What is arccosine
Arccosine (written arccos or cos⁻¹) is the inverse of cosine: it answers the question of which angle has this cosine. Since cos(60°) = 0.5, arccos(0.5) = 60°. The output ranges from 0° to 180°.
Cosine on its own is not one-to-one — it repeats every 360° and is symmetric about 0°, so infinitely many angles share the same cosine value. To make a genuine inverse function possible, cosine is restricted to the interval from 0° to 180°, where it falls steadily from 1 down to −1. Arccosine reverses exactly that restricted piece, which is why every valid input produces exactly one answer. Most tables and calculators return the angle in radians, which the radians to degrees calculator will turn into degrees.
- arccos(1) = 0°
- arccos(0) = 90°
- arccos(0.5) = 60°
- arccos(−1) = 180°
How to use this calculator
Enter a value between −1 and 1 and read the angle in degrees. Values outside that range have no real arccosine and are rejected. To go the other way, use the cosine calculator.
The most common mistake is entering an angle instead of a cosine value. Arccosine takes a ratio — for instance the adjacent side divided by the hypotenuse in a right triangle — and hands back the angle that produces it. If an earlier step in your work leaves you with something like 1.0001 through rounding, clamp it to 1 before taking the arccosine, otherwise the input is out of range.
Output range
Arccosine returns its principal value from 0° to 180°, covering both acute and obtuse angles. It pairs with arcsine through arccos(x) = 90° − arcsin(x).
Because that range includes obtuse angles, arccosine is the natural choice for the law of cosines, where the largest angle in a triangle can exceed 90°. Arcsine cannot do this job — it only returns angles from −90° to 90°. Negative inputs give obtuse results: a cosine of −0.5 corresponds to 120°.
Frequently asked questions
- What is arccos(0.5)?
- arccos(0.5) = 60°, the principal angle whose cosine is 0.5.
- What is the output range of arccosine?
- The principal value of arccosine ranges from 0° to 180°.
- Why must the input be between −1 and 1?
- Cosine never goes outside −1 to 1, so values beyond that range have no real arccosine.
- How are arcsine and arccosine related?
- For any valid x, arccos(x) = 90° − arcsin(x).
- Does this calculator return degrees or radians?
- It returns degrees. If your work needs radians, multiply the answer by π ÷ 180 — most programming languages and calculus formulas expect radians, not degrees.