Tan Inverse Calculator
Enter a tangent value - or the opposite and adjacent sides of a right triangle - to find the angle whose tangent equals that ratio. Results appear instantly in degrees, radians, gradians, and as a pi fraction. The show-your-work panel traces every step, and the reference table lists all standard angles at a glance.
Formula
Worked example
Example: find the angle whose tangent is 1. arctan(1) = π/4 radians = 45°. To verify: tan(45°) = 1. For the two-sides input with opposite = 3 and adjacent = 4: tan θ = 3/4 = 0.75, so θ = arctan(0.75) ≈ 36.8699°.
What is the inverse tangent (arctan)?
The inverse tangent function, written arctan(x) or tan⁻¹(x), answers the question: "what angle has a tangent equal to x?" Because the tangent function is periodic, the inverse is defined only on the principal range from -90° to 90° (equivalently -π/2 to π/2 radians), which is the range this calculator always returns for single-argument mode. In programming and engineering, the two-argument form atan2(y, x) extends the range to the full circle from -180° to 180° by using the signs of both coordinates to determine the correct quadrant - you can select that mode above.
How to use this calculator
Select an input mode from the dropdown. "Tangent value" is the simplest: type any real number and the angle appears instantly. "Opposite / Adjacent" is useful in geometry and surveying: enter the two legs of a right triangle and the calculator computes the ratio for you. "Two-argument atan2(y, x)" is the form used in most programming languages and handles angles beyond the standard -90° to 90° range. Results appear in four formats - degrees, radians, gradians, and a pi fraction - so you can use whichever unit your context requires. Adjust the decimal-places selector for the precision you need.
Degrees, radians, gradians and pi fractions
Degrees (°) divide a full circle into 360 equal parts. Radians measure angle as the ratio of arc length to radius: a full circle is 2π radians. Gradians (also called gon or grad) divide a full circle into 400 parts, making a right angle exactly 100 grad, which simplifies some surveying calculations. A pi fraction expresses the radian value as a multiple of π - for example, π/4 instead of 0.785398 - which is the clearest form for exact algebraic work. To convert: degrees = radians × 180/π; gradians = degrees × 10/9.
Right-triangle geometry and the atan2 form
In a right triangle, the tangent of an angle equals the length of the opposite side divided by the length of the adjacent side. Entering those two lengths in "Opposite / Adjacent" mode lets the calculator find the angle without you needing to divide manually. The atan2(y, x) form is the version used in most software: unlike the single-argument arctan, it inspects both y and x to place the angle in the right quadrant. For example, atan2(1, -1) = 135° (second quadrant), while arctan(1 / -1) = arctan(-1) = -45°. Use atan2 whenever the point might lie in the second or third quadrant.
Common arctan reference values
| Tangent value (x) | Degrees | Radians (pi) | Gradians |
|---|---|---|---|
| -∞ | -90.000° | -π/2 | -100.000 grad |
| -1.7321 (−√3) | -60.000° | -π/3 | -66.667 grad |
| -1.0000 (−1) | -45.000° | -π/4 | -50.000 grad |
| -0.5774 (−1/√3) | -30.000° | -π/6 | -33.333 grad |
| 0 | 0.000° | 0 | 0.000 grad |
| 0.5774 (1/√3) | 30.000° | π/6 | 33.333 grad |
| 1.0000 (1) | 45.000° | π/4 | 50.000 grad |
| 1.7321 (√3) | 60.000° | π/3 | 66.667 grad |
| +∞ | 90.000° | π/2 | 100.000 grad |
Standard inverse tangent values for angles encountered in trigonometry and geometry.
Frequently asked questions
What is the difference between tan⁻¹ and 1/tan?
In modern notation tan⁻¹(x) always means the inverse tangent (arctan), not the reciprocal. The reciprocal of tan is the cotangent: cot(x) = 1/tan(x). Although the "⁻¹" superscript might look like a reciprocal, in function notation f⁻¹ denotes the inverse function, not 1/f. Most calculators label the key "tan⁻¹" or "arctan" and mean the inverse.
What is the range of arctan?
The standard (principal value) range of arctan is from -90° to 90° (exclusive), or -π/2 to π/2 radians. It never actually reaches those endpoints because tangent has vertical asymptotes there. The two-argument form atan2 extends the range to -180° to 180° (inclusive of -180°) by using the sign of both inputs.
How do I convert arctan from radians to degrees?
Multiply the radian value by 180/π (approximately 57.2958). For example, arctan(1) = π/4 radians; π/4 × 180/π = 45°. This calculator does the conversion automatically and shows both formats at once.
Can I enter a very large or negative number?
Yes. arctan is defined for all real numbers from negative infinity to positive infinity. As x approaches +∞ the result approaches 90° (π/2 rad), and as x approaches -∞ it approaches -90° (-π/2 rad). The function is continuous and smooth everywhere, so any finite number you enter will produce a valid angle.
When should I use atan2 instead of arctan?
Use atan2(y, x) when you need the angle of a vector or point and it might lie outside the first or fourth quadrant. A classic case is converting from Cartesian to polar coordinates: the correct angle for the point (-1, 1) is 135°, but arctan(1 / -1) = arctan(-1) = -45°. atan2 uses the individual signs of y and x to get the quadrant right. In single-argument mode (arctan), both (-1, 1) and (1, -1) map to the same result, losing quadrant information.
What is arctan used for in real life?
Arctan appears in slope calculations (converting rise/run to an angle), navigation and bearing calculations, computer graphics (rotating vectors, computing view angles), physics (calculating angles of incline or refraction), and engineering (phase angles in AC circuits). Any situation where you know a ratio and need the angle calls for the inverse tangent.