Arctan Calculator (Tan Inverse)
Source: NIST Digital Library of Mathematical Functions, §4.23 (Inverse Trigonometric Functions) · Source verified August 14, 2026
An arctan calculator evaluates the inverse tangent, written arctan(x), tan⁻¹(x), or atan(x). It answers the reverse of the tangent question: given a ratio, which angle produces it? The notation tan⁻¹ marks a function inverse rather than a reciprocal power, so it is not the same as cotangent, cot(x) = 1/tan(x). Arctan accepts every real number, unlike arcsin and arccos, and returns the principal value in the open interval from -π/2 to π/2 radians.
Enter a tangent ratio and get the angle back. This tan inverse calculator returns the principal value in degrees and radians, names the exact angle for standard ratios, and shows the general solution.
Quick Answer
Find the angle whose tangent is a given number. Enter the ratio and read the principal value in degrees and radians, strictly between -90° and 90°.
Any real number, positive or negative. Unlike arcsin and arccos there is no limit on how large x can be. · e.g. 1
Common ratios
Show the angle in
arctan(1) in degrees
45°
x = 1 is a standard ratio, so the angle is exactly π/4 rad = 45°.
Examples
x = 1
45° · π/4 rad · exact
x = √3
60° · π/3 rad · exact
x = -1
-45° · -π/4 rad · exact
x = 2.5
68.1986° · 1.19029 rad · no exact form
How it works
Formula · θ = arctan(x), the angle whose tangent is x · principal value -90° < θ < 90°
Tangent turns an angle into a ratio. Arctangent runs that backwards: give it the ratio and it returns the angle.
Inverse · θ = arctan(x)
Forward · tan(θ) = x
In a triangle · θ = arctan(opposite ÷ adjacent)
Principal value: -90° < θ < 90° · -π/2 < θ < π/2
What tan inverse actually means
Every notation for this function names the same thing: tan⁻¹(x), arctan(x), and atan(x) are interchangeable. All three ask one question: which angle has a tangent of x?
The superscript in tan⁻¹ is the source of nearly every error with this function. It marks a function inverse, the way f⁻¹ does, and it is not an exponent. Reading it as a power gives 1 ÷ tan(x), which is a genuinely different function with its own name, cotangent, and its own notation, cot(x). Cotangent takes an angle and returns a ratio. Arctangent takes a ratio and returns an angle. They point in opposite directions, so substituting one for the other does not produce a slightly wrong number, it produces the wrong kind of quantity.
If you want cotangent rather than the inverse, the sin cos tan calculator computes cot alongside sin, cos, tan, csc, and sec.
Why the answer stops short of 90 degrees
Tangent repeats every 180°, so tan(45°), tan(225°), and tan(-135°) are all exactly 1. Infinitely many angles share that ratio, and a function is only allowed to return one value, so one of them has to be chosen. The convention is the principal value: the single answer lying strictly between -90° and 90°.
The interval is open at both ends. Large ratios push the angle toward 90° without reaching it, because tan(90°) is undefined rather than infinite. So no finite ratio you can type here will ever return exactly 90°.
That is a statement about arctan, not about vertical lines. A vertical line does have a geometric inclination of 90°; what it does not have is a finite slope to feed into arctan, since its run is zero. The two facts sit side by side rather than in conflict, and the slope calculator shows both: it reports the slope of a vertical line as undefined while still listing its angle as 90°.
To recover the angles the principal value discards, add whole multiples of half a turn:
θ = arctan(x) + 180°n · θ = arctan(x) + πn
Exact values worth knowing
| Ratio x | arctan(x) in degrees | In radians |
|---|---|---|
| 0 | 0° | 0 |
| 1/√3 | 30° | π/6 |
| 1 | 45° | π/4 |
| √3 | 60° | π/3 |
| -1/√3 | -30° | -π/6 |
| -1 | -45° | -π/4 |
| -√3 | -60° | -π/3 |
The negative half follows from the positive half without any extra work, because arctan is an odd function: arctan(-x) = -arctan(x).
Working it out by hand
Worked example: a ramp rises 3 metres over a horizontal run of 5 metres. What angle does it make with the ground?
- Form the ratio. Tangent is opposite over adjacent, so x = 3 ÷ 5 = 0.6.
- Check the standard ratios. 0.6 is not 0, 1/√3, 1, or √3, so there is no exact angle and the answer will be a decimal.
