Cotangent Calculator

What this calculator does

So, you have an angle and you want its cotangent. This tool takes the angle, in degrees or radians, and returns the cotangent rounded to three decimal places, or tells you when it is not defined.

One box for the angle, a dropdown for degrees or radians, and a Calculate button.

How to use it

1. Type your angle.
2. Choose degrees or radians to match it.
3. Press Calculate.

Match the unit to your value. Where the cotangent has no value, the tool returns "Not defined" instead of a misleading figure.

Cotangent is one over tangent

Cotangent is the reciprocal of tangent, and that is precisely how the tool finds it: it computes the tangent of your angle and divides 1 by it.

cot(angle) = 1 / tan(angle)

So where the tangent is steep and large, the cotangent is small, and where the tangent is small, the cotangent is large. It is the third of the reciprocal trigonometric functions, sitting alongside secant, which flips cosine, and cosecant, which flips sine.

The other way to see it: cosine over sine

There is a second, often handier form. Since tangent is sine over cosine, flipping it turns cotangent into cosine over sine. Where tangent is height over width, cotangent is width over height, the same two lengths the other way up. That makes cotangent the natural measure when you care about how far across a thing goes for each unit of height, rather than the other way round. Both forms, one over tangent and cosine over sine, give the identical value; the tool uses the reciprocal of tangent.

Where cotangent is undefined

Cotangent is cosine over sine, so it breaks down wherever sine is zero. That happens at 0 degrees, at 180 degrees, at 360 degrees, and at every whole multiple of 180. At those angles you would be dividing by zero, so the cotangent is undefined, and the tool reports "Not defined" there. Notice these are different angles from where tangent and secant fail. Tangent and secant break where cosine is zero, at odd multiples of 90; cotangent breaks where sine is zero, at multiples of 180. Each function is undefined exactly where the quantity on its bottom vanishes.

A worked example

Take 45 degrees. The tangent of 45 degrees is 1, so the cotangent is 1 divided by 1, which is 1.000. At 30 degrees the tangent is about 0.577, so the cotangent is about 1.732, its reciprocal. And at 0 degrees, where the sine is zero, the tool returns Not defined.

The "co": cotangent and the complement

Like cosine, cotangent carries a "co" for a reason. Just as cosine is the sine of the complementary angle, cotangent is the tangent of the complementary angle, the one that completes it to 90 degrees. So the cotangent of 20 degrees equals the tangent of 70 degrees. This is the same complement pattern that links every "co" function to its plain partner, and it is why cotangent behaves like a mirror image of tangent, large where tangent is small and undefined where tangent is zero.

Questions people ask

What is the cotangent of an angle?

The reciprocal of the tangent, 1 divided by the tangent. Equivalently, cosine divided by sine.

Which is it, one over tangent or cosine over sine?

Both. They are equal. Flipping sine over cosine gives cosine over sine, which is the same as one over tangent.

When is cotangent undefined?

Wherever sine is zero: at 0 degrees, 180 degrees, and every multiple of 180. There the division has no value.

How is it different from tangent's undefined points?

Tangent fails where cosine is zero, at odd multiples of 90. Cotangent fails where sine is zero, at multiples of 180. Different angles, because they divide by different things.

Why the "co" in cotangent?

Because it is the tangent of the complementary angle. The cotangent of an angle equals the tangent of the angle that completes it to 90 degrees.

References

On cotangent as the reciprocal of tangent. Cotangent is 1 over the tangent, equally cosine over sine, one of the reciprocal circular functions, undefined where sine vanishes.

1. Eric W. Weisstein, "Trigonometric Functions," from MathWorld, a Wolfram resource, on the cotangent and the full set of circular functions.
2. "Trigonometric functions," MacTutor History of Mathematics, on tangent and cotangent developing together from shadow measurement.