Inverse Tan Calculator

Find any inverse tangent in seconds. Enter a tangent value and this inverse tan calculator returns the angle in both degrees and radians, with step-by-step working, the exact π form, the degrees-minutes-seconds breakdown, and the sine and cosine of the angle.

Three modes cover every way inverse tan comes up in real work: a plain tangent value, the opposite and adjacent sides of a right triangle (for example rise over run on a slope), and a reverse mode that computes tan of an angle so you can check your answer both ways.

How to use the inverse tan calculator

Pick the tab that matches your problem. In Tangent value mode, type any real number into the value field, choose whether the main result should display in degrees or radians, and press Calculate. Negative numbers and decimals are all fine, because the domain of arctan is every real number.

In Opposite / Adjacent mode, enter the two side lengths of a right triangle and the tool divides them, then returns the angle. This is the fastest way to get a slope angle from rise over run, and it pairs well with the slope calculator when you need the full line analysis. Note the adjacent side cannot be zero: a zero adjacent side means a vertical slope, and the calculator tells you the angle is undefined there.

In atan2(y, x) mode, enter the two components of a point or vector and the tool returns the full-circle angle from -180 to 180 degrees with its quadrant, which plain arctan cannot do. In Reverse: tan(angle) mode, enter an angle and its unit and the tool computes the tangent of that angle, with a verification step showing arctan of the answer. It is the quickest way to sanity-check a result: run arctan forward, then run the answer back through the reverse mode and confirm you land where you started.

Every calculation fills a result panel that always shows both degrees and radians, the exact π form, the degrees-minutes-seconds breakdown, and the sine and cosine of the resulting angle. The steps box under the panel shows the arithmetic line by line. Copy my results puts the whole summary on your clipboard, and Share results opens your device share sheet so you can send it anywhere.

The math: what arctan really means

The arctangent is the inverse of the tangent function. If θ = arctan(x), then tan(θ) = x. It answers the question: which angle has this tangent value? Since tan(45°) = 1, it follows that arctan(1) = 45°.

The domain of arctan is all real numbers, from negative infinity to positive infinity. The tangent function repeats every 180°, so to make the inverse single-valued it is restricted to the principal range: −90° to 90° (equivalently −π/2 to π/2 radians), endpoints excluded. Every input maps to exactly one angle in that range.

Three facts make arctan easy to work with. It is an odd function, so arctan(−x) = −arctan(x): flipping the sign of the input flips the sign of the answer. Round trips are exact: tan(arctan(x)) = x for every real x. And for positive x, arctan(x) + arctan(1/x) = π/2. The calculator computes Math.atan(x) in radians first, then converts with degrees = radians × 180/π, which is exactly how scientific calculators and spreadsheet functions like Excel ATAN work internally.

Verified worked example

Example 1: arctan(1). Enter 1 in Tangent value mode. Step 1: x = 1. Step 2: θ = arctan(1) = 0.78539816 radians. Step 3: convert to degrees: 0.78539816 × 180/π = 45.00000000°. The exact form is π/4, and the DMS readout is 45° 0' 0.00". As a check, sin(45°) = 0.707107 and cos(45°) = 0.707107, and tan(45°) = 1, which closes the loop. Every figure here was produced by running the actual tool logic, not copied from a table. An angle of 45° from equal opposite and adjacent sides is the classic setup of the 45-45-90 triangle.

Example 2: a ramp angle from side lengths. A ramp rises 3 m over a horizontal run of 4 m. Switch to Opposite / Adjacent mode, enter 3 and 4. The tool divides: 3 / 4 = 0.75, then computes arctan(0.75) = 36.86989765° (0.64350111 radians). That is the ramp's angle of inclination, accurate to eight decimals.

Common arctan values

Memorizing these seven reference points lets you estimate any arctan result at a glance, and they double as a quick accuracy check for the tool.

arctan and atan2: why your programming language has both

arctan(x) takes one argument, a single ratio, and always returns an angle between −90° and 90°. That is perfect when you already have the ratio, but it cannot tell which quadrant a 2D vector points in: the ratios 1/1 and −1/−1 both give arctan(1) = 45°, even though the vectors point in opposite directions.

