What is the inverse cosine of x?
Type a number x from −1 to 1. The inverse cosine calculator shows arccos x, the angle whose cosine is x, in degrees and radians, and the exact angle for values such as 1/2 and −√3/2.
- arccos x
- 60°
arccos(0.5) = 60°.
- In radians
- 1.047198
- Exact angle
- π/3 = 60° (cos 60° = 1/2)
arccos x: 60°. arccos(0.5) = 60°.
How to calculate
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.
Example with the default inputs (Value x 0.5): arccos(0.5) = 60°.
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.
- 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
Each example is checked against the calculator on every build.
- Value x 0.5 gives arccos x 60°, In radians 1.047198, Exact angle 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
- Value x -1 gives arccos x 180°, Exact angle 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
- Value x -0.866025 gives arccos x 150°, Exact angle 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
- Value x 0.3 gives arccos x 72.542397°.Source: NIST DLMF §4.23 (inverse trigonometric functions, principal values), https://dlmf.nist.gov/4.23
How it works
arccos x is the angle θ with cos θ = x and 0° ≤ θ ≤ 180° (0 ≤ θ ≤ π). This range holds exactly one angle for each x from −1 to 1.
Exact angles. The calculator checks the cosines of the multiples of 15° from 0° to 180°: 1, (√6 + √2)/4, √3/2, √2/2, 1/2, (√6 − √2)/4, 0, and the same values negated. If x is within 10⁻¹⁴ of one of them, the angle is that multiple of 15° exactly, and the page shows it as a multiple of π. Otherwise the angle is computed in 64-bit floating point.
Output format. The angle shows in degrees. "In radians" shows the same angle in radians as a decimal. The exact angle shows as text, such as π/3 = 60° (cos 60° = 1/2), with - for negative values.
Assumptions
- x is from −1 to 1. A value outside that range is not accepted, because no real angle has that cosine.
- The answer is the principal value; the FAQ shows how to find the other angles.
Worked examples by hand
x = 0.5. cos 60° = 1/2 and 60° is between 0° and 180°, so arccos 0.5 = 60° = π/3 ≈ 1.0471975511965976 rad.
x = −1. cos 180° = −1, so arccos(−1) = 180° = π ≈ 3.141592653589793 rad.
x = −0.8660254037844386 (−√3/2). cos 150° = −cos 30° = −√3/2, so arccos x = 150° = 5π/6 ≈ 2.6179938779914944 rad.
x = 0.3. 0.3 is not a special value, so arccos 0.3 ≈ 1.266103672779499 rad ≈ 72.5424°.
Other questions people ask
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°.