Arcus Tangent (arctan) Calculator
Enter any real number to find its arcus tangent. The result appears in both degrees and radians as you type. Switch to reverse mode to go the other way: enter the angle and recover the tangent. The steps panel shows the exact arithmetic and the curve visual places your angle on the classic S-shaped arctan graph.
What is the arcus tangent (arctan)?
The arcus tangent, written arctan(x) or tan^-1(x), is the inverse of the tangent function. It answers the question: "which angle has this tangent value?" For example, if tan(45 degrees) = 1, then arctan(1) = 45 degrees. Because the tangent function repeats every 180 degrees, the inverse is defined on a restricted interval so that every input produces exactly one output. The standard (principal-value) range is from -90 to +90 degrees, or -pi/2 to +pi/2 radians. This means arctan always returns an angle in that range, even though the original angle could have been in any other period.
How to use this calculator
In the default "arctan(x)" mode, type any real number into the tangent value field. The calculator returns the angle in degrees and in radians, plus the sine and cosine of that angle. Switch to "tan(angle) - reverse" mode to go the other way: enter an angle and recover the tangent value. Use the angle unit selector to choose whether angles are expressed in degrees or radians. The steps panel walks through the arithmetic so you can verify or copy the work into your own notes. The chart shows the full S-shaped arctan curve from x = -10 to x = 10 so you can see where your value lands on it.
The arctan formula and right-triangle interpretation
For a right triangle with an angle theta, the tangent is defined as opposite / adjacent, so theta = arctan(opposite / adjacent). This makes arctan the go-to tool for finding angles from side lengths. For example, if a ramp rises 3 m over a horizontal distance of 4 m, the angle of inclination is arctan(3/4) = arctan(0.75) = 36.87 degrees. The calculator handles the arithmetic; you just supply the ratio. The three trig identities worth knowing: tan(arctan(x)) = x for all real x; arctan(-x) = -arctan(x) (arctan is an odd function); and arctan(x) + arctan(1/x) = pi/2 for x greater than 0.
Degrees vs. radians: which to use?
Degrees (0 to 360) are the everyday convention and the best choice for navigation, construction, and most applied problems. Radians (0 to 2*pi) are the natural unit in calculus and physics because the derivative of tan(x) is the clean form sec^2(x) only when x is in radians. The conversion is straightforward: multiply degrees by pi/180 to get radians, or multiply radians by 180/pi to get degrees. The principal value of arctan spans -90 to 90 degrees (equivalently -pi/2 to pi/2 radians).
Common arctan values
| Tangent x | arctan(x) in degrees | arctan(x) in radians | Exact radians |
|---|---|---|---|
| -infinity | -90 | -1.570796 | -pi/2 |
| -1.732051 (-sqrt(3)) | -60 | -1.047198 | -pi/3 |
| -1 | -45 | -0.785398 | -pi/4 |
| -0.577350 (-1/sqrt(3)) | -30 | -0.523599 | -pi/6 |
| 0 | 0 | 0 | 0 |
| 0.577350 (1/sqrt(3)) | 30 | 0.523599 | pi/6 |
| 1 | 45 | 0.785398 | pi/4 |
| 1.732051 (sqrt(3)) | 60 | 1.047198 | pi/3 |
| infinity | 90 | 1.570796 | pi/2 |
Exact or rounded arctan values for the most frequently used tangent inputs. These correspond to the standard angles used in trigonometry.
Frequently asked questions
What is arctan(1)?
arctan(1) = 45 degrees = pi/4 radians. This is because tan(45 deg) = 1, so the inverse returns 45 degrees. It corresponds to an isoceles right triangle where the two legs are equal in length.
What is arctan(0)?
arctan(0) = 0 degrees = 0 radians. The tangent of 0 degrees is 0, so the inverse function returns 0.
Can arctan ever equal 90 degrees?
No. The principal value range of arctan is strictly between -90 and 90 degrees (exclusive). The function approaches but never reaches +/-90 degrees, because that would require the tangent to be infinite. As x approaches infinity, arctan(x) approaches pi/2 (90 degrees) as a horizontal asymptote.
What is the difference between arctan and atan2?
arctan(x) takes a single argument and always returns a value between -90 and 90 degrees. atan2(y, x) takes two arguments (the y and x components of a vector) and returns an angle anywhere from -180 to 180 degrees, correctly handling all four quadrants. Use atan2 in programming when you need the full angle of a 2D vector; use arctan when you already have a single tangent ratio.
How do I use arctan to find an angle in a right triangle?
Identify the side opposite the angle and the side adjacent to it. Divide: x = opposite / adjacent. Then arctan(x) gives the angle. For example, if opposite = 5 and adjacent = 12, then the angle = arctan(5/12) = arctan(0.4167) = approximately 22.62 degrees.
Why does arctan have a restricted domain?
The tangent function is not one-to-one across all real numbers: it repeats with period 180 degrees. An inverse function can only exist if each output corresponds to exactly one input. By restricting the range to (-90, 90) degrees, every tangent value maps to exactly one angle, which is the principal value that arctan returns.