What this calculator does
Inverse cos, written arccos or cos⁻¹, answers the question a plain cosine calculation cannot: given a cosine ratio, what angle produced it? Where cosine takes an angle and returns a ratio between -1 and 1, arccos takes that ratio and returns the angle back, restricted to the range 0° to 180° so the answer is unambiguous.
Arccos is the third of the standard inverse trigonometric functions alongside arcsin and arctan. It comes up whenever a triangle side ratio, a dot product, or a physics calculation gives you a cosine value and the actual angle is what is needed next.
The formula
Enter a value between -1 and 1, since cosine never produces a ratio outside that range. The calculator returns the angle whose cosine equals that value, in both degrees and radians, and shows the cosine of that angle recomputed as a check that it matches the value entered.
| Term | Meaning |
|---|---|
| Arccos | The inverse cosine function, also written cos⁻¹, which returns an angle from a cosine ratio. |
| Domain | The valid input range for arccos: -1 to 1. |
| Range | The angles arccos can return: 0° to 180° (0 to π radians). |
The inputs explained
| Field | What to enter |
|---|---|
| Value (the cosine ratio, between -1 and 1) | The cosine ratio, a number between -1 and 1 inclusive. |
When to use it
Finding an angle from a right triangle's sides
If the adjacent side and the hypotenuse of a right triangle are known, their ratio is the cosine of the angle between them, and arccos converts that ratio back into the angle itself.
Finding the angle between two vectors
The cosine of the angle between two vectors comes from their dot product divided by the product of their magnitudes; arccos of that result gives the angle directly.
Checking a trigonometry answer
After computing cos(θ) for a known angle, entering that result back into arccos should return the original angle, which is a quick way to check the working.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
Arccos across the full valid input range
A spread of cosine values across the entire domain arccos accepts.
| Value (cosine ratio) | Angle (degrees) | Angle (radians) |
|---|---|---|
| -1 | 180.000° | 3.1416 |
| -0.5 | 120.000° | 2.0944 |
| 0 | 90.000° | 1.5708 |
| 0.5 | 60.000° | 1.0472 |
| 0.866 | 30.003° | 0.523650 |
| 1 | 0° | 0 |
Questions
What is the domain of arccos?
Arccos only accepts inputs from -1 to 1 inclusive, because cosine itself never produces a value outside that range for any real angle. Entering a value outside -1 to 1 has no valid answer.
Why does arccos only return angles from 0° to 180°?
Cosine is not one-to-one over all angles, since cos(60°) and cos(-60°) are both 0.5, so arccos is defined to always return the angle in the 0° to 180° range to give one unambiguous answer rather than infinitely many.
How is arccos different from arcsin and arctan?
All three take a ratio and return an angle, but from different trigonometric ratios: arcsin from the sine ratio, arctan from the tangent ratio, and arccos from the cosine ratio. Their output ranges also differ: arcsin and arctan return -90° to 90°, while arccos returns 0° to 180°.
What is arccos(0.5) in degrees?
Exactly 60°, since cos(60°) = 0.5. This is one of the standard angle values worth recognising, alongside arccos(0) = 90° and arccos(1) = 0°.
For the inverse sine function, see the arcsin calculator, and for inverse tangent, see the arctan calculator.