What is the inverse tangent of x?
Type any number x. The inverse tangent calculator shows arctan x, the angle whose tangent is x, in degrees and radians, and the exact angle for values such as 1 and √3.
- arctan x
- 45°
arctan(1) = 45°.
- In radians
- 0.785398
- Exact angle
- π/4 = 45° (tan 45° = 1)
arctan x: 45°. arctan(1) = 45°.
How to calculate
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.
Example with the default inputs (Value x 1): arctan(1) = 45°.
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.
- 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
Each example is checked against the calculator on every build.
- Value x 1 gives arctan x 45°, In radians 0.785398, Exact angle 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
- Value x -1.732051 gives arctan x -60°, Exact angle -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
- Value x 2 gives arctan x 63.434949°.Source: NIST DLMF §4.23 (inverse trigonometric functions, principal values), https://dlmf.nist.gov/4.23
- Value x -0.25 gives arctan x -14.036243°.Source: NIST DLMF §4.23, https://dlmf.nist.gov/4.23
How it works
arctan x is the angle θ with tan θ = x and −90° < θ < 90° (−π/2 < θ < π/2). This range holds exactly one angle for each real x.
Exact angles. The calculator checks the tangents of the multiples of 15° from −75° to 75°: 0, ±(2 − √3), ±√3/3, ±1, ±√3 and ±(2 + √3). If x is within 10⁻¹⁴ × max(1, |value|) of one of them, the angle is that multiple of 15° exactly, and the page shows it as a multiple of π. Otherwise the angle is computed in 64-bit floating point.
Output format. The angle shows in degrees. "In radians" shows the same angle in radians as a decimal. The exact angle shows as text, such as π/4 = 45° (tan 45° = 1), with - for negative values.
Assumptions
- x is any number from −10¹⁵ to 10¹⁵.
- The answer is the principal value; the FAQ shows how to find the other angles.
Worked examples by hand
x = 1. tan 45° = 1 and 45° is between −90° and 90°, so arctan 1 = 45° = π/4 ≈ 0.7853981633974483 rad.
x = −1.7320508075688772 (−√3). tan(−60°) = −tan 60° = −√3, so arctan x = −60° = −π/3 ≈ −1.0471975511965976 rad.
x = 2. 2 is not a special value, so arctan 2 ≈ 1.1071487177940906 rad ≈ 63.4349°.
x = −0.25. arctan(−0.25) ≈ −0.24497866312686414 rad ≈ −14.0362°.
Other questions people ask
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.