The arccos calculator returns the angle whose cosine is the value you enter, in degrees or radians, with the exact form shown wherever one exists. Its defining property — and the reason it deserves a page separate from arcsin and arctan — is its range: [0, π], or 0° to 180° inclusive.
Arb Digital built the three inverse trigonometric tools as separate pages because that range difference is not a technicality. Arccosine is the only one of the three whose output spans the complete set of angles a triangle's interior angle can take. That single property makes it the safe choice in a whole class of problems where arcsine quietly loses the answer, and it is why the law of cosines never produces an ambiguous triangle while the law of sines routinely does.
What This Arccos Calculator Does
Enter a cosine value between −1 and 1 and it returns the principal arccosine in both degrees and radians, the exact form at the standard values, and the arcsine of the same input for direct comparison. That comparison panel is deliberate: seeing arccos(−0.5) = 120° sitting next to arcsin(−0.5) = −30° makes the range difference concrete in a way that a written definition does not.
Inputs outside [−1, 1] are rejected with an explanation rather than a NaN. Cosine is the horizontal coordinate of a point on a circle of radius 1, so it cannot exceed 1 or fall below −1, and an input outside that band nearly always means a ratio was built wrongly or the underlying measurements are inconsistent.
How to Use It
- Enter the cosine value. This is commonly adjacent over hypotenuse in a right triangle, a rearranged law-of-cosines expression, or a normalised dot product between two vectors.
- Pick degrees or radians. Degrees is the default. Vector and physics work usually wants radians.
- Set the decimal places to match the precision of your inputs.
- Read the principal value. Unlike arcsine, this is the complete answer for any interior angle — no supplement check is needed.
- Compare with the arcsine panel if you are deciding which inverse function a problem calls for.
The Formula: How It's Calculated
Cosine, like sine, is not one-to-one: cos 60° and cos 300° are both 0.5, and so are infinitely many other angles. To define an inverse, one branch is chosen — the stretch from 0° to 180°, where cosine falls steadily from 1 down to −1 without ever repeating a value.
So y = arccos(x) means cos y = x with y restricted to [0, π]. Domain [−1, 1], range [0, π]. Working the default input: arccos(0.5) asks which angle in that band has a horizontal coordinate of one half on the unit circle, and the answer is π⁄3 radians, or 60°. Now try the negative: arccos(−0.5) = 120°, an obtuse angle returned directly, with no adjustment required. Arccosine is a decreasing function — as the input rises from −1 to 1, the output falls from 180° to 0° — which is the opposite of arcsine's behaviour and worth holding in mind as a sanity check. The Wolfram MathWorld entry on the inverse cosine states the branch convention formally.
Why Arccosine Is the Safe Choice for Triangle Angles
A triangle's interior angle can be anything strictly between 0° and 180°. Arccosine's range is exactly [0°, 180°]. The match is perfect, and the consequence is large: when you find a triangle angle by arccosine, the value you get is the value that is actually there. There is no second candidate to consider and nothing to check.
Arcsine's range stops at 90°, so it cannot express an obtuse angle at all. Ask it for the angle in a triangle that happens to be 120° and it will return 60° instead, because both share the same sine. Nothing warns you. This is the entire mechanism behind the ambiguous SSA case documented on our law of sines calculator, and it is why the law of cosines calculator — which solves for angles through arccosine — always returns exactly one triangle. If you have a choice of route to an unknown angle, and the angle might be obtuse, take the cosine route.
The Angle Between Two Vectors
The most common professional use of arccosine has nothing to do with triangles as such. The angle between two vectors u and v is θ = arccos((u · v) ÷ (|u| |v|)): take the dot product, divide by the product of the two magnitudes, and take the inverse cosine of the result. The division normalises the dot product into [−1, 1], which is precisely arccosine's domain — the two fit together by design rather than by coincidence.
The range fits too. The angle between two directions is naturally reported between 0° and 180°: zero when they point the same way, 90° when they are perpendicular and the dot product is zero, and 180° when they are exactly opposed. Arcsine could not express the "opposed" case at all. This formula sits underneath lighting calculations in computer graphics, similarity scoring in machine learning, joint angles in robotics, and force resolution in mechanics. When a normalised dot product comes out as 1.0000001 through floating-point drift, clamping it to 1 before the arccosine is standard practice — otherwise a perfectly ordinary calculation throws a domain error.
Why Inputs Outside −1 to 1 Have No Answer
Cosine is the x‑coordinate of a point on the unit circle, and that circle extends from −1 to 1 horizontally. Nothing on it sits further left or right, so no real angle has a cosine of 1.2. The calculator says so plainly instead of returning NaN.
