How this instrument works
Inverse tangent, written atan(x) or tan⁻¹(x), answers 'which angle has a tangent of x?' It reverses ordinary tangent, and unlike its two siblings, inverse sine and inverse cosine, it places no restriction on the input at all — because plain tangent itself can output any real number, from deeply negative to deeply positive, atan(x) accepts every real number as a valid input in return.
The principal value convention here runs from −90° to 90°, matching inverse sine's range but for a different underlying reason: tangent itself is undefined exactly at ±90° (where cosine, its denominator, hits zero) and repeats its full range of values between each pair of those undefined points, so restricting to one such open interval keeps atan single-valued.
A two-argument cousin, atan2(y, x), extends this idea to full 2D direction-finding — where plain atan(x) alone can't distinguish a point from its exact opposite (since −3 ⁄ 4 and 3 ⁄ −4 give the same ratio), atan2 reads both the x and y components' individual signs and correctly places the angle in the right quadrant, the tool actually used behind vector-direction and polar-coordinate calculations elsewhere on this site.
- Enter any real number into the x field.
- Read atan(x): the sheet returns the corresponding angle, always between −90° and 90°.
- Try large positive and negative values to see the result approach, but never quite reach, ±90°.
Worked example — atan(1)
atan(1) works out to exactly 45° — the tangent of 45° is 1, since an isosceles right triangle's opposite and adjacent sides are equal by definition, making their ratio exactly 1.
atan(0) lands on exactly 0°, the trivial reference point. And atan(√3), approximately 1.732, works out to exactly 60° — one of the handful of clean trigonometric values tied to the 30-60-90 triangle's fixed side ratios, worth recognizing on sight rather than recomputing each time.
Questions
What does inverse tangent mean?
It answers 'which angle produces this tangent value?' — the reverse of ordinary tangent. atan(x) = θ means tan(θ) = x, restricted to a single well-defined angle between −90° and 90°.
Why doesn't inverse tangent restrict its input, unlike inverse sine or cosine?
Because ordinary tangent itself is unbounded — it can output any real number, unlike sine and cosine, which are both confined to −1 through 1. Since tangent's range covers every real number, inverse tangent's domain covers every real number in return.
What is atan2, and how is it different from atan?
atan2(y, x) is a two-argument version that reads both a point's x and y components separately, correctly placing the resulting angle in the right quadrant. Plain atan(x) alone can't distinguish a ratio from its point's exact opposite, since both give the identical ratio despite pointing in reverse directions.
What happens as the input grows very large?
atan(x) approaches, but never quite reaches, 90° as x grows toward positive infinity, and approaches −90° as x grows toward negative infinity — tangent's own vertical asymptotes at ±90° become inverse tangent's horizontal asymptotes instead.
Where is inverse tangent used practically?
Any time a slope needs converting to an angle — recovering a line's angle of inclination from its rise-over-run slope, or finding a ramp's steepness from its height-to-length ratio, both apply inverse tangent directly to that ratio.