Unit Circle Lab
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Inverse Tangent Calculator

This inverse tangent calculator finds arctan(x) in degrees and radians, with exact values for special inputs. Switch to atan2(y, x) to get the angle of a point in the correct quadrant.

try

accepts 1 · −√3 · 1/√3 · 2.5 · sqrt(3)/3

xy45°

arctan(1)

45° π/4 rad Principal value, in (−90°, 90°)
Degrees
45°
Radians
π/4
Principal range
(−90°, 90°) · (−π/2, π/2)
Solutions in [0°, 360°)
45° and 225°
General solution
θ = 45° + 180°·k

Steps

rounded to 3–4 decimals
  1. arctan accepts any real number, so x = 1 is fine.
  2. Look for the angle θ in the principal range (−90°, 90°), which is (−π/2, π/2) in radians, with tan θ = 1.
  3. That is a special angle: tan 45° = 1, and 45° is in the range. So arctan(1) = 45° = π/4.
  4. Tangent repeats, so arctan gives only one answer. All solutions of tan θ = 1: θ = 45° + 180°·k.

How it works

One input or two, depending on what you know.

  1. 01 · ratio

    arctan for a ratio

    Give the tangent value (a slope, or y/x). arctan returns the angle between −90° and 90°.

  2. 02 · point

    atan2 for a point

    Give y and x separately. atan2 uses their signs to place the angle in the right quadrant, from −180° to 180°.

  3. 03 · draw

    See it on the circle

    The diagram shows the tangent segment on the line x = 1, or the point (x, y) and its angle.

§1What arctan means

Tangent turns an angle into a ratio: tan 45° = 1. Inverse tangent does the opposite: arctan(1) = 45°, or π/4. On the unit circle, tan θ = y/x, and it is also the height where the terminal side meets the vertical line x = 1. The violet segment in the diagram is that height.

§2Domain and range

Tangent can be any real number, so the domain of arctan is all real numbers: there is no input it rejects. The range is the open interval (−90°, 90°), or (−π/2, π/2). The endpoints are left out because tan 90° is undefined. As x grows, arctan(x) gets closer and closer to 90° without reaching it: arctan(1000) ≈ 89.94°.

§3Exact values of arctan

Inverse tangent of the special values
xarctan x (deg)arctan x (rad)
−√3−60°−π/3
−1−45°−π/4
−√3/3−30°−π/6
00°0
√3/330°π/6
145°π/4
√360°π/3

√3/3 and 1/√3 are the same number, so both give 30°.

§4All solutions of tan θ = x

Tangent repeats every 180°, not every 360°: the opposite point on the circle has the same ratio y/x. So the solutions are θ = arctan(x) + 180°·k.

tan θ = √3. arctan(√3) = 60° → θ = 60° + 180°·k → 60° and 240° in one turn

§5atan2(y, x) and the quadrant

arctan only sees the ratio y/x, so it cannot tell (1, 1) from (−1, −1): both have ratio 1 and arctan gives 45°. But (−1, −1) is in Quadrant III, at −135° (or 225°). The two-argument function atan2(y, x) keeps the signs of x and y apart and returns the true direction of the point, between −180° and 180°.

atan2(1, −1). arctan(−1) = −45°, but x < 0 and y > 0 → Quadrant II → −45° + 180° = 135°

atan2(−√3, −1). arctan(√3) = 60°, but the point is in Quadrant III → 60° − 180° = −120° (240°)

Most programming languages, including JavaScript and Python, take the arguments in the order atan2(y, x). Spreadsheets such as Excel and Google Sheets use ATAN2(x, y), with x first, so check the order before you copy a formula.

§6Angle of a slope

A line with slope m rises m units for every unit it runs, so its angle with the x-axis is arctan(m). A slope of 1 is a 45° incline, and a road with a 6% grade has slope 0.06 and rises at arctan(0.06) ≈ 3.43°.

§7Common mistakes

  • Using arctan(y/x) for a point with x < 0. The answer lands in the wrong quadrant; use atan2.
  • Thinking arctan(x) = 1/tan x. The reciprocal of tangent is cotangent.
  • Forgetting that the period is 180°: the solutions of tan θ = 1 are 45° and 225°, not 45° and 135°.
  • Expecting arctan to reach 90°. It never does; tan 90° is undefined.

§8FAQ

What is arctan of infinity?

arctan(x) approaches 90° (π/2) as x grows without bound, but infinity is not a number you can enter, and the limit is never reached.

What is atan2(0, 0)?

It is undefined in mathematics: the origin has no direction. Some programming languages return 0 by convention, but this calculator reports it as undefined.

Why does atan2 return negative angles?

Its range is (−180°, 180°], so points below the x-axis get negative angles. Add 360° to get the angle in [0°, 360°); the result panel shows both.

How is arctan related to arcsin and arccos?

All three undo a trig function and return a principal value. See the inverse sine and inverse cosine calculators for their ranges.