# What is the inverse tangent of x?

Finds arctan x, the angle between −90° and 90° whose tangent is x, in degrees and radians, exact as a multiple of π for the special values.

- Page: https://www.acalculator.org/math/inverse-tangent-calculator
- JSON spec: https://www.acalculator.org/math/inverse-tangent-calculator.json
- Version: 715f482854b9

## Default answer

Example with the default inputs (Value x 1): arctan(1) = 45°.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| x | Value x | Any number: the tangent of the angle. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| angle | arctan x | The principal angle whose tan is x, in degrees. |
| radians | In radians | The same angle in radians. |
| exact | Exact angle | The angle as a multiple of π, when x is the value at a multiple of 15°. |

## Method

arctan x is the angle θ between −90° and 90° (−π/2 and π/2) with tan θ = x. When x is within 10⁻¹⁴ (relative, for |x| > 1) of the tangent of a multiple of 15° in that range, the angle is that multiple exactly; otherwise it is computed in 64-bit floating point.

## Assumptions

- x is any number from −10¹⁵ to 10¹⁵.
- The answer is the principal value. Other angles with the same tangent are θ plus whole half turns (θ + 180°k).

## Worked examples

1. x = 1 gives angle = 0.785398, radians = 0.785398, exact = π/4 = 45° (tan 45° = 1). Source: OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arctan: domain all reals, range (−π/2, π/2)). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions.
2. x = -1.732051 gives angle = -1.047198, exact = -π/3 = -60° (tan -60° = -√3). Source: OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arctan: domain all reals, range (−π/2, π/2)). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions.
3. x = 2 gives angle = 1.107149. Source: NIST DLMF §4.23 (inverse trigonometric functions, principal values), https://dlmf.nist.gov/4.23.
4. x = -0.25 gives angle = -0.244979. Source: NIST DLMF §4.23, https://dlmf.nist.gov/4.23.

## FAQ

### What is inverse tangent?

Inverse tangent, written arctan x or tan⁻¹ x, undoes the tangent: it gives the angle whose tangent is x. Because many angles share a tangent, it gives the principal value, the one between −90° and 90°.

### What is arctan 1?

arctan 1 = 45° = π/4 radians, because tan 45° = 1. Also arctan √3 = 60° and arctan(√3/3) = 30°.

### Can I take the arctan of any number?

Yes. The tangent takes every real value between −90° and 90°, so arctan x exists for every x. As x grows, arctan x gets close to 90° but never reaches it.

### How do I find an angle from a slope?

A slope of rise ÷ run is the tangent of the angle the line makes with the x-axis. Type the slope as x. A slope of 1 is 45°; a slope of 0.25 is about 14.04°.

### How do I find all the angles with a given tangent?

Take θ = arctan x. The angles with tangent x are θ + 180°k for any whole number k. For x = 1: 45°, 225°, −135°, and so on.

### Is tan⁻¹ x the same as 1 ÷ tan x?

No. tan⁻¹ x is the inverse function, arctan x. The reciprocal 1 ÷ tan x is the cotangent, cot x.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arctan: domain all real numbers, range (−π/2, π/2)). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions (retrieved 2026-10-01)
- OpenStax, Algebra and Trigonometry 2e, §7.4 The Other Trigonometric Functions (special values of tangent). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-4-the-other-trigonometric-functions (retrieved 2026-10-01)
- NIST Digital Library of Mathematical Functions, §4.23 Inverse Trigonometric Functions (principal values). https://dlmf.nist.gov/4.23 (retrieved 2026-10-01)