Arccosine Calculator
Get the angle from its cosine.
Result
How it is calculated
You enter a cosine value v, a unitless ratio. The tool returns θ = arccos(v), the single angle between 0° and 180° whose cosine equals v. It is computed in radians and shown to 6 decimal places, then converted with θdeg = θrad × 180 ÷ π and shown to 4 decimal places. Cosine never leaves the range −1 to 1, so a value such as 1.01 or −3, and an empty box, give no result. The ends behave as expected: v = 1 gives 0°, v = 0 gives 90° and v = −1 gives 180°. A positive v always gives an acute angle and a negative v an obtuse one. Unlike arcsine, the result is never negative.
Worked example
Two vectors have a normalized dot product of −0.5, meaning the dot product divided by the product of their lengths. The angle between them is arccos(−0.5). The tool returns 2.094395 radians, and 2.094395 × 180 ÷ π gives 120.0000 degrees. The negative cosine already tells you the angle is obtuse, and 120° is greater than 90°. Compare arccos(0.5), which gives the acute 60°.
Common mistakes
- Entering the angle where the cosine value belongs. The box wants a ratio from −1 to 1, not 60.
- Dropping the minus sign. arccos(0.5) is 60° but arccos(−0.5) is 120°; the sign decides whether the angle is acute or obtuse.
- Forgetting to normalize a dot product. For vectors you must divide by both lengths first; the raw dot product is usually larger than 1 and will be refused.
- Expecting a negative angle like arcsine gives. The output here always runs from 0° to 180°.
- Reading the radians line as degrees. 1.047198 is 60° expressed in radians, not about one degree.
About this calculator
The arccosine function, written cos⁻¹(x) or arccos(x), is the inverse of cosine: given a ratio between −1 and 1, it returns the angle (in the range 0° to 180°, or 0 to π radians) whose cosine produces that ratio. Because cosine itself repeats every 360°, arccos is defined only on this restricted range so that each input has exactly one output — this is the "principal value" convention used throughout mathematics and engineering.
Arccos shows up constantly wherever an angle needs to be recovered from a dot product or a ratio of sides. In physics and 3D graphics, the angle between two vectors is found by taking arccos of their normalized dot product; in surveying and navigation, it appears in bearing and triangulation formulas; in mechanical engineering, it is used to solve for joint angles in linkages and robotic arms from known link lengths. It is also the natural complement to arcsine — the two are related by arccos(x) = 90° − arcsin(x).
This calculator takes a cosine value between −1 and 1 and returns the corresponding angle in both degrees and radians, handling the domain restriction automatically so you don't have to consult a unit circle by hand.
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