047198, exact = π/3 = 60° (cos 60° = 1/2). Source: OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arccos: domain [−1, 1], range [0, π]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions.
2. x = -1 gives angle = 3.141593, exact = π = 180° (cos 180° = -1). Source: OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arccos: domain [−1, 1], range [0, π]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions.
3. x = -0.866025 gives angle = 2.617994, exact = 5π/6 = 150° (cos 150° = -√3/2). Source: OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arccos: domain [−1, 1], range [0, π]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions.
4. x = 0.3 gives angle = 1.266104. Source: NIST DLMF §4.23 (inverse trigonometric functions, principal values), https://dlmf.nist.gov/4.23.

## FAQ

### What is inverse cosine?

Inverse cosine, written arccos x or cos⁻¹ x, undoes the cosine: it gives the angle whose cosine is x. Because many angles share a cosine, it gives the principal value, the one from 0° to 180°.

### What is arccos 0.5?

arccos 0.5 = 60° = π/3 radians, because cos 60° = 1/2. The angle −60° also has cosine 1/2, but it is not the principal value.

### Why does arccos 1.5 give an error?

The cosine of a real angle is always from −1 to 1, so no angle has cosine 1.5. The calculator accepts x from −1 to 1 only.

### Why is the range of arccos 0° to 180° and not −90° to 90°?

Cosine takes every value from −1 to 1 exactly once between 0° and 180°, and it is even, so cos(−θ) = cos θ. A range from −90° to 90° would give each positive value twice and never reach the negative ones.

### How do I find all the angles with a given cosine?

Take θ = arccos x. The angles with cosine x are ±θ + 360°k for any whole number k. For x = 1/2: 60°, −60°, 300°, 420°, and so on.

### How do I use arccos to find an angle in a triangle?

With three sides a, b and c, the law of cosines gives cos C = (a² + b² − c²) ÷ (2ab). Type that value here to get angle C. For sides 3, 4 and 5, cos C = 0, so C = 90°.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arccos: domain [−1, 1], range [0, π]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions (retrieved 2026-10-01)
- OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-01)
- NIST Digital Library of Mathematical Functions, §4.23 Inverse Trigonometric Functions (principal values). https://dlmf.nist.gov/4.23 (retrieved 2026-10-01)