The domain for arccos(x) is -1 ≤ x ≤ 1.
What is an inverse cosine calculator?
An inverse cosine calculator finds the angle whose cosine is a given number. In math notation, this is written as arccos(x) or cos-1(x). If you already know a cosine value and need the original angle, inverse cosine is the function you use.
For example, if cos(θ) = 0.5, then θ = arccos(0.5). The principal answer is 60° (or π/3 radians).
How to use this inverse cosine calculator
- Enter a cosine value x between -1 and 1.
- Select your preferred output unit: radians or degrees.
- Set the number of decimal places for rounding.
- Click Calculate arccos(x).
The calculator returns the principal inverse cosine value and also shows the equivalent angle in the other unit for convenience.
Domain and range of arccos
Domain (allowed input values)
The input to arccos must satisfy:
-1 ≤ x ≤ 1
Any value outside that interval is invalid for real-number inverse cosine.
Range (output angles)
The principal range of arccos is:
0 to π radians (equivalently, 0° to 180°)
This means each valid input x has exactly one principal arccos output within this interval.
Quick reference values
- arccos(1) = 0
- arccos(0.5) = π/3 = 60°
- arccos(0) = π/2 = 90°
- arccos(-0.5) = 2π/3 = 120°
- arccos(-1) = π = 180°
Degrees vs radians
Trigonometry often uses radians in higher math, calculus, and programming APIs. Degrees are more intuitive in geometry and day-to-day angle measurement. This tool supports both, and converts automatically.
Conversion formulas:
- Degrees = Radians × (180 / π)
- Radians = Degrees × (π / 180)
Common mistakes to avoid
- Entering values greater than 1 or less than -1.
- Confusing arccos(x) with 1 / cos(x). They are different operations.
- Forgetting that arccos returns only the principal angle (0 to π).
- Mixing degree mode and radian mode when comparing answers.
Where inverse cosine is used
Inverse cosine appears in geometry, physics, engineering, graphics, and data science:
- Finding unknown angles in triangles.
- Calculating the angle between vectors using the dot product.
- Signal processing and phase relationships.
- Computer graphics for lighting, camera orientation, and rotation.
FAQ
Why does my input return an error?
Inputs outside [-1, 1] are invalid for real arccos results. Check your value and rounding.
Can arccos have more than one angle?
Cosine is periodic, so there are infinitely many coterminal angles in full trigonometry. However, the inverse cosine function returns one principal value in [0, π].
Is this calculator accurate?
Yes. It uses JavaScript’s built-in Math.acos(), which is precise for typical web and educational calculations.