Inverse Tan Calculator

Enter a tangent value and this inverse tan calculator returns the angle in degrees and radians, with step-by-step working, the exact pi form, the degrees-minutes-seconds breakdown, and the sine and cosine of the angle. Four modes cover the ways inverse tan actually comes up: a plain tangent value, opposite over adjacent for right triangles and slopes, a reverse mode that computes tan of an angle so you can check your answer both ways, and atan2(y, x) for quadrant-aware angles from vector components.

Quick Answer

How to use the inverse tan calculator, mode by mode

Pick the tab that matches the form your problem arrives in, enter the values, and press Calculate. Results stay hidden until you calculate, and switching tabs hides the previous result so a stale answer never sits next to new inputs.

Tangent value mode

Type any real number: positive, negative, decimal or zero. Choose whether the large main result displays in radians or degrees. The unit choice changes only that headline figure; the panel below it always lists both units, the pi form, the DMS breakdown, and the sine and cosine of the angle. If you are working through the rest of a triangle, the cosine calculator covers the adjacent-over-hypotenuse side of the same angle.

Opposite / Adjacent mode

Enter the two side lengths and the tool divides opposite by adjacent, then applies arctan. This is the fastest route from a slope to its angle: a ramp, a roof pitch, a road grade. Rise 3 over run 4 gives the worked example below. The adjacent side cannot be zero, because a vertical slope has no finite ratio; the tool reports “undefined” with an explanation instead of a made-up angle. For the gradient itself rather than its angle, an average slope calculator works in rise-over-run terms over a whole route.

Reverse: tan(angle) mode

Enter an angle and its unit, and the tool computes tan of that angle, then shows a check line: arctan of the answer, back in degrees. Run an arctan result through this mode and you should land where you started, which catches degree/radian mix-ups before they reach your homework or drawing.

atan2(y, x) mode

Enter the vertical and horizontal components of a point or vector. The tool returns the angle in degrees, from -180 to 180, names the quadrant (or the axis, when x is 0), and shows the radian value in the steps. Use this mode for bearings, rotations, and any coordinate work where direction, not just steepness, matters.

The math: what arctan really means

Arctan is the inverse of tangent on a restricted branch: if theta = arctan(x), then tan(theta) = x, with theta kept inside the principal range -90 to 90 degrees (-pi/2 to pi/2 radians), endpoints excluded. The NIST Digital Library of Mathematical Functions, in its inverse trigonometric functions chapter, defines the principal values the same way, and the calculator follows that definition through JavaScript’s Math.atan, converting with degrees = radians x 180/pi.

Three properties do most of the practical work. Arctan is an odd function, so arctan(-x) = -arctan(x): flip the sign of the input and the answer flips with it. Round trips are exact, so tan(arctan(x)) = x for every real x. And for positive x, arctan(x) + arctan(1/x) = pi/2, which is why arctan(0.75) and arctan(1.333…) sum to 90 degrees: they are the two acute angles of the same right triangle. The companion inverses follow the same pattern on their own branches: this site’s inverse sine calculator and inverse cosine calculator handle the ratios built from the hypotenuse.

Worked examples, verified against the tool

Every figure in this section was produced by running the calculator’s own code, not copied from a table. Enter the same inputs and you will get the same eight-decimal output.

Example 1: arctan(1), the pre-filled run

Step 1: x = 1. Step 2: theta = arctan(1) = 0.78539816 radians. Step 3: 0.78539816 x 180/pi = 45.00000000 degrees. The pi form is pi/4, the DMS readout is 45 deg 0′ 0.00″, and sin and cos both print 0.707107. Equal opposite and adjacent sides, a 45-45-90 triangle, is exactly the geometry behind this answer.

Example 2: a ramp angle from rise and run

A ramp rises 3 m over a horizontal run of 4 m. In Opposite / Adjacent mode the tool divides 3 / 4 = 0.75, then computes arctan(0.75) = 36.86989765 degrees (0.64350111 radians, DMS 36 deg 52′ 11.63″). The panel’s sin 0.600000 and cos 0.800000 are the giveaway that this is the 3-4-5 triangle in disguise: 3/5 and 4/5.

