Cotangent Calculator
Enter any angle and choose degrees, radians, or multiples of pi. The calculator returns the exact cotangent value, shows the equivalent tangent (reciprocal), and walks through the computation step by step. A reference table