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Arccos(x), written cos⁻¹(x), is the inverse of the cosine function: it answers "what angle has this cosine value?" Since cosine only ever produces values between -1 and 1, arccos is only defined on that same input range, and by convention it always returns an angle between 0° and 180° (its principal value range). Arccos shows up whenever you know a ratio of sides or vectors and need the angle that produced it: finding the angle between two vectors from their dot product and magnitudes, working out a triangle's angle from the Law of Cosines, computing the angle of a robotic arm joint from known link lengths in inverse kinematics, or determining the phase angle between two out-of-sync waveforms in signal processing.
How it works
Cosine takes an angle and returns the ratio of the adjacent side to the hypotenuse in a right triangle (or, more generally, the x-coordinate on the unit circle). Arccos reverses that process: given a ratio x between -1 and 1, it finds the unique angle θ in the range 0° to 180° such that cos(θ) = x. Because cosine is one-to-one (strictly decreasing) over that 0°-to-180° range, exactly one angle in that range has any given cosine value, which is why arccos can return a single unambiguous answer without needing extra context like quadrant information.
- Enter value (between -1 and 1).
- Click Calculate to see your results.
Examples
arccos(0.5)
arccos(0.5) = 60°, because cos(60°) = 0.5.
arccos(-1)
arccos(-1) = 180°, the maximum possible arccos output, since cos(180°) = -1.
arccos(0)
arccos(0) = 90°, because cos(90°) = 0 — the midpoint of arccos's output range.
Who should use it
- Finding the angle between two vectors from their dot product.
- Solving a triangle's angle with the Law of Cosines.
- Inverse kinematics for robotic arms and animation rigs.
Industry applications
- Engineering and physics
- Robotics and computer animation
- Trigonometry and pre-calculus education
Advantages
- Instant, exact results in both degrees and radians.
- Clear principal-range convention avoids ambiguity.
Limitations
- Only returns the principal value (0°-180°) — cannot directly express angles outside that range.
Common mistakes to avoid
- Entering a value outside [-1, 1], which has no valid real-number arccos.
- Expecting a negative-angle result — arccos never returns a negative angle, unlike arcsin.
- Confusing arccos(x) with secant (1/cos(x)) — they are unrelated operations despite the similar-looking notation.
Best practices
- Double-check whether your downstream calculation needs the answer in degrees or radians before using the result.
- Remember the co-function shortcut arcsin(x) + arccos(x) = 90° as a quick sanity check on paired calculations.
- When working with vectors or the Law of Cosines, confirm your cosine ratio is computed correctly before taking its arccos, since a small input error is amplified into an angle error.
Tips
- Need the inverse sine or inverse tangent instead? Use the Arcsin Calculator or Arctan Calculator.