The two-argument variant atan2(y, x) fixes this by taking the components separately and returning the full angle from −180° to 180°, handling all four quadrants correctly. Use atan2 in code (JavaScript Math.atan2, Python math.atan2, Excel ATAN2) whenever you are working with vector components, bearings, or rotations. Use arctan when you already have one tangent ratio and just want its angle. For the single-argument cases on the sine and cosine side, the inverse sine calculator and the inverse cosine calculator cover the same ground.

What other inverse tan calculators miss

Most competing tools do exactly one thing: take a single value and print one angle in one unit. After testing the top-ranking pages, here is what this calculator adds that they do not combine in one place:

• Four input modes. A plain value mode, an opposite-over-adjacent mode for slope and right-triangle problems, a reverse tan-of-angle mode for checking work, and an atan2(y, x) mode with quadrant readout for vectors, bearings, and rotations. The strongest competitors offer at most one of these extras.
• Both units, every time. The main result honors your unit choice, and the panel always shows the other unit too, so you never convert twice.
• sin and cos of the angle. Useful the moment you need the full trig picture of the angle you just found.
• Exact π form and DMS. Results like π/4 and 45° 0' 0.00" appear automatically where the fraction is simple, which is what students and engineers actually write down.
• Step-by-step working. Every mode prints the arithmetic line by line, so the answer can be copied into homework or a report.
• Honest edge handling. A zero adjacent side or a 90° input in reverse mode says "undefined" and explains why, instead of printing a misleading giant number.

Limitations

This tool returns the principal value of arctan: arctan(1) is always 45°, never 225°, even though tan(225°) also equals 1. If you need full-circle angles, add or subtract multiples of 180° yourself, or use atan2(y, x) in code. The result can approach but never equal ±90°, since that would require an infinite tangent. In reverse mode, tan is undefined at 90° plus any multiple of 180°, and the tool says so explicitly. Displayed values are rounded to eight decimals; the exact π form is shown wherever the fraction reduces to a simple one.

FAQs

Is tan⁻¹ the same as 1 divided by tan?

No. The superscript −1 in tan⁻¹(x) means inverse function, not reciprocal: it returns the angle whose tangent is x. The reciprocal 1/tan(x) is a different function entirely, the cotangent. So tan⁻¹(1) = 45°, while 1/tan(1°) is about 57.29.

Can arctan ever equal 90 degrees?

No. The range of arctan is the open interval (−90°, 90°). As x grows toward infinity, arctan(x) approaches 90° as a horizontal asymptote but never reaches it, because reaching it would require the tangent to be infinite. The same holds for −90° on the negative side.

Why did I get a negative angle?

Arctan is an odd function: a negative ratio always produces a negative angle between −90° and 0°. For example, arctan(−1) = −45°. If your application needs a 0° to 360° bearing, add 360° to the result (−45° becomes 315°).

How do I calculate arctan on a scientific calculator?

Press Shift (or 2nd on TI models), then the tan key: the display shows tan⁻¹ or atan. Type your value and press equals. Check the angle mode first: the same input returns 0.7854 in radian mode and 45 in degree mode, and mixing them up is the most common student error.

What is the difference between arctan and atan2?

arctan(x) takes one ratio and returns an angle between −90° and 90°. atan2(y, x) takes the two components separately and returns the full angle between −180° and 180°, correctly identifying the quadrant. Use atan2 for vectors, bearings, and rotations; use arctan when you already have a single tangent ratio.

How do I compute arctan in Excel or Google Sheets?

Use =ATAN(value), which returns radians. For degrees, wrap it: =DEGREES(ATAN(value)). Both programs also support =ATAN2(x_num, y_num) for the two-argument quadrant-aware version.

Which angle has a tangent of √3?

60°, or π/3 radians. Since tan(60°) = √3, arctan(√3) = 60°. It is one of the seven reference values in the table above, and a quick way to check any arctan tool: type 1.73205081 and confirm you get 60°.