Compute inverse cosine for input in [−1, 1] with result in degrees or radians.
Common values of arccos
| x | arccos(x) in degrees | arccos(x) in radians |
|---|---|---|
| 1 | 0° | 0 |
| √3∕2 ≈ 0.8660 | 30° | π/6 |
| √2∕2 ≈ 0.7071 | 45° | π/4 |
| 0.5 | 60° | π/3 |
| 0 | 90° | π/2 |
| −0.5 | 120° | 2π/3 |
| −√2∕2 ≈ −0.7071 | 135° | 3π/4 |
| −√3∕2 ≈ −0.8660 | 150° | 5π/6 |
| −1 | 180° | π |
How arccos relates to cosine and arcsin
Inverse of cosine. If cos(θ) = x, then arccos(x) = θ, but the cosine function is not one-to-one on all real numbers — so arccos is defined to return the principal value, the unique angle in the interval [0, π] (that is, [0°, 180°]).
Complement identity. arccos and arcsin are tied by the equation arccos(x) = π/2 − arcsin(x). For example, arcsin(0.5) = 30°, so arccos(0.5) = 90° − 30° = 60°.
Why the range is [0, π]. Restricting the angle to [0, π] ensures arccos is a proper function — every x in [−1, 1] maps to exactly one angle. Outside that range, cosine would repeat values and the inverse would be ambiguous.
FAQ
What is arccos and when is it used?
Arccos (also written acos or cos−1) is the inverse cosine function. Given a ratio x between −1 and 1, it returns the angle θ whose cosine equals x. It is used when you know the adjacent side and hypotenuse of a right triangle (x = adjacent / hypotenuse) and need the angle between them, in vector geometry to recover the angle between two vectors from their dot product, and in physics problems involving projections.
What is the domain and range of arccos?
The domain is x ∈ [−1, 1] — the cosine of any real angle stays inside that interval, so values outside it have no real arccos. The range is [0, π] radians, equivalent to [0°, 180°]. This is called the principal branch: every valid input yields exactly one output angle inside that range.
How does arccos relate to arcsin?
The two are complementary: arccos(x) + arcsin(x) = π/2 (or 90°) for every x in [−1, 1]. So arccos(x) = π/2 − arcsin(x). For example, arcsin(0.7071) ≈ 45°, so arccos(0.7071) ≈ 45°; arcsin(0) = 0°, so arccos(0) = 90°. The identity is a direct consequence of the cofunction rule cos(θ) = sin(π/2 − θ).
Can arccos(x) be negative?
No. By definition arccos returns values only in [0, π], so the output is always between 0° and 180° inclusive — it is never negative. If you compute arccos of a negative number such as arccos(−0.5), the result is 120°, which is still positive. This is different from arcsin, whose range is [−90°, 90°] and can be negative.
What happens if x is outside [−1, 1]?
The calculator shows “Undefined”. In the real numbers, cosine never produces values above 1 or below −1, so arccos(1.5) or arccos(−2) has no real-valued answer. Such inputs typically signal a setup error — for instance, a ratio like adjacent/hypotenuse that exceeds 1 means the triangle’s dimensions are inconsistent. (Complex-valued extensions exist, but are outside this tool’s scope.)
How do I convert the result between degrees and radians?
Multiply radians by 180/π ≈ 57.2958 to get degrees; multiply degrees by π/180 ≈ 0.017453 to get radians. Example: arccos(0.5) = 60° = 60 × π/180 = π/3 ≈ 1.04719755 rad. The toggle above switches the main output instantly, and the secondary panel always displays both units.
Compute the inverse cosine (arccos, acos, cos⁻¹) of any value x in the interval [−1, 1]. The result is the angle θ whose cosine equals x, returned on the principal branch — between 0° and 180°, or 0 and π radians. Switch the output unit with one click and see both representations, including the exact π-fraction when one applies (π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π). Worked examples: arccos(0.5) = 60°, arccos(−1) = 180°, arccos(√2/2) ≈ 45°, arccos(0) = 90°. Typical uses include recovering an angle from the ratio adjacent/hypotenuse in a right triangle, or finding the angle between two vectors from their dot product. Values outside [−1, 1] are flagged as undefined since real cosine never leaves that range.