The diagnostic value of that error is high. In a right-triangle problem, a cosine above 1 means the adjacent side came out longer than the hypotenuse, which is geometrically impossible — check which side you called the hypotenuse using the Pythagorean theorem calculator. In a law-of-cosines problem, it means the three sides violate the triangle inequality and do not form a triangle at all. In vector work, it usually means the normalisation step was skipped or a magnitude was computed wrongly. Each cause is easy to find once you know the error is telling you about your data rather than about the calculator.
Exact Values and the Complementary Identity
The exact values worth recognising run in the opposite direction to arcsine's: arccos(1) = 0, arccos(√3⁄2) = π⁄6 (30°), arccos(√2⁄2) = π⁄4 (45°), arccos(1⁄2) = π⁄3 (60°), arccos(0) = π⁄2 (90°). For negative inputs the pattern reflects around 90° rather than around zero: arccos(−1⁄2) = 120°, arccos(−√2⁄2) = 135°, arccos(−1) = 180°.
Linking the two functions is a clean identity that holds for every valid input: arcsin(x) + arccos(x) = 90°, or π⁄2 in radians. Check it against the comparison panel above with any value you like — arccos(0.5) = 60° and arcsin(0.5) = 30°, and they sum to 90°. It works even for negative inputs, where arcsine goes negative and arccosine goes obtuse, and the two still sum to 90°. That identity is the fastest way to convert between the two by hand, and a quick way to verify any result on this page. The trigonometric functions calculator takes you in the forward direction if you want to confirm a cosine value, and the angle converter covers gradians if you need them.
What Arccosine Cannot Tell You
Because cosine is an even function — cos(−θ) equals cos(θ) — arccosine permanently discards the sign of the original angle. Feed it the cosine of −60° and it returns 60°, since both angles share the cosine 0.5. Direction is lost. That is exactly the right behaviour for an unsigned quantity like the angle between two vectors or a triangle's interior angle, where a negative answer would be meaningless.
It is the wrong behaviour when the sign carries information — a bearing, a rotation direction, or the polar angle of a point that could sit below the axis. For those, work from the coordinates rather than a single ratio; the two-argument inverse tangent on our arctan calculator keeps the sign and returns the correct quadrant across a full turn. The unit circle calculator is a useful companion for seeing which angles share a cosine value in the first place. The NIST Digital Library of Mathematical Functions documents the branch structure of all six inverse functions.
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Browse All Free Tools Contact Arb DigitalCommon Mistakes to Avoid
- Expecting a negative output — arccosine never returns one. Its range starts at 0°. If you need a signed angle, you need a different function.
- Forgetting to clamp a normalised dot product — floating-point drift can push it to 1.0000001, which is outside the domain and throws an error in otherwise correct code.
- Assuming arccosine increases with its input — it decreases. A larger cosine means a smaller angle.
- Using arcsine where the angle could be obtuse — arccosine covers the full interior-angle range and arcsine does not.
- Treating cos-1 as 1÷cos — the superscript means inverse function. The reciprocal of cosine is secant.
Related Free Tools From Arb Digital
Compare the ranges on the arcsin calculator and the arctan calculator, solve triangles with the law of cosines calculator or the law of sines calculator, evaluate forwards with the trigonometric functions calculator, and see the geometry on the unit circle calculator. The full free online tools hub lists everything else.
Frequently Asked Questions
[0, π] in radians, or 0° to 180° in degrees, inclusive at both ends. It is the only inverse trigonometric function whose range covers the full span of a triangle's interior angle.
No. Its range begins at 0°, so every result is zero or positive. If your problem needs a signed angle that distinguishes clockwise from anticlockwise, use the two-argument inverse tangent instead.
Cosine is the horizontal coordinate of a point on a circle of radius 1, so it can never exceed 1 or fall below −1. An out-of-range input means the ratio was built wrongly or the underlying measurements are inconsistent.
Divide the dot product of the two vectors by the product of their magnitudes, then take the arccosine of the result. The division always produces a value in [−1, 1], which matches arccosine's domain exactly.
For any valid input x, arcsin(x) + arccos(x) = 90°, or π⁄2 radians. That identity holds for negative inputs too and is the quickest way to convert between the two by hand.
2π⁄3 radians, which is 120°. Arccosine returns obtuse angles directly, with no supplement adjustment needed.
It decreases. As the input rises from −1 to 1, the output falls from 180° to 0°. This is the reverse of arcsine and a useful sanity check on any result.
This tool is provided for education and reference. Verify any result used in engineering, graphics or navigation work against an independent calculation.
