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Cosine Calculator
Use this cosine calculator to find cos of any angle, or work backwards with arccos (inverse cosine) to find the angle from a ratio. Switch between degrees and radians, and watch the unit circle update as you type so you can see exactly where the angle sits.
Cos is always between −1 and 1, for any angle.
For cos, enter the angle. For arccos, enter a number from −1 to 1 and the answer is the angle whose cosine is that number.
How it works
Cosine is one of the three basic trigonometric ratios, alongside sine and tangent. In a right triangle, cos of an angle is the length of the side next to the angle (the adjacent side) divided by the length of the hypotenuse, the longest side. If the adjacent side is 1 and the hypotenuse is 2, cos of that angle is 1 ÷ 2 = 0.5. On the unit circle, where the radius is 1, cos is simply the horizontal run of the radius — how far across the circle the angle reaches.
Degrees and radians are two ways of measuring the same angles. A full circle is 360 degrees, or 2π radians (about 6.2832). To convert degrees to radians, multiply by π ÷ 180; to convert radians to degrees, multiply by 180 ÷ π. So 60 degrees is π ÷ 3 radians, about 1.0472, and 1 radian is about 57.2958 degrees. School maths and everyday work usually use degrees, while calculus and physics often use radians — this calculator lets you use either, and the answer is given in the same unit you chose.
Arccos, or inverse cosine, runs the process backwards. Cos takes an angle and gives a ratio; arccos takes a ratio and gives the angle. Arccos(0.5) is 60 degrees, because cos(60°) = 0.5. Arccos only accepts numbers from −1 to 1, since no cosine is ever outside that range, and it always returns an angle between 0° and 180°. That range matters: cos repeats every 360 degrees and is symmetric, so many angles share the same cosine, and arccos picks the one between 0° and 180°. If you enter a number outside −1 to 1, this calculator shows a friendly message instead of an error.
How cos changes with the angle is easy to picture on the unit circle. At 0°, the radius points right, the horizontal run is the full radius, and cos is 1. As the angle grows, the run shrinks: cos(30°) ≈ 0.866, cos(45°) ≈ 0.7071, cos(60°) = 0.5, and at 90° the radius points straight up, the run is zero, and cos is 0. Past 90°, cos turns negative, reaching −1 at 180°, then climbs back to 0 at 270° and to 1 at 360°. Cos is a smooth wave that repeats every 360 degrees, which is why it describes anything that goes back and forth, from sound waves to the swing of a pendulum.
Common values are a good way to check your answers. Cos(0°) = 1, cos(30°) ≈ 0.866, cos(45°) ≈ 0.7071, cos(60°) = 0.5, cos(90°) = 0, cos(120°) = −0.5, cos(180°) = −1, and cos(270°) = 0. The 45° value, about 0.7071, is 1 divided by the square root of 2, and it appears whenever a square is cut along its diagonal. The 30° and 60° values come from the 30-60-90 triangle, where the sides are in the ratio 1 : √3 : 2.
The law of cosines extends cos beyond right triangles. For any triangle with sides a, b, and c, where C is the angle opposite side c, the rule is c² = a² + b² − 2ab·cos(C). When C is 90°, cos(C) is 0, the last term vanishes, and the rule becomes Pythagoras' theorem — so the law of cosines is really Pythagoras' theorem with a correction for triangles that are not right-angled. Surveyors use it to find a distance they cannot measure directly, and navigation uses it to work out how far east or west a journey goes when travelling on a bearing.
Worked example
Find cos(60°). This is the classic first cosine problem, and it comes straight from the 30-60-90 triangle.
Step 1: Picture a right triangle with angles 30°, 60°, and 90°. In this triangle the sides are in the ratio 1 : √3 : 2.
Step 2: The side next to the 60° angle is the short leg, length 1. The hypotenuse is the longest side, length 2.
Step 3: Cos is adjacent divided by hypotenuse, so cos(60°) = 1 ÷ 2 = 0.5.
Step 4: Check on the unit circle. At 60°, the horizontal run of the radius is half the radius, which is 0.5 when the radius is 1.
Try it yourself: choose "Cosine cos(x)", leave the unit on Degrees, enter 60, and press Calculate. The tool gives exactly the same answer — cos(60°) = 0.5. Then switch to "Inverse cosine arccos(x)", enter 0.5, and you get 60° back, confirming that cos and arccos undo each other.
FAQ
What is cos of an angle?
Cos of an angle is the ratio of the adjacent side to the hypotenuse in a right triangle. On the unit circle, it is the horizontal run of the radius. For example, cos(60 degrees) equals 0.5, because in a 30-60-90 triangle the side next to the 60-degree angle is half the hypotenuse. Cos starts at 1 when the angle is 0 degrees and falls to 0 at 90 degrees.
How do I switch between degrees and radians?
Use the Degrees / Radians selector above the angle box. In degrees, a full circle is 360 degrees; in radians, a full circle is 2 pi, about 6.2832 radians. To convert, multiply degrees by pi divided by 180 to get radians, or multiply radians by 180 divided by pi to get degrees. For example, 60 degrees equals pi divided by 3, which is about 1.0472 radians.
What is arccos (inverse cosine)?
Arccos is the reverse of cos: it takes a ratio and returns the angle that has that cosine. Arccos(0.5) is 60 degrees, because cos(60 degrees) equals 0.5. Arccos only accepts numbers from -1 to 1, since no cosine is outside that range, and it always returns an angle between 0 and 180 degrees. It is the tool to reach for when you know the ratio of two sides and need the angle.
Why is cos(90 degrees) equal to 0?
Cos is the horizontal run of the radius on the unit circle. At 90 degrees the radius points straight up, so there is no horizontal run at all — the run is 0. As the angle grows past 90 degrees, cos turns negative, because the radius now leans to the left of the vertical axis. Cos(180 degrees) is -1, cos(270 degrees) is 0 again, and cos(360 degrees) is back to 1.
Where is cosine used in real life?
Cosine is used wherever a direction has to be split into parts. Engineers use it to find the horizontal part of a force pushing at an angle, and the law of cosines finds the third side of any triangle from two sides and the angle between them. Navigation uses cosine to work out how far east or west a journey goes when travelling on a bearing, and sound and light waves are described with cosine curves. In construction, cos turns a sloped length into its horizontal run.