Tan-1 Calculator

Enter a value x to find the angle θ whose tangent equals x (θ = tan⁻¹(x)), shown in degrees, radians, and degrees-minutes-seconds.

Quick Facts

Formula
θ = tan⁻¹(x) = arctan(x)
θ is the angle whose tangent equals x.
Principal range
-90° < θ < 90°
Equivalently -π/2 to π/2 radians; tan⁻¹ never returns exactly ±90°.
Degree-radian conversion
radians = degrees × π/180
π radians = 180°, so 1 radian ≈ 57.2958°.
Not a reciprocal
tan⁻¹(x) ≠ 1/tan(x)
tan⁻¹ is the inverse function (arctan); 1/tan(x) is the cotangent, cot(x).

Your Results

Calculated
Angle
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Primary result, in your preferred unit
Angle (alternate unit)
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Same angle, other unit
Degrees-Minutes-Seconds
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DMS notation
Valid Range
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Principal value bounds

Ready

Enter a value for x, then press Calculate.

Formula and Method for tan⁻¹ (Inverse Tangent / Arctan)

The inverse tangent, written tan⁻¹(x) or arctan(x), undoes the tangent function: it takes a ratio x and returns the angle θ whose tangent is x, so that tan(θ) = x. If a right triangle has an opposite side of length y and an adjacent side of length x, the ratio y/x is the tangent of the angle at that vertex, and θ = tan⁻¹(y/x) recovers the angle from the ratio. This calculator takes any real number x and computes θ = tan⁻¹(x) directly in degrees, radians, and degrees-minutes-seconds (DMS).

How the calculation works

Enter the value x (this can be any real number — positive, negative, zero, or very large — since tangent's range is all real numbers). The calculator evaluates θ = arctan(x) in radians using the standard math library function, then converts to degrees with θ° = θ_rad × 180/π. The decimal-degree result is further broken into degrees, minutes, and seconds: the whole-number part is degrees, the decimal part times 60 gives minutes, and the decimal part of that times 60 gives seconds.

Common mistakes

  • Confusing tan⁻¹(x) with 1/tan(x): tan⁻¹ is the inverse function (arctan) and returns an angle; 1/tan(x) is the cotangent, cot(x), and returns a ratio — they are not the same thing.
  • Ignoring the principal range: tan⁻¹(x) always returns a value strictly between -90° and 90°. If you know the angle actually lies in the second or fourth quadrant (for example, from negative x and y triangle legs), add or subtract 180° to get the true angle.
  • Mixing degrees and radians: always check which unit a downstream formula expects before plugging the result in — most calculus and physics formulas expect radians.

Real-world applications

  • Finding a slope's angle of inclination from rise/run in construction, roofing, and road grading.
  • Computing a bearing or heading in navigation and surveying from north/east displacement components.
  • Recovering an angle from a velocity or force's vector components (v_y/v_x) in physics and engineering.
  • Solving right triangles when two legs are known but the angle is not.

Frequently Asked Questions

What does tan⁻¹ mean?
tan⁻¹(x), also written arctan(x), is the inverse tangent function. It answers the question: what angle θ has a tangent equal to x? So if tan(θ) = x, then θ = tan⁻¹(x).
What is the range of tan⁻¹(x)?
The principal value of tan⁻¹(x) always falls strictly between -90° and 90° (-π/2 and π/2 radians), no matter how large or small x is, because that is the range mathematicians chose to make the inverse function single-valued.
Is tan⁻¹(x) the same as 1/tan(x)?
No. tan⁻¹(x) is the inverse function (arctan) that returns an angle. 1/tan(x) is the reciprocal of the tangent, which is the cotangent, cot(x), and returns a ratio, not an angle. Mixing these up is a very common mistake.
How do I convert the decimal-degree result to degrees, minutes, and seconds?
Keep the whole number of degrees, multiply the remaining decimal by 60 to get minutes, keep the whole number of minutes, and multiply the new remainder by 60 to get seconds. For example, 36.87° becomes 36° 52′ 12″.

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