Arctan Calculator (tan⁻¹)
Turn a tangent value back into an angle. Enter any real number — a trig ratio, a slope, a rise-over-run — and get tan⁻¹ in degrees, radians, and multiples of π, always within the principal range (−90°, 90°).
Example: with Value x (tangent or slope) 1 → arctan(x) in degrees: 45°.
- In radians0.785398 rad
- As a multiple of π0.25π rad
- Slope readingA line with slope 1 makes a 45° angle with the horizontal
Computed by the calculator below using its default values. Change any input to see your own numbers.
arctan (tan⁻¹) inverts the tangent: it returns the angle in (−90°, 90°) whose tangent is x. arctan(1) = 45° because tan(45°) = 1.
What tan⁻¹ actually returns
Tangent takes an angle and returns a ratio — opposite over adjacent, or rise over run. Arctan runs it backward: give it the ratio, get the angle. Because infinitely many angles share each tangent value (tan repeats every 180°), arctan must pick one, and the convention is the principal value in (−90°, 90°). arctan(1) = 45°, never 225°, even though tan(225°) is also 1.
That restriction is what the ⁻¹ notation means here: inverse function, not reciprocal. tan⁻¹(x) is the angle; 1/tan(x) is the cotangent, a completely different quantity. Mixing those up is the single most common arctan error.
Slopes, grades, and vectors
Any time you know a rise and a run, arctan converts them to an angle: a roof that rises 6 inches per foot of run sits at arctan(6/12) ≈ 26.57°; a 10% highway grade is arctan(0.10) ≈ 5.71°. In coordinate work, arctan(y/x) recovers a vector's direction — with the caveat that for points in the second and third quadrants you must add or subtract 180°, which is why programming languages provide atan2(y, x).
How it’s calculated
θ = arctan(x), computed with Math.atan, returning the principal value in (−π/2, π/2) radians, i.e. (−90°, 90°). Degrees = radians × 180/π with π = 3.141592653589793. The π-multiple row divides the radian value by π. Whole-degree results display exactly (45°); others show 4 decimals.
Principal value only — if your angle lies in the second or third quadrant (x-component negative), add or subtract 180° or use a two-argument atan2 with the signs of both coordinates.
Arctan landmarks
| x | arctan(x) degrees | radians |
|---|---|---|
| 0 | 0° | 0 |
| 0.5774 (√3⁄3) | 30° | π/6 ≈ 0.5236 |
| 1 | 45° | π/4 ≈ 0.7854 |
| 1.7321 (√3) | 60° | π/3 ≈ 1.0472 |
| 2 | 63.4349° | 1.1071 |
| x → ∞ | approaches 90° | π/2 ≈ 1.5708 |
Computed with θ = arctan(x); exact forms from the 30-60-45 special triangles.
Common mistakes
- Treating tan⁻¹(x) as 1/tan(x) — the first is an angle (inverse function), the second is cotangent. On a calculator they live on different keys.
- Expecting an answer outside (−90°, 90°): arctan alone cannot say whether a vector points into quadrant II or III; adjust by 180° using the signs of x and y.
- Reading the result in the wrong mode — 0.7854 radians misread as degrees turns a 45° angle into less than 1°.
- Feeding in an angle instead of a ratio: arctan wants the slope or tangent value (rise ÷ run), not degrees.
Frequently asked questions
What is arctan?
The inverse tangent function: arctan(x), also written tan⁻¹(x), returns the angle θ in (−90°, 90°) with tan(θ) = x. Give it a ratio or slope, get the angle back.
What is tan⁻¹ of 1?
45°, or π/4 ≈ 0.7854 radians. A right triangle with equal legs has opposite/adjacent = 1, and its base angles are 45°.
Is tan⁻¹ the same as 1/tan?
No. The −1 is function-inverse notation: tan⁻¹(x) is an angle. The reciprocal 1/tan(θ) is the cotangent, a ratio. tan⁻¹(2) ≈ 63.43°, while 1/tan(2 rad) ≈ −0.4577.
How do I convert a slope or grade to an angle?
Angle = arctan(slope). A 100% grade (rise equals run) is arctan(1) = 45°; a 10% grade is arctan(0.1) ≈ 5.71°. For rise and run measurements, use arctan(rise ÷ run).
Why does my answer never reach 90 degrees?
Because tangent blows up to infinity as the angle approaches 90°. Arctan of even a huge number like 1,000 is 89.94° — the function approaches 90° asymptotically but never lands on it.