- Apply the inverse: θ = arctan(0.6). On a calculator this is usually shift or 2nd followed by the tan key, with the mode set to degrees.
- Read the result: θ ≈ 30.9638°, or about 0.54042 radians.
- Verify by going forward again. tan(30.9638°) ≈ 0.6, which is the ratio you started with, so the answer is consistent.
Step 5 is worth keeping as a habit. Recomputing the tangent of your answer catches the two most frequent slips at once: leaving the calculator in radian mode when you wanted degrees, and pressing tan instead of its inverse.
Limitations and invalid cases
- No domain restriction. Every real number has an arctangent, so the only entry this page rejects is one that is not a number.
- One angle out of infinitely many. The result is the principal value. If your problem lives in the second or third quadrant, add 180° to reach the angle you actually want.
- A ratio loses quadrant information. The points (1, 1) and (-1, -1) both give the ratio 1, so both return 45°. If you started from coordinates rather than a single ratio, use atan2 or work from the signs by hand.
- 90° is never returned. A vertical relationship has no finite tangent, so no input produces exactly 90°, however large. That is a limit of what arctan can be given, not a claim that vertical lines lack a 90° inclination.
- Decimals are rounded. The calculator labels the familiar standard ratios in the table above with their exact special-angle forms. Everything else is displayed to six decimal places.
Related trigonometry calculators
- Sin cos tan calculator for the forward direction, angle to ratio, plus csc, sec, and cot.
- SOHCAHTOA calculator when you have two side lengths rather than a ratio.
- Unit circle calculator for where an angle lands and what its coordinates are.
- Reference angle calculator for reducing an angle outside the principal range.
- Right triangle calculator to solve every side and angle at once.
- Angle converter for degrees, radians, and gradians.
- Scientific calculator for atan inside a longer expression.
- All education calculators.
Sources
- NIST Digital Library of Mathematical Functions, §4.23 Inverse Trigonometric Functions for the definition of arctan and of the principal branch.
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Frequently asked questions
No, and this is the single most common mistake with the notation. The superscript -1 in tan⁻¹(x) marks a function inverse, not a reciprocal power. tan⁻¹(x) is arctan(x), the angle whose tangent is x. The reciprocal 1/tan(x) is a different function called cotangent, written cot(x). Compare tan⁻¹(1) = 45° with cot(1 radian) ≈ 0.642: they are not the same number and they are not even the same kind of quantity, because arctan returns an angle and cot returns a ratio.
Strictly between -90° and 90°, or equivalently between -π/2 and π/2 radians. Both endpoints are excluded. That interval is called the principal value, and it exists because tangent repeats forever: infinitely many angles share any given tangent, so a function has to pick one to return. As x grows without bound the angle creeps toward 90° but never arrives, so no finite ratio returns exactly 90°. A vertical line still has a geometric inclination of 90°; what it lacks is a finite slope to feed in, because its run is zero.
No. Arctan accepts every real number, positive or negative, however large. This is what separates it from arcsin and arccos, which reject anything outside -1 to 1. A tangent is opposite divided by adjacent, and either side of a right triangle can be the longer one, so the ratio has no ceiling.
Add whole multiples of 180°, not 360°. The general solution is θ = arctan(x) + 180°n for any integer n, or θ = arctan(x) + πn in radians. Tangent has a period of half a turn rather than a full turn because it is sine over cosine and both flip sign together 180° apart. Writing + 360°n here is a common way to lose marks.
The familiar standard ratios from the 30-60-90 and 45-45-90 triangles: x = 0 gives 0, x = 1/√3 gives 30° (π/6), x = 1 gives 45° (π/4), and x = √3 gives 60° (π/3). Negatives mirror these, because arctan(-x) = -arctan(x). Other inputs are shown numerically, which is a statement about what this tool recognises rather than a claim that no other ratio has any exact symbolic form. The calculator names the exact form only when your input really is that value. Typing 1.732 instead of √3 gives 59.9993°, and the calculator shows that decimal rather than claiming an exact 60°.
When you have a coordinate pair rather than a single ratio. Arctan sees only y/x, so it cannot tell the point (1, 1) from (-1, -1): both give the ratio 1 and both return 45°, even though the second point is in the third quadrant. The two-argument atan2(y, x) keeps the signs separate and returns the correct angle across all four quadrants. For a triangle where you know two side lengths, the SOHCAHTOA calculator is the better starting point.
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