Arctan Calculator

What arctan does

The arctangent is the inverse of the tangent: if tan θ = x, then arctan x = θ. It is also written atan(x) or tan⁻¹(x). Since the tangent of an angle is the ratio opposite ÷ adjacent, which is also the slope of a line, arctan converts a slope or a ratio of two lengths into an angle.

• Domain: every real number, with no restrictions, unlike arcsine and arccosine.
• Range: strictly between −90° and 90° (−π/2 and π/2 radians). The endpoints are never reached; the angle only creeps towards them for very large positive or negative inputs.
• Symmetry: arctan(−x) = −arctan(x).
• Growth: it climbs steeply near 0, where arctan x ≈ x (in radians), and flattens out for large |x|.

Only one of the infinitely many angles with a given tangent is reported, the principal value. Others differ by whole multiples of 180°, because the tangent repeats every half turn.

How to use the calculator

Type a value in the Tangent box and press Calculate. A decimal (0.5, -6) or a fraction written with a slash and no spaces (3/4, -4/3) is accepted. You get two lines, both labelled arctan(x): the angle in degrees, then the angle in radians.

Results are rounded to six decimals. A whole number is shown bare (arctan 1 = 45 degrees), a result with at most two decimals gets two, three or four decimals are padded to four (arctan 0.001 = 0.0010 radians) and everything else keeps six (26.565051 degrees for an input of 0.5).

A fraction with a zero denominator is read as 0, and so is text that is not a number. Scientific notation such as 1e1 is read as a number (10).

Arctan of selected values

Worked examples: from slope to angle

Road grade. A grade of 8% means the road climbs 8 m for every 100 m travelled horizontally, a slope of 0.08. The angle is arctan 0.08 = 4.573921° (0.079830 radians). Notice that grade is a tangent, not a sine: the horizontal distance, not the distance along the surface, is in the denominator.

Roof pitch. A 5-in-12 roof rises 5 units for every 12 horizontal. Enter 5/12 to see arctan = 22.619865°. A 6-in-12 roof has slope 0.5, which is 26.565051°.

Right triangle. Legs of 3 and 4: the angle opposite the side of 3 is arctan(3/4) = 36.869898°, and the other acute angle is arctan(4/3) = 53.130102°. The two add up to 90°, as they must.

Estimating arctan by hand

For |x| not larger than 1, the Gregory–Leibniz series gives the angle in radians:

arctan x = x − x³/3 + x⁵/5 − x⁷/7 + …

Example: x = 0.2. The terms are 0.2, −0.002667, +0.000064 and −0.000002. Their sum is 0.197396 radian, which converts to 11.3099° after multiplying by 180 ÷ π. Setting x = 1 shows that π ÷ 4 = 1 − 1/3 + 1/5 − …, though that series converges far too slowly to be practical. For |x| above 1, use arctan x = 90° − arctan(1/x) when x is positive.

The quadrant pitfall

Arctan only sees the ratio y ÷ x, not the signs of x and y separately, so it can never return an angle outside −90° to 90°. If a point has coordinates (−1, 1), the ratio is −1 and arctan gives −45°, but the point actually lies at 135° from the positive x-axis. The remedy is to add 180° whenever x is negative: −45° + 180° = 135°. Programming languages package this fix as the two-argument function atan2(y, x).

Related tools: the tangent calculator goes from angle to ratio, the trigonometry calculator solves complete triangles, and arccotangent handles the reciprocal case. Browse the whole trigonometry section for the rest.