Arccos Calculator
Enter a value between −1 and 1 and get its inverse cosine — the angle whose cosine equals that value — in degrees, radians, multiples of π and gradians.
arccos(0.5)
60°
- Degrees
- 60°
- Radians
- 1.047198 rad
- Radians as a multiple of π
- 0.333333 × π
- Gradians
- 66.666667 gon
arccos returns the angle between 0° and 180° (0 and π radians) whose cosine equals your value. It is the inverse of cosine on that interval.
How to use this calculator
Type the cosine value you want to invert — any number from −1 to 1, decimals and negatives included. Pick whether the headline result should show degrees or radians; the breakdown always lists both, plus the radian value written as a multiple of π and the angle in gradians. If you enter a value outside −1 to 1 the calculator tells you the input is out of range rather than returning a wrong answer, because no real angle has a cosine beyond that interval.
How the calculation works
Cosine takes an angle and returns a ratio between −1 and 1; arccos runs that in reverse. Because cosine repeats every 360° and is symmetric, it only has a true inverse when restricted to one interval — by convention 0° to 180° (0 to π radians). arccos(x) is therefore defined as the unique angle θ in [0, π] with cos θ = x, the principal value used by every scientific calculator and programming language. Degrees convert from radians by multiplying by 180/π, and gradians (also called gons, where a right angle is 100) by multiplying by 200/π. All other angles sharing the same cosine are ±θ plus whole multiples of 360°.
Worked example
Take x = 0.5. The angle between 0° and 180° whose cosine is 0.5 is 60°, so arccos(0.5) = 60° — exactly π/3 radians (≈ 1.047198) or 66.666667 gradians. You can verify it in reverse: cos(60°) = 0.5. For a negative input the answer lands in the second quadrant: arccos(−0.5) = 120° = 2π/3 radians, because cosine is negative for angles between 90° and 180°.
Frequently asked questions
What is arccos?
Arccos (also written cos⁻¹ or acos) is the inverse of the cosine function. Where cosine takes an angle and gives you a ratio, arccos takes the ratio and gives you back the angle. arccos(x) returns the unique angle between 0° and 180° (0 and π radians) whose cosine equals x. For example, arccos(0.5) = 60° because cos(60°) = 0.5.
Why does arccos only accept values between −1 and 1?
Because cosine itself only ever produces values in that range — it is the ratio of the adjacent side to the hypotenuse in a right triangle (or the x-coordinate on the unit circle), and neither can exceed the hypotenuse or radius. Asking for arccos(2) means asking "which angle has a cosine of 2?", and no real angle does, so the result is undefined.
Why does arccos only return angles between 0° and 180°?
Cosine repeats every 360° and satisfies cos(−θ) = cos(θ), so infinitely many angles share any given cosine value. A function can only return one answer, so arccos is defined on the interval where cosine passes through every value in [−1, 1] exactly once: 0° to 180°. This is called the principal value. Every other solution is ±arccos(x) plus a whole multiple of 360°.
What is the difference between arccos and cos⁻¹?
Nothing — they are two notations for the same function. cos⁻¹ is common on calculator keys and in textbooks, arccos in mathematics and programming (acos in most languages). Note that cos⁻¹(x) means the inverse function, not 1/cos(x) — the reciprocal of cosine is a different function called secant.
What are the exact arccos values worth remembering?
The unit-circle specials: arccos(1) = 0°, arccos(√3/2) = 30°, arccos(√2/2) = 45°, arccos(1/2) = 60°, arccos(0) = 90°, arccos(−1/2) = 120°, arccos(−√2/2) = 135°, arccos(−√3/2) = 150°, and arccos(−1) = 180°. In radians those are 0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6 and π.
How do arccos, arcsin and arctan relate?
They are the inverses of the three main trig functions, each restricted to its own principal range: arccos returns angles in [0°, 180°], while arcsin and arctan return angles in [−90°, 90°]. For any x in [−1, 1] they satisfy the identity arcsin(x) + arccos(x) = 90° (π/2 radians), so you can always get one from the other by subtracting from 90°.