Arccos Calculator - Inverse Cosine (Degrees and Radians)
Enter a cosine value between -1 and 1 to find the angle whose cosine equals that value. The calculator returns the result in degrees and radians simultaneously, shows the step-by-step working, and plots the arccos curve so you can see exactly where your value falls. A built-in reference table lists every common exact value from arccos(-1) = 180 degrees to arccos(1) = 0 degrees.
Formula
Worked example
cos(60 deg) = 0.5, so arccos(0.5) = 60 deg = pi/3 rad. Verify: cos(pi/3) = 0.5. For x = -0.5: arccos(-0.5) = 120 deg = 2*pi/3 rad, an obtuse angle because the input is negative.
What is arccos?
Arccos (written arccos, acos, or cos^-1) is the inverse cosine function. Given a number x between -1 and 1, arccos(x) returns the unique angle theta in the range 0 to 180 degrees (0 to pi radians) whose cosine equals x. For example, because cos(60 deg) = 0.5, arccos(0.5) = 60 deg. The restriction to 0-180 degrees is called the principal value; it is necessary because cosine repeats itself every 360 degrees, so without a restriction the inverse would not be unique.
How to calculate arccos
Enter any cosine value from -1 to 1 in the input above. The calculator applies the inverse cosine function and returns the angle in both degrees and radians. It also shows the exact pi-fraction for all standard unit-circle values. The "Show your work" panel below the result walks through each arithmetic step: confirming the input is in the valid domain, applying Math.acos(), converting radians to degrees, and verifying the answer by computing cos(result) to confirm it equals your original x.
Degrees vs radians - which should you use?
Degrees (0 to 180) are the natural choice for geometry, construction, and navigation, where angles are described in everyday language. Radians (0 to pi) are used in calculus, physics, and programming because they make derivative and integral formulas for trig functions much simpler - the derivative of cos(x) in radians is just -sin(x), with no scale factor. Many programming languages and spreadsheets default to radians in their math libraries, so if you are coding, confirm whether the function expects degrees or radians before using the result.
Domain, range, and the inverse-cosine graph
The cosine function maps any angle to a value between -1 and 1. Arccos reverses this, but must be restricted to produce a single answer per input. The standard domain for arccos is x in [-1, 1] and its range (the output) is y in [0 deg, 180 deg] or equivalently [0, pi] in radians. The arccos curve is strictly decreasing: arccos(-1) = 180 deg, arccos(0) = 90 deg, and arccos(1) = 0 deg. This means larger cosine values correspond to smaller angles, which can feel counterintuitive at first. The graph of arccos is the reflection of the portion of the cosine curve from 0 to 180 degrees across the diagonal line y = x.
Real-world applications
Inverse cosine appears in many practical fields. In geometry and trigonometry, you use arccos to find an unknown angle in a triangle when you know two side lengths (via the law of cosines: cos(C) = (a^2 + b^2 - c^2) / (2ab), then C = arccos(...)). In physics, it determines the angle between two vectors using the dot-product formula: theta = arccos(A.B / (|A||B|)). In computer graphics, arccos is used to compute rotation angles, lighting normals, and joint constraints. In navigation and geodesy, the great-circle distance formula uses arccos to convert a dot product back into an arc angle. Engineers also use arccos when finding the contact angle in gear design, the incline of ramps, and the pitch of roofs.
Common arccos values
| x | arccos(x) in degrees | arccos(x) in radians |
|---|---|---|
| -1 | 180 deg | pi |
| -0.8660254 | 150 deg | 5pi/6 |
| -0.7071068 | 135 deg | 3pi/4 |
| -0.5 | 120 deg | 2pi/3 |
| 0 | 90 deg | pi/2 |
| 0.5 | 60 deg | pi/3 |
| 0.7071068 | 45 deg | pi/4 |
| 0.8660254 | 30 deg | pi/6 |
| 1 | 0 deg | 0 |
Exact inverse cosine values for the standard unit-circle angles.
Frequently asked questions
What is the domain of arccos?
The domain is all real numbers x with -1 <= x <= 1. Any input outside this range has no real arccos value because the cosine function itself never exceeds 1 or falls below -1, so no real angle can produce a cosine outside that interval. Entering a value like 1.5 yields an undefined (NaN) result.
What is the range of arccos?
The output (range) of the real arccos function is always between 0 and 180 degrees (or 0 to pi radians). This is called the principal value. If you need an angle in a different quadrant (for example, 270 degrees), you need to know the sign context of the original problem and use a full-circle inverse like atan2 instead.
What is the difference between arccos and cos^-1?
They mean the same thing. arccos(x), acos(x), and cos^-1(x) are all standard notations for the inverse cosine. The superscript -1 denotes the inverse function, not a reciprocal (1/cos is the secant, not arccos). Most programming languages use the name acos (for example, Math.acos() in JavaScript, numpy.arccos() in Python).
Why is arccos(0.5) equal to 60 degrees and not also 300 degrees?
Both 60 and 300 degrees have a cosine of 0.5, but the arccos function is defined to return only one value per input, specifically the angle in [0, 180] degrees. By convention this unique output is called the principal value. 300 degrees is outside that range (it is equivalent to -60 degrees), so arccos returns 60 degrees. If the full-circle solution matters, add 360 - 60 = 300 deg as a second solution manually.
How do I use arccos to find an angle in a triangle?
Use the law of cosines: c^2 = a^2 + b^2 - 2*a*b*cos(C). Rearrange to cos(C) = (a^2 + b^2 - c^2) / (2*a*b), then enter that fraction into this calculator to get angle C. For a right triangle you can also use the simpler ratio: cos(theta) = adjacent / hypotenuse, so theta = arccos(adjacent / hypotenuse).
How do I convert arccos output from radians to degrees?
Multiply the radian value by 180 and divide by pi. For example, pi/3 radians x (180/pi) = 60 degrees. Equivalently, divide by pi and multiply by 180. This calculator shows both units simultaneously, so you never need to convert by hand.
What is arccos of a number greater than 1 or less than -1?
There is no real solution. The real cosine function only produces values in [-1, 1], so its inverse is only defined on that interval. In complex number theory an extension of arccos exists for inputs outside [-1, 1], but for all real-world geometric purposes an input outside [-1, 1] indicates a calculation error (often rounding noise pushing a ratio slightly beyond 1).