Cos-1 Calculator - Inverse Cosine (Arccosine)
Enter a cosine value between -1 and 1 to find the angle whose cosine equals that value. The result appears in both degrees and radians, with exact fraction notation where possible. Use the triangle mode to solve for an unknown angle directly from the adjacent side and hypotenuse.
Formula
Worked example
Find the angle whose cosine is 0.5: arccos(0.5) = 60 degrees = pi/3 radians. Verify: cos(60 deg) = 0.5. For a right triangle with adjacent = 3 and hypotenuse = 5: cos(theta) = 3/5 = 0.6, so theta = arccos(0.6) = 53.13 degrees.
What is cos-1 (arccosine)?
The cos-1 function, written arccos(x) or cos^{-1}(x), is the inverse of the cosine function. It answers the question: "which angle has a cosine equal to x?" Because cosine is not one-to-one over all real numbers, the inverse is defined only on the principal range from 0 degrees to 180 degrees (0 to pi radians), which means the calculator always returns a value in that range. The input x must lie between -1 and 1, since cosine of any real angle never leaves that interval. This makes arccos useful whenever you know a ratio and need the corresponding angle, such as finding the angle in a triangle from two known sides.
The arccosine formula and how to use it
The core relationship is: if cos(theta) = x, then theta = arccos(x). In a right triangle this translates directly to theta = arccos(adjacent / hypotenuse). To use this calculator, select "Enter cosine value" mode and type any number between -1 and 1. The angle is returned in both degrees and radians. If you have two side lengths instead, switch to "Right-triangle sides" mode: enter the adjacent side and the hypotenuse, and the calculator computes the ratio for you before applying arccos. For example, a triangle with adjacent side 3 and hypotenuse 5 gives cos(theta) = 0.6, so theta = arccos(0.6) = 53.13 degrees.
Degrees vs radians and exact values
Degrees divide a full rotation into 360 equal parts. Radians relate the angle to the radius of a circle: one radian is the angle at the center subtended by an arc equal in length to the radius, and a full circle is 2*pi radians. To convert between them: degrees = radians * (180 / pi), radians = degrees * (pi / 180). For the standard arccosine values, the radian result is always a simple fraction of pi. For example, arccos(0.5) = pi/3 radians, arccos(0) = pi/2 radians, arccos(-1) = pi radians. The reference table above lists all the common exact values, which are worth memorising for exams and quick mental checks.
Properties and common mistakes
A few properties are worth knowing. First, the identity cos(arccos(x)) = x holds for all x in [-1, 1], but arccos(cos(theta)) equals theta only if theta is already in [0, pi], because of the principal range restriction. Second, arccos(-x) = pi - arccos(x), so for example arccos(-0.5) = pi - arccos(0.5) = pi - pi/3 = 2*pi/3 (120 degrees). Third, the most frequent mistake is entering a value outside [-1, 1]: this returns an error because no real angle has a cosine outside that range. Finally, cos-1 as written on some calculators means the multiplicative inverse 1/cos(x) = sec(x), not the inverse function. This calculator always uses the inverse-function interpretation.
Common arccosine values
| x (cosine) | arccos(x) in degrees | arccos(x) in radians |
|---|---|---|
| -1 | 180 deg | pi |
| -sqrt(3)/2 (-0.866) | 150 deg | 5*pi/6 |
| -sqrt(2)/2 (-0.707) | 135 deg | 3*pi/4 |
| -1/2 (-0.5) | 120 deg | 2*pi/3 |
| 0 | 90 deg | pi/2 |
| 1/2 (0.5) | 60 deg | pi/3 |
| sqrt(2)/2 (0.707) | 45 deg | pi/4 |
| sqrt(3)/2 (0.866) | 30 deg | pi/6 |
| 1 | 0 deg | 0 |
Exact values for the most frequently used cosine inputs.
Frequently asked questions
What is the difference between cos-1 and 1/cos?
They are different things, and the notation can be confusing. cos-1(x) in the context of inverse trigonometry means the arccosine function: the angle whose cosine is x. The result is an angle (in degrees or radians). By contrast, 1/cos(x) is the reciprocal of cosine, called the secant function (sec x). This calculator computes the inverse trigonometric function, not the reciprocal.
Why is the domain of arccos limited to -1 to 1?
The cosine of any real angle always produces a value between -1 and 1. No real angle can have a cosine outside that interval, so asking for arccos(2), for example, has no real answer. If you enter a value outside [-1, 1], the calculator will return an error because the result would be a complex number, which is outside the scope of this tool.
Why does arccos only return angles between 0 and 180 degrees?
Because cosine is symmetric: cos(50 deg) = cos(-50 deg) = cos(310 deg), so infinitely many angles share the same cosine. To define a true inverse function, mathematicians restrict the output to one interval where cosine is one-to-one. The standard choice is [0 deg, 180 deg] (or [0, pi] radians), called the principal range. If you need an angle outside that range, add or subtract multiples of 360 degrees as appropriate for your problem.
How do I use arccos to find an angle in a triangle?
In a right triangle, cosine of an angle equals the adjacent side divided by the hypotenuse. If you know those two side lengths, compute the ratio (adjacent / hypotenuse) and then take the arccos of that ratio. Switch to triangle mode in this calculator to skip the manual ratio step. For non-right triangles, use the law of cosines: cos(A) = (b^2 + c^2 - a^2) / (2bc), then take arccos of that expression.
What is arccos(0)?
arccos(0) = 90 degrees = pi/2 radians. This is because cos(90 deg) = 0, so the angle whose cosine equals zero is 90 degrees.
How do I convert the result from radians to degrees?
Multiply the radian value by 180 divided by pi (approximately 57.2958). For example, pi/3 radians times (180/pi) = 60 degrees. This calculator shows both units automatically, so you do not need to convert by hand.