Inverse Tan Calculator
Find the arctan (tan⁻¹) of any number instantly. Enter a value and the calculator returns the angle in degrees and radians, along with its quadrant and a quick-reference table.
Input Value
Enter a number to compute its inverse tan
x can be any real number — positive, negative, zero, or a decimal. arctan is defined for all real numbers.
Choose whether the result is shown in degrees (°) or radians (rad).
Your Inverse Tan Result
Angle in degrees, radians, and quadrant info
Enter a value above and click Calculate Inverse Tan to reveal the angle whose tangent equals that value.
Understanding Inverse Tan
Inverse tan (arctan, or tan⁻¹) answers the question: “what angle has this tangent value?” It is the mirror image of the tangent function, swapping inputs and outputs.
Definition
If tan(θ) = x, then arctan(x) = θ. The function reverses what tangent does, returning an angle from a ratio.
Principal Range
arctan always returns a value between -90° and 90° (-π/2 to π/2 radians), regardless of how large or small x is.
Symmetry
arctan is an odd function: arctan(-x) = -arctan(x). Negative inputs always produce negative angles within the principal range.
Asymptotic Behaviour
As x approaches infinity, arctan(x) approaches 90°. As x approaches negative infinity, it approaches -90°, but never quite reaches it.
Common Arctan Values
A quick lookup of frequently used inverse tan values in both degrees and radians.
| x | tan⁻¹(x) Degrees | tan⁻¹(x) Radians | Notes |
|---|---|---|---|
| −√3 | −60° | −π/3 | Common exact value |
| −1 | −45° | −π/4 | Common exact value |
| −1/√3 | −30° | −π/6 | Common exact value |
| 0 | 0° | 0 | arctan is an odd function |
| 1/√3 | 30° | π/6 | Common exact value |
| 1 | 45° | π/4 | Common exact value |
| √3 | 60° | π/3 | Common exact value |
| → +∞ | → 90° | → π/2 | Asymptotic limit, never reached |
Inverse Tan FAQ
Everything you need to know about arctan, its range, and how it relates to the tangent function.
Inverse tan, written as tan⁻¹(x) or arctan(x), is the inverse function of tangent. Given a ratio x, arctan(x) returns the angle whose tangent equals that ratio. For example, arctan(1) = 45° because tan(45°) = 1.
The range of the principal value of arctan(x) is between -90° and 90° (or -π/2 and π/2 radians), exclusive. This is because tangent is periodic, so arctan always returns the angle within this restricted range, even though infinitely many angles share the same tangent value.
Inverse tan cannot generally be calculated by hand for arbitrary values; it requires a calculator, lookup table, or series approximation such as the Taylor series for arctan. Common values like arctan(1) = 45° or arctan(0) = 0° can be recalled from the unit circle.
arctan(x) takes a single ratio and only returns angles between -90° and 90°, so it cannot distinguish between points in different quadrants that share the same ratio. atan2(y, x) takes both the y and x coordinates separately and returns the correct angle across the full -180° to 180° range, properly accounting for the quadrant.
Arctan is used to find angles from slope or ratio data in fields such as engineering, navigation, computer graphics, and physics. Common applications include calculating the angle of elevation or depression, determining the slope angle of a ramp or roof, and computing bearings from coordinate differences.
No. arctan(x), or tan⁻¹(x), is the inverse function of tangent and returns an angle. 1/tan(x) is the reciprocal of tangent, also known as cotangent, and returns a ratio, not an angle. The ‘-1’ superscript in tan⁻¹ denotes the inverse function, not an exponent.