Quick answer
Inverse cosine returns the angle whose cosine is x: θ = arccos(x), for −1 ≤ x ≤ 1. arccos(0.5) = 60°, or 1.0472 radians.
Enter a value between −1 and 1. The calculator returns the angle in degrees and radians.
Related tools: Cosine Calculator: cos of an angle in degrees or radians. Inverse Sine Calculator: arcsin (inverse trigonometric function) with the result in degrees and radians. Inverse Tangent Calculator: arctan (inverse trigonometric function) with the result in degrees and radians.
How to use the Inverse Cosine Calculator
- Enter value (between -1 and 1).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Value | A number | ≥ -1, ≤ 1 |
Formula and method
Formula
θ = arccos(x)
IEEE-754 double precision; shown to at most 6 decimal places.
Principal value of the inverse function. asin/atan return −90°…90° and acos returns 0°…180°.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Value = 0.5.
| Result | Value |
|---|---|
| arccos(0.5) | 60° |
| Radians | 1.047198 |
Example 2
Inputs: Value = -1.
| Result | Value |
|---|---|
| arccos(-1) | 180° |
| Radians | 3.141593 |
The first line is the headline result (arccos(0.5)). Below it the inverse Cosine Calculator lists radians, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Range
The principal value of arccos lies between 0° and 180° (0 to π). arccos(1) = 0°, arccos(0) = 90° and arccos(−1) = 180°. Outside −1 to 1 there is no real angle.
Uses
Find an angle from the sides of a right triangle (adjacent ÷ hypotenuse) or from the law of cosines. For example a 3-4-5 triangle's angle next to the side 4 is arccos(0.8) = 36.87°.
Domain, range and angles between vectors
arccos accepts −1 to 1 and returns 0° to 180°. arccos(0.5) = 60°, arccos(0) = 90° and arccos(−0.5) = 120°.
The angle between two vectors is arccos of their dot product divided by the product of their lengths.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Value: entering "2" shows "Value must be at most 1." beside that field and no result.
Choosing between the Inverse Cosine Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Cosine Calculator](/math/cosine-calculator/) | angle and unit | cos(60°) |
| [Inverse Sine Calculator](/math/inverse-sine/) | value | arcsin(0.5) |
| [Inverse Tangent Calculator](/math/inverse-tangent-calculator/) | value | arctan(1) |
| [Cosine Rule Calculator](/math/cosine-rule-calculator/) | side b, side c and included angle A | Side a |
| [Inverse Trigonometry Calculator](/math/inverse-trigonometry-calculator/) | function and value | Angle |
Background: trigonometry
In a right triangle, sin θ is the opposite side over the hypotenuse, cos θ the adjacent side over the hypotenuse and tan θ the opposite over the adjacent. Their reciprocals are cosecant, secant and cotangent. At 30° the values are sin = 0.5, cos = 0.866025 and tan = 0.577350.
Check the angle unit before you calculate: degrees convert to radians by multiplying by π/180, so 30° is about 0.5236 rad. Some values are undefined, for example tan 90°, because the cosine is zero there and the ratio would divide by zero.
Assumptions and limitations
- Value must be between -1 and 1
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for value to confirm this.
Frequently asked questions
What is arccos(−0.5)?
arccos(−0.5) = 120°, in the second quadrant.
Is arccos the same as 1/cos?
No, 1/cos is the secant.
Why only −1 to 1?
Cosine never leaves that range.
Why does arccos reject values above 1?
No real angle has a cosine above 1 or below −1.
What does the Inverse Cosine Calculator calculate?
arccos (inverse trigonometric function) with the result in degrees and radians. It returns arccos(0.5) and Radians.
What formula does the Inverse Cosine Calculator use?
θ = arccos(x). Principal value of the inverse function.
What do I need to enter in the inverse Cosine Calculator?
You enter Value.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Value must be between -1 and 1.
What is an example of a inverse Cosine result?
With value 0.5, the arccos(0.5) is 60°.
Related calculators
- Cosine Calculator — cos of an angle in degrees or radians.
- Inverse Sine Calculator — arcsin (inverse trigonometric function) with the result in degrees and radians.
- Inverse Tangent Calculator — arctan (inverse trigonometric function) with the result in degrees and radians.
- Cosine Rule Calculator — Apply the cosine rule to find a missing side or angle of a triangle from the sides and the included angle.
- Inverse Trigonometry Calculator — Arcsine, arccosine or arctangent in degrees and radians.
- Trigonometry Calculator — All six trigonometric functions of an angle.
Sources
- Standard trigonometric definitions on the unit circle (NIST DLMF §4.14)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.