Arcus Tangent Calculator - Arctan and arctan2 Solver

An arcus tangent calculator turns a tangent value into the angle that produced it. It applies the inverse of the tangent function so you can read an angle in degrees or radians from any real number, and it offers the two-argument arctan2 form for the correct quadrant when you know both x and y. Use it whenever you know a ratio of opposite to adjacent sides and need the matching angle, or whenever you need the bearing of a vector from its components.

• • Find an angle from a slope: If a roof pitch, ramp, or graph rises 3 units for every 4 units it runs, the arcus tangent of 3/4 gives the angle in degrees and radians.
• • Recover an angle from coordinates: Given (x, y), arctan2 returns the direction from the origin without confusing opposite-side tangents that look identical to arctan alone.
• • Convert trigonometry homework: When a problem gives you tan(θ) = 0.5774, the arcus tangent calculator returns θ = 30° without redoing the geometry by hand.
• • Compute a bearing or heading: Arctan2 of north and east components gives the heading of a moving object or a vector field point without manual quadrant checks.

In practice, the arcus tangent is one of the four basic inverse trig functions, along with arcsin, arccos, and arccot. Most scientific calculators label it as atan or tan⁻¹, but the layout of the keys does not change what the function does: it takes a ratio and returns an angle.

The single-argument form always returns an angle between -90° and 90° because the tangent function repeats every 180° and is not one-to-one across the full circle. The two-argument form, written arctan2(y, x) on this page, looks at the sign of both inputs so it can return a full -180° to 180° answer. The calculator offers both so you can pick the one that matches your inputs.

If you actually want the missing side of a triangle, the arcus tangent calculator is one half of the workflow, and the Right Triangle Calculator handles the Pythagorean and side-solving half.

Internally the calculator applies JavaScript's Math.atan or Math.atan2 to your inputs and then converts radians to degrees with the standard 180/π factor. Every result is shown in both units so you can paste it directly into a degree-based or radian-based formula.

• x: Real tangent value in arctan mode, or horizontal (east) component in arctan2 mode. Any real number is valid.
• y: Vertical (north) component used only in arctan2 mode. Combined with x to choose the correct quadrant.
• θ (theta): Resulting angle. In arctan mode it lives in (-90°, 90°); in arctan2 mode it lives in (-180°, 180°].

Math.atan2 is the standard two-argument inverse tangent used by virtually every programming language and graphics library. It returns a value in the closed interval (-π, π] (that is, -180° < θ ≤ 180°), and the four sign combinations of (x, y) place the result in quadrant I, II, III, or IV, which is exactly why the calculator also surfaces a Quadrant row.

The arcus tangent calculator surfaces the matching quadrant label as a third result so you can sanity-check the angle at a glance. If you need the same angle converted to DMS or to gradians for surveying, switch to the angle converter rather than re-applying the radian-to-degree factor by hand.

According to Wolfram MathWorld, the principal value of arctan(x) lives in the open interval (-π/2, π/2), while atan2(y, x) extends the range to (-π, π] using the signs of y and x to choose the correct quadrant.

According to Wikipedia (Atan2), atan2(1, -1) returns 3π/4 (135°) because the angle lies in quadrant II where y is positive and x is negative.

Once you have the radian result, Angle Converter lets you switch the same angle to gradians, DMS, or revolutions without re-doing the inverse tangent step.

These four ideas are enough to use this calculator correctly on any input you throw at it.

The principal value convention is what makes arctan a true function (one input, one output) instead of a relation. The trade-off is that some inputs collapse what would otherwise be two answers into one. That is why the calculator exposes arctan2 as a separate mode rather than overloading a single input box.

If you are working on a physics or engineering problem that expects an answer in radians, the radians row is your source of truth. If you are sketching a roof or laying out a ramp, the degree row is the practical one to copy.

For an in-depth walkthrough of the radian-to-degree factor behind the second output row, Radians to Degrees Calculator shows the same conversion in isolation with extra precision and a step-by-step breakdown.

Five short steps cover every workflow this calculator supports, from a simple slope to a vector bearing.

1. 1 Pick the mode: Choose arctan if you only have a tangent value, or arctan2 if you also know the vertical component y.
2. 2 Enter x: Type the tangent value or the horizontal component. Decimals, negatives, and large numbers are all accepted.
3. 3 Enter y when in arctan2: Type the vertical component. The calculator ignores y when the mode is arctan.
4. 4 Read both rows: The angle in degrees is the primary result; the radians row is the same angle in radian form for math and physics formulas.
5. 5 Check the quadrant label: In arctan2 mode the Quadrant row says I, II, III, or IV, which is the fastest way to confirm the result is on the side of the plane you expected.

A surveyor stands at the origin and needs the bearing of a tower at coordinates (-120, 95). Set mode to arctan2, type x = -120 and y = 95, and read off 141.6° plus the Quadrant = II label, which is exactly the northwest direction the tower is in.

The calculator saves time and avoids quadrant errors in workflows that already use arctan and arctan2 by hand.

• • Both arguments supported: Switch between arctan and arctan2 in one click instead of retyping the same numbers into a different tool.
• • Degrees and radians at the same time: The primary result is in degrees and the secondary result is in radians, so you do not have to remember the 180/π factor mid-calculation.
• • Quadrant label built in: Arctan2 mode returns a roman numeral I, II, III, or IV so you can confirm the bearing at a glance without drawing a coordinate plane.
• • Domain errors are explained: Instead of a silent NaN, the calculator tells you when atan2(0, 0) is undefined so you can fix the input.
• • Pairs with related angle tools: Use the result directly with the angle converter for gradians, the right triangle calculator to solve a triangle, or the radians-to-degrees calculator for one-off conversions.

If you are working through a problem set and bouncing between arctan and arctan2, the calculator removes the chance of using the wrong branch and getting an answer that is exactly 90° off. The same is true on a job site, where a 90° mistake in a roof pitch is a re-do, not a rounding error.

Because the calculator is just JavaScript running in the browser, it updates as you type and tolerates the negatives, fractions, and large slopes you would otherwise paste into a spreadsheet or type into a scientific calculator with the wrong mode set.

When the use case is a vector or coordinate pair, Vector Magnitude Calculator shows the matching magnitude workflow that turns the same x and y into a length.

Three things change the answer you should expect, plus two practical caveats about how inverse tangent is usually defined.

• • The single-argument arctan cannot return angles of exactly ±90° (the limit is approached but never reached), so a vertical line whose slope is infinite is outside its range; arctan2 handles that case correctly with x = 0.
• • The calculator reports the principal value of the inverse tangent. Other valid angles, such as 45° + 180°k for any integer k, are not shown, and you must add them yourself if the problem allows multiple solutions.

When you copy the radians value into a formula that expects a different unit, double-check that the next function in the chain does not also assume degrees. A common mistake is to feed a degree value into a sin() function in code, which silently treats it as radians and returns a small number.

If the result of arctan2 looks correct but does not match the textbook answer, the textbook may be using a 0° to 360° convention instead of -180° to 180°. In that case, add 360° to a negative result to land in the textbook range.

According to Wikipedia (Inverse trigonometric functions), the principal branch of arctan is defined for all real numbers with output range (-π/2, π/2), which corresponds to (-90°, 90°).

If the angle from this calculator is the central angle of a circular arc, Arc Length Calculator uses that angle together with the radius to return the arc length and chord length.