The inverse cosine function is defined only for x in the interval [-1, 1].
What this arccos calculator does
This tool computes arccos(x), also called inverse cosine or cos-1(x). If cosine tells you the ratio for a given angle, arccos works in reverse: it gives you the angle that has cosine value x.
Example: if x = 0.5, then arccos(0.5) = 60° (or π/3 radians), because cos(60°) = 0.5.
How to use the calculator
- Enter a number between -1 and 1.
- Select whether you want the primary result in radians or degrees.
- Choose your preferred decimal precision.
- Click Calculate arccos(x).
The calculator shows both radians and degrees, plus a quick cosine check so you can confirm the result.
Domain and range of arccos
Domain (valid input)
For real-number output, arccos only accepts values from -1 to 1, inclusive. Any input outside this interval does not produce a real angle.
Range (principal output)
The principal value of arccos is always in:
- [0, π] in radians
- [0°, 180°] in degrees
This is why inverse cosine returns one main angle, even though many coterminal angles have the same cosine value.
Common arccos values
| x | arccos(x) in radians | arccos(x) in degrees |
|---|---|---|
| 1 | 0 | 0° |
| 0.5 | π/3 | 60° |
| 0 | π/2 | 90° |
| -0.5 | 2π/3 | 120° |
| -1 | π | 180° |
Why arccos is useful
Inverse cosine appears in many fields:
- Geometry: finding angles in triangles from side ratios.
- Physics: resolving vectors and measuring directions.
- Computer graphics: angle calculations for 3D rotations and shading.
- Engineering: signal analysis and trigonometric modeling.
Common mistakes to avoid
- Entering a value less than -1 or greater than 1.
- Confusing arccos(x) with 1 / cos(x) (they are not the same).
- Mixing radians and degrees without converting.
- Forgetting that arccos returns the principal angle in [0, π].
Quick formula reference
If θ = arccos(x), then:
- cos(θ) = x
- θ in radians is between 0 and π
- θ in degrees = θ × 180 / π
You can use this calculator as a fast check when solving trig homework, building models, or validating angle calculations in code.