Arctan Calculator: Find Angle From Tangent Ratio Fast
Enter any tangent value and get the angle behind it: in degrees, in radians, with its quadrant and reference angle labeled. Built for students checking homework and anyone converting between ratios and angles. New: flip the direction and the tool computes tan(angle) from an angle instead.
For critical calculations, verify input values and double-check results before relying on them.
What arctan actually is
The arctangent answers a reversed question. Tangent takes an angle and returns a ratio (opposite over adjacent). Arctangent takes the ratio and returns the angle. Formally, arctan maps every real number to an angle between negative 90 and positive 90 degrees, because the tangent function climbs strictly from negative infinity to positive infinity across that open interval, which makes the inverse well defined. Its derivative is 1/(1 + x squared), a fact that shows up constantly in calculus.
What this tool shows you
Type a value and press Calculate (or hit Enter). You get five things: the angle in degrees to 4 decimal places, the angle in radians to 4 decimal places, the quadrant the principal angle falls in, a plain-language description of that quadrant, and the reference angle (the acute angle to the x-axis, which equals the absolute value of the degree result). Below that, a breakdown panel restates your input, the formula used (atan of your value), the raw radian result to 8 decimals, and the degree conversion, so you can follow every step. Flip the direction selector to tan(angle) and the same panel shows the tangent value instead, with the exact form for notable angles and an automatic reverse check.
New: the reverse direction, tan(angle)
Some problems hand you the angle and ask for the ratio. The reverse mode takes an angle in degrees or radians and returns tan(angle) to 6 decimals. For the eight notable angles (0, 30, 45, 60, 120, 135, 150, 180 degrees) it also names the exact value, so tan(30°) shows 0.577350 alongside 1/√3. At 90° and its odd multiples the tool reports undefined with an explanation instead of a misleading enormous number, because tangent has a vertical asymptote there. Every reverse result carries a reverse check: the tool runs atan on the answer and shows it lands back on your angle.
Why you only ever see quadrants I and IV
This surprises students, so here is the honest explanation. The standard arctangent returns only the principal value: an angle strictly between -90 and +90 degrees. Positive inputs land in Quadrant I, negative inputs in Quadrant IV, and zero sits at the origin. That is not a limitation of this calculator; it is the definition of the function, and it is exactly what the inverse-tangent key on a scientific calculator returns (Purdue’s MA 15400 course notes state the same rule). If you need the full 360-degree picture, for example the direction of a point in quadrants II or III, you need the two-argument atan2(y, x) function, which spans -180 to +180 degrees. This tool computes single-value atan, and says so.
Worked example: arctan(1)
Enter 1. The tool computes atan(1) = 0.78539816 radians (shown to 8 decimals in the breakdown), converts to degrees: 0.78539816 times (180 / pi) = 45.0000°. Since the input is positive, the quadrant reads I (Positive ratio) and the reference angle is 45.0000°. You can verify it backwards: tan(45°) = 1. Try -1 next and you will get -45.0000°, -0.7854 radians, Quadrant IV, which is the same magnitude mirrored below the x-axis.
Worked example: tan(45°), the reverse
Switch the direction to Value from angle: tan(angle), keep degrees, and enter 45. The tool returns 1.000000 with the exact value 1, and the reverse check reads atan(1.000000) = 45.0000°. Try 90 next: the tool reports Undefined and explains the vertical asymptote instead of printing a huge artifact like 1.633e+16.
Values worth memorizing
A short table of exact values covers most homework and most real use. These come from the standard reference values:
| x | arctan(x) in degrees | arctan(x) in radians |
|---|---|---|
| -1 | -45° | -pi/4 |
| -1/√3 | -30° | -pi/6 |
| 0 | 0° | 0 |
| 1/√3 | 30° | pi/6 |
| 1 | 45° | pi/4 |
| √3 | 60° | pi/3 |
Notice the range never touches ±90°: as x grows toward infinity, arctan(x) approaches 90° from below but never reaches it.
Frequently asked questions
Is arctan the same as tan⁻¹?
Yes. Arctan, atan, tan⁻¹, and inverse tangent all name the same function: the angle whose tangent equals the input value.
Why does arctan never return 90 degrees?
The principal range is the open interval (-90°, 90°). Tangent is undefined exactly at ±90°, so no finite input maps there; huge inputs like arctan(1,000,000) approach 90° from below but never arrive.
What is the difference between atan and atan2?
atan(x) takes one value and returns an angle between -90° and 90°. atan2(y, x) takes a point’s coordinates and returns the full quadrant-aware direction between -180° and 180°. Use atan2 for navigation, robotics, and game development; use atan when you only have the ratio.
Can I enter a fraction or a negative number?
Yes. Any real number works: fractions like 0.5774 (which gives 30°), negatives (which give negative angles in Quadrant IV), and zero (which gives 0° at the origin). Non-numeric input shows a “Please enter a valid number” message.
How do I convert the radian result to degrees myself?
Multiply radians by 180/pi (about 57.2958). The tool does this for you: degrees = radians × (180 / π).
What is a reference angle?
The reference angle is the acute angle between the terminal side and the x-axis. For arctan results it always equals the absolute value of the degree answer, so for arctan(-1) the angle is -45° and the reference angle is 45°.
Can the calculator go the other way, angle to tangent?
Yes. Switch the direction selector to tan(angle), enter the angle in degrees or radians, and the tool returns the tangent value with exact forms for notable angles and an automatic reverse check.
What does tan(90°) return?
Undefined, with an explanation. Tangent has a vertical asymptote at 90° plus any multiple of 180°, so there is no finite value to show.
Related calculators
Continue with the inverse sine calculator and the cosine calculator, find the missing angle of a right triangle with the 45-45-90 triangle calculator, or measure rotation with the central angle calculator. For the formal definitions of the inverse trig functions, see the Purdue MA 15400 lesson notes.