Example 3: atan2(-1, -1), where plain arctan is not enough

The ratio -1 / -1 is 1, so plain arctan reports 45 degrees, pointing up and to the right. The point (-1, -1) actually sits down and to the left. atan2 mode returns -135.00000000 degrees (-2.35619449 radians, exact form -3pi/4) and labels it Quadrant III. That 180 degree difference is the entire reason atan2 exists.

Common arctan values worth memorizing

These seven reference points let you sanity-check any arctan result at a glance. If your input sits between two rows, your answer must sit between their angles; if it does not, a unit or sign error has crept in.

One typing caution: the table’s angles are exact for the exact values. If you type the truncated decimal 1.73205081 instead of the full √3, the tool honestly reports 60.00000003 degrees, because that is the arctan of the number you actually entered. Type more digits (1.7320508075688772) when the last decimals matter.

arctan or atan2: choosing the right one

Use arctan when you already hold a single ratio and only need its angle: a tangent value, a slope, a grade. Its -90 to 90 degree range is a feature there, because a slope angle steeper than 90 degrees is not a slope any more. Use atan2 when you hold the two components separately and direction matters: vectors, bearings, rotations, screen coordinates, robotics. Programmers meet the same split in code, where, as the MDN documentation for Math.atan2 puts it, the two-argument form returns the angle of the point (x, y) from the positive x-axis, quadrant included.

Spreadsheet users get both as well: =ATAN(value) returns radians, =DEGREES(ATAN(value)) returns degrees, and =ATAN2 takes its arguments in the spreadsheet order (x first, then y), the reverse of the mathematical atan2(y, x) order used here and in most programming languages. Swapping them is a classic silent error; the quadrant label in this tool’s steps makes the swap visible.

What other inverse tan calculators miss

Most rival tools take one value and print one angle in one unit. After checking the top-ranking pages for this keyword, here is what they leave out, and what this page combines:

• Four modes, not one. The strongest rival we found pairs plain arctan with an atan2 mode, and one adds a graph. None of the top results combines value, opposite/adjacent, reverse tan checking, and atan2 with a quadrant readout on a single page.
• Worked answers that expose bad rival examples. One widely ranked rival states arctan results to only six decimals and gives no steps at all; another restricts its opposite/adjacent discussion to positive lengths without saying what a negative ratio does. Here, negative inputs are accepted, explained, and shown: -3 over 4 returns -36.86989765 degrees.
• The full panel. Degrees, radians, exact pi form, DMS, and sin/cos of the angle arrive together. Rivals typically stop at degrees plus radians, so the pi form students must quote, and the DMS surveyors use, need a second tool.
• Honest edges instead of giant numbers. A zero adjacent side and a 90 degree reverse input both print “undefined” with the reason. Several rival tools print numbers like 16331239353195370 for tan(90 degrees) because their code never checks the cosine.
• Steps you can copy. Every mode prints its arithmetic line by line, including the ratio division in slope mode and the quadrant line in atan2 mode.

Limitations and quirks, stated plainly

The tool returns the principal value only: arctan(1) is always 45 degrees, never 225 degrees, even though tan(225 degrees) also equals 1. For full-circle answers, add or subtract multiples of 180 degrees yourself, or use atan2 mode with the actual components. The result can approach but never equal 90 or -90 degrees, since that would need an infinite tangent.

Slope mode is ratio-only. A negative adjacent side flips the sign of the ratio exactly as a negative opposite side does (3 over -4 and -3 over 4 both return -36.86989765 degrees); it does not place the angle in another quadrant. If the direction of the vector matters, that is atan2’s job, and the atan2 quadrant label has one rough edge of its own: a point on the negative x-axis, such as (x, y) = (-3, 0), is labelled “Quadrant II” because the code groups y = 0 with the upper half-plane, even though the point sits on the axis itself at 180 degrees.

Two display quirks are preserved from the original tool and worth knowing. At x = 0 the exact pi form prints as “0pi” rather than “0”. And the DMS readout rounds seconds to two decimals without carrying a 60.00 into the minutes: 89.999999 degrees displays as 89 deg 59′ 60.00″. The angle and the other readouts are unaffected; only the seconds field shows the uncarried rounding. In reverse mode, the undefined test is |cos(angle)| below 0.0000000001, so 89.9999999999 degrees reports “undefined” while 89.999999 degrees returns the genuine, enormous tan value 57295779.81445014.

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