Arctan Calculator

What this calculator does

So, you know the tangent of an angle and you want the angle. This tool reverses tangent: enter a tangent value and it returns the angle, in radians and in degrees, to three decimal places.

One input box holds the tangent value. It is the tangent calculator run backwards, and it has one feature the arcsine and arccosine tools do not share, which is worth a section of its own.

How to use it

1. Type the tangent value, any number at all.
2. Press Calculate.

The answer comes in both radians and degrees together, ready to use in whichever your work calls for.

Arctangent undoes tangent

Tangent turns an angle into a ratio, the slope of the line at that angle. Arctangent, written arctan or tan with a small -1, turns that slope back into the angle. The name reads as "the angle whose tangent is this number." So arctan(1) means "the angle whose tangent is 1", which is 45 degrees. Whenever you have a slope or a ratio of rise to run and want to know the angle it corresponds to, arctangent is the operation that gives it to you.

Why it accepts any number

This is the real difference between arctangent and the arcsine or arccosine tools. Those two refuse anything outside -1 to 1, because sine and cosine are boxed into that range. Tangent is not. Tangent is height over width, a slope, and a slope can be anything: gentle like 0.1, steep like 50, or negative for a downhill line. There is no largest tangent. So arctangent happily accepts any number you give it, with no restriction, because every real number really is the tangent of some angle. That is why this tool never rejects your input the way the arcsine one does.

Recovering an angle from a slope

Here is where arctangent quietly earns its keep. The slope of a line, its rise over run, is exactly the tangent of the angle the line makes with the horizontal. So if you know a slope and want that angle, you take its arctangent. A slope of 1 gives 45 degrees, a slope of 0 gives a flat 0 degrees, and a steep slope gives an angle approaching, but never quite reaching, 90 degrees. This is why the answer always stays between -90 and 90 degrees: no straight line, however steep, is fully vertical, so its angle never quite reaches a right angle. It is the bridge from the gradient of a line back to the tilt you can picture.

A worked example

Enter 1. The tool returns the angle whose tangent is 1, which is 45 degrees, or about 0.785 radians. Enter 0 and it returns 0 degrees. Enter about 1.732 and it returns 60 degrees, a steeper line. Enter a large number like 1000 and it returns an angle just under 90 degrees, close to vertical but not quite there.

Radians and degrees

Both units are given together. Degrees are the everyday measure; radians measure by arc length on the unit circle and are standard in higher maths. For arctangent, the degree answer always falls strictly between -90 and 90, and the radian answer strictly between minus pi over 2 and pi over 2, approaching those limits for very large inputs but never reaching them.

Questions people ask

What does arctangent do?

It takes a tangent value, or a slope, and returns the angle that has that tangent. It is the inverse of the tangent function.

Why does it accept any number when arcsine does not?

Because tangent has no upper limit. A slope can be any size, so every real number is the tangent of some angle, and arctangent never needs to reject an input.

How do I get an angle from a slope?

Take the arctangent of the slope. Since a line's slope is the tangent of its angle, arctangent turns the slope straight back into that angle.

Why does the answer stay between -90 and 90 degrees?

Because no straight line is perfectly vertical, so its angle never reaches a right angle. Large slopes give angles close to 90 degrees but never equal to it.

Which unit is the answer in?

Both radians and degrees, shown at once, so you can use whichever you need.

References

On the inverse tangent and its range. Arctangent is the inverse of the tangent, defined for every real number, returning a principal angle strictly between -90 and 90 degrees, which is also the angle of a line with a given slope.

1. Eric W. Weisstein, "Inverse Trigonometric Functions," from MathWorld, a Wolfram resource, on the inverse trigonometric functions, including the inverse tangent, and their ranges.
2. "Inverse Trigonometric Functions," Calculus I, Lumen Learning, on the domain and range of the arctangent and the other inverse trigonometric functions.