Arcus Tangent (arctan)
Compute inverse tangent (arctan) of any real x. Arcus tangent calculator returns angle in degrees and radians with formula breakdown and domain range info.
About This Calculator
What Is Arcus Tangent?
Arcus tangent (arctan or tan⁻¹) is the inverse of the tangent function. Given any real number x, it returns the angle theta whose tangent equals x. Unlike the tangent function which repeats every pi radians, the arcus tangent is defined to return the principal value in the range (-pi/2, pi/2) radians -- or (-90 deg, 90 deg) in degrees.
Mathematically: arctan(x) = theta where tan(theta) = x and theta in (-pi/2, pi/2)
The arctan function is odd: arctan(-x) = -arctan(x). Its graph passes through the origin and approaches horizontal asymptotes at y = +/-pi/2 as x -> +/-∞.
How to Use This Calculator
- Enter any real number in the input field (e.g. 1, -2.5, 0.5).
- Click Calculate arctan to compute the inverse tangent.
- View the result in both degrees and radians with a step-by-step breakdown.
- Share your calculation by copying the page URL -- all inputs are saved in the query string.
Common Arcus Tangent Values
| x | arctan(x) in degrees | arctan(x) in radians |
|---|---|---|
| 0 | 0 deg | 0 |
| 1 | 45 deg | pi/4 |
| sqrt3 ≈ 1.732 | 60 deg | pi/3 |
| ∞ (limit) | 90 deg | pi/2 |
| -1 | -45 deg | -pi/4 |
| -∞ (limit) | -90 deg | -pi/2 |
Frequently Asked Questions
What is arcus tangent (arctan)?
Arcus tangent, also written as arctan or tan⁻¹, is the inverse trigonometric function of the tangent. For any real number x, arctan(x) returns the angle whose tangent is x. The output range is (-pi/2, pi/2) radians or (-90 deg, 90 deg) degrees.
What is the difference between arcus tangent and arctan?
Arcus tangent and arctan are the same function -- the inverse tangent. Arcus tangent is the Latin-derived full name, while arctan is the common abbreviation. Both refer to tan⁻¹(x), returning the angle for a given tangent value.
What values can I input for arcus tangent?
The arcus tangent function accepts any real number -- positive, negative, zero, fractions, or decimals. Unlike arcsin or arccos, there is no domain restriction for arctan. The output angle always falls between -90 deg and +90 deg (or -pi/2 to pi/2 radians).
How do you convert arcus tangent from radians to degrees?
To convert arctan(x) from radians to degrees, multiply the radian result by 180/pi. For example, arctan(1) = pi/4 radians = 45 deg. This calculator shows both units automatically so you do not need to convert manually.
What is arctan(0), arctan(1), and arctan(∞)?
arctan(0) = 0 deg (0 radians). arctan(1) = 45 deg (pi/4 radians). As x approaches +∞, arctan(x) approaches 90 deg (pi/2 radians). As x approaches -∞, arctan(x) approaches -90 deg (-pi/2 radians). These are the horizontal asymptotes of the arctan curve.
How is arcus tangent used in real life?
Arctan is widely used in engineering, physics, and computer graphics to compute angles from slopes or ratios. Applications include robot arm joint angles, camera field-of-view calculations, signal phase detection, and converting rise-over-run to pitch angle in construction.
Is the arcus tangent calculator free?
Yes, this arcus tangent calculator is completely free to use with no registration or download required. You can compute arctan for any real number instantly and share results via URL.