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Formula Used
Principal inverse cosine: θ = arccos(x), where −1 ≤ x ≤ 1.
Degree conversion: degrees = radians × 180 ÷ π.
Gradian conversion: gradians = radians × 200 ÷ π.
Triangle mode: θ = arccos(adjacent ÷ hypotenuse).
Vector mode: θ = arccos((a · b) ÷ (|a||b|)).
General equation: θ = ±arccos(x) + 2πk.
Inverse cosine returns the principal angle matching a cosine value. Its real input domain extends from negative one through positive one. The principal output always lies between zero and pi radians.
This calculator also handles triangles, vectors, equations, and batches. Exact values appear when the input matches a familiar angle. Complex mode extends calculations beyond the ordinary real-number domain safely.
How to Use This Calculator
- Select single, triangle, vector, equation, or batch mode.
- Enter numbers, fractions, pi, or supported square-root expressions.
- Choose a preferred unit and decimal precision.
- Enable complex results only when advanced output is needed.
- Select the calculation button to view results and steps.
- Copy, export CSV, or print the finished result.
Use single mode for one cosine value or expression. Triangle mode needs an adjacent side and a valid hypotenuse. Vector mode accepts two-dimensional or three-dimensional vector components.
Equation mode finds every solution inside a chosen interval. Batch mode processes many entries and marks invalid rows clearly. Browser history stores recent summaries only on your device locally.
Example Data Table
| Input or setup | Mode | Principal result | Exact form |
|---|---|---|---|
| 0.5 | Single | 60° | π/3 |
| 0 | Single | 90° | π/2 |
| −sqrt(2)/2 | Single | 135° | 3π/4 |
| Adjacent 3, hypotenuse 5 | Triangle | 53.130102° | Approximate |
| a=(1,0), b=(0,1) | Vector | 90° | π/2 |
Common Inverse Cosine Values
| Cosine value | Degrees | Radians | Gradians |
|---|---|---|---|
| 1 | 0° | 0 | 0 gon |
| sqrt(3)/2 | 30° | π/6 | 33.3333 gon |
| sqrt(2)/2 | 45° | π/4 | 50 gon |
| 1/2 | 60° | π/3 | 66.6667 gon |
| 0 | 90° | π/2 | 100 gon |
| −1/2 | 120° | 2π/3 | 133.3333 gon |
| −1 | 180° | π | 200 gon |
Frequently Asked Questions
What is cosine inverse?
Cosine inverse, written arccos or cos⁻¹, returns an angle whose cosine equals the supplied value.
Is cosine inverse the same as secant?
No. Arccos finds an angle, while secant equals one divided by cosine.
Why must a real input stay between −1 and 1?
Real cosine values never exceed that closed interval, so other inputs require complex-number results.
What is the range of arccos?
The principal real range is zero through π radians, or zero through 180 degrees.
How are radians converted to degrees?
Multiply the radian value by 180 and divide the result by π.
Can cosine equations have multiple answers?
Yes. Equation mode uses periodic solutions and returns every root inside your chosen interval.
How does triangle mode work?
It divides the adjacent side by the hypotenuse before applying the inverse cosine.
Can I enter negative cosine values?
Yes. Negative real values produce principal angles in Quadrant II or on an axis.
Why might a calculator show a domain error?
The input may exceed the real domain, contain invalid syntax, or create an undefined ratio.