How the Arccos Calculator (Inverse Cosine) works
Arccos, written arccos(x) or cos⁻¹(x), is the inverse of the cosine function. Given a value x that represents a cosine ratio, arccos answers the question "which angle has this cosine?" Because cosine oscillates and repeats, it is not one-to-one over all angles, so mathematicians restrict its inverse to a single well-defined output range: arccos always returns the one angle θ between 0° and 180° (0 and π radians) whose cosine equals x.
Formula and method
The calculator takes your input x, checks that it falls in the valid domain −1 ≤ x ≤ 1 (cosine itself can never produce a value outside that range), and computes θ = arccos(x) using the browser's built-in Math.acos() function, which returns the result in radians. That radian value is converted to degrees with degrees = radians × (180 / π). Two supporting values are also shown: sin(θ) = √(1 − x²), which follows directly from the Pythagorean identity sin²θ + cos²θ = 1 and is always non-negative because θ stays within 0°–180°; and a verification value, cos(θ) recomputed from the resulting angle, which should equal your original x within rounding.
Common sources of error
- Value outside [−1, 1]: arccos is undefined for any x less than −1 or greater than 1, since no real angle has a cosine outside that range.
- Degrees vs. radians confusion: Math.acos() and most programming languages return radians, not degrees — always confirm which unit a downstream formula expects.
- Mixing up arccos with secant: arccos(x) is the inverse function of cosine, not the reciprocal 1/cos(x) (that is secant); the two are unrelated operations that happen to look similar.
Checking your result
A fast sanity check: plug the resulting angle back into cosine. If cos(θ) does not come back out close to your original x, something was entered or converted incorrectly — this calculator performs that check automatically and shows it as the fourth result. Also confirm the angle falls between 0° and 180°; any arccos output outside that band signals an error, since that is the entire range of the function.
Applications
Arccos shows up whenever you know a ratio of sides or a dot product and need the angle behind it: finding an interior angle of a triangle from the Law of Cosines, computing the angle between two vectors from their dot product and magnitudes (θ = arccos(a·b / (|a||b|))), navigation and surveying bearings, computer graphics and robotics for orienting objects, and physics problems involving projections and components.