Arccos calculator
Enter a number between −1 and 1 and get the angle whose cosine it is, in degrees or radians. For example cos⁻¹(0.5) = 60°.
Fill in the fields; the answer appears here straight away.
How it's worked out
How it works
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Enter the values
Type the numbers or the formula. Decimals may use a point or a comma. No idea? Click Fill in an example.
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Instant answer
The answer appears as you type, with the most important intermediate values.
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See the working
Under "How it's worked out" you see the steps, handy for checking your own calculation.
What does arccos do?
Arccos (cos⁻¹ or acos) gives the angle that belongs to a cosine. Because several angles have the same cosine (cos 60° = cos 300°), the calculator gives the principal value, between 0° and 180°.
A common use
In a triangle with three known sides, the cosine rule gives the cosine of an angle; arccos then gives the angle itself. Only numbers from −1 to 1 have an arccos. See also the arcsin and arctan calculators.
Examples
| Input | arccos in degrees |
|---|---|
| 1 | 0° |
| 0.5 | 60° |
| 0 | 90° |
| −0.5 | 120° |
| −1 | 180° |
An angle from three sides
A triangle with sides 5, 6 and 7. To find the angle opposite the side of 7, use the cosine rule: cos C = (5² + 6² − 7²) ÷ (2 × 5 × 6) = 12 ÷ 60 = 0.2. Then C = arccos(0.2) ≈ 78.46°. Do this for the other two sides to find the remaining angles, which add up to 180°. See the cosine calculator for the forward direction.
Rounding near 1
Close to 1 and −1, arccos changes very quickly: arccos(0.9999) is about 0.81°, while arccos(0.99) is about 8.1°. A tiny rounding error in the input can shift the angle noticeably, so keep as many digits as you have.
Frequently asked questions
What is the range of arccos?
The principal value lies between 0° and 180° (0 to π radians). The cosine is negative for angles above 90°, so arccos(−0.5) = 120°.
Why is arccos of 2 an error?
Cosines only take values between −1 and 1, so there is no real angle with a cosine of 2. The calculator says so instead of giving a number.
How do I find the angle between two vectors?
Divide their dot product by the product of their lengths and take the arccos of the result. Vectors (1, 0) and (1, 1) give 1 ÷ √2 ≈ 0.7071, so the angle is 45°.
Is there a relationship with arcsin?
Yes: arcsin(x) + arccos(x) = 90°. For x = 0.5 that is 30° + 60°.