Arccos Calculator
FreeCalculate the arccosine (inverse cosine) of a value in degrees or radians, with special-angle detection.
Runs entirely in your browser.
General Solution
Recent Activity
Visible to everyone. Last 20 uses across all visitors, newest first.
arccos(0.5) = 60°
1 day ago
What this tool does
This tool calculates the arccosine (inverse cosine) of a value between -1 and 1, right in your browser, in degrees or radians, with exact fraction-of-π values for common special angles and a unit circle visualization.
How It Works
By definition, θ = arccos(x) means cos(θ) = x. The principal value returned is always between 0 and π radians (0° to 180°). Converting to degrees uses θ(degrees) = θ(radians) × (180/π). Because cosine is periodic, the complete set of angles sharing the same cosine is described by the general solution θ = ±θ₀ + 2πk radians (or ±θ₀ + 360°k in degrees), where θ₀ is the principal value and k is any integer.
Problems it solves
- Finding the angle that produces a known cosine value, for trigonometry homework or engineering calculations.
- Getting exact fraction-of-π values for common special angles instead of long decimals.
- Converting an inverse cosine result between degrees and radians.
- Visualizing where an angle sits on the unit circle.
Frequently asked questions
›Why must the input be between -1 and 1?
Cosine values never fall outside that range for real angles, so there is no real angle whose cosine equals a number outside [-1, 1], which is why arccos is only defined there.
›Why does the result always fall between 0 and 180 degrees?
Cosine repeats and mirrors itself across the full circle, so infinitely many angles share the same cosine. Arccos returns just one of them, the "principal value" between 0° and 180° (0 and π radians), by convention.
›What does the general solution formula mean?
Since cosine is periodic and symmetric, every angle with the same cosine as the principal value can be written as ±θ₀ plus any whole multiple of a full turn, that is what the ±θ₀ + 360°k (or +2πk in radians) formula captures.
›What is a "special angle"?
Certain cosine values, like 0, 0.5, or √2/2, correspond to angles that have simple, exact expressions such as 0°, 60°, or 45°, rather than long decimals. This tool highlights those cases automatically when your input matches one closely enough.
›How is this different from arcsin or arctan?
They all answer a similar question for a different trig function; arccos asks "what angle has this cosine," arcsin asks the same for sine, and arctan for tangent. Each has its own valid input domain and principal output range.
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