Arccot Calculator

What this calculator does

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

One input box holds the cotangent value. It is the cotangent calculator run backwards, and because cotangent is a reciprocal, the way it is computed is worth spelling out.

How to use it

1. Type the cotangent value, any number.
2. Press Calculate.

The result appears in both radians and degrees at once, ready for whichever your work uses.

Arccotangent undoes cotangent

Cotangent turns an angle into a ratio, the reciprocal of its tangent. Arccotangent, written arccot or cot with a small -1, turns that ratio back into an angle. The name reads as "the angle whose cotangent is this number." So arccot(1) means "the angle whose cotangent is 1", which is 45 degrees. Like arctangent, it accepts any number, since cotangent, being a slope of sorts turned upside down, can take any real value.

How this tool computes it

Rather than treat arccotangent as a brand-new function, the tool builds it from arctangent, using the relationship

arccot(x) = arctan(1 / x)

It flips your cotangent value into its reciprocal, then takes the arctangent of that. This works because cotangent is one over tangent, so the angle whose cotangent is x is the same as the angle whose tangent is one over x. It is a clean shortcut: any tool that can do arctangent can do arccotangent by inverting the input first.

The two conventions, and which one you are seeing

Arccotangent is one of those functions where different textbooks genuinely disagree, and it is only fair to be upfront about it. Because it is built here as the arctangent of the reciprocal, this tool returns an angle between -90 and 90 degrees, matching arctangent's own range. Some textbooks and references instead define arccotangent with a range from 0 to 180 degrees, which keeps the function continuous across all inputs. Neither is wrong; they are two accepted conventions. So if an answer here differs from one in a particular textbook by 180 degrees for a negative input, that is the convention gap, not an error. For a positive cotangent value, the common everyday case, both conventions agree.

A worked example

Enter 1. The tool flips it to 1 and takes the arctangent, giving the angle whose cotangent is 1, which is 45 degrees, or about 0.785 radians. Enter about 1.732 and it flips to roughly 0.577, whose arctangent is 30 degrees. Enter 0 and, since one over zero is treated as very large, it returns 90 degrees, which is right, because the cotangent of 90 degrees is 0.

Radians and degrees

Both units come back together. Degrees are the everyday measure; radians measure by arc length on the unit circle and are standard in higher maths. Under this tool's convention, the degree answer falls between -90 and 90, mirroring the arctangent it is built from.

Questions people ask

What does arccotangent do?

It takes a cotangent value and returns the angle that has that cotangent. It is the inverse of the cotangent function.

How does the tool work it out?

It uses arccot(x) = arctan(1 / x): it flips your value into its reciprocal and takes the arctangent of that.

Why might my textbook give a different answer?

Because arccotangent has two accepted range conventions. This tool returns -90 to 90 degrees; some texts use 0 to 180. For negative inputs the two can differ by 180 degrees, though for positive inputs they agree.

Does it accept any number?

Yes. Cotangent can take any real value, so arccotangent accepts any input, just like arctangent.

Which unit is the answer in?

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

References

On the inverse cotangent and its conventions. Arccotangent is the inverse of the cotangent, here built as the arctangent of the reciprocal, and its range depends on the convention chosen.

1. Eric W. Weisstein, "Inverse Cotangent," from MathWorld, a Wolfram resource, which notes that care is needed because the range and form differ by convention.
2. Eric W. Weisstein, "Inverse Trigonometric Functions," from MathWorld, a Wolfram resource, on the inverse trigonometric functions and the differing conventions for their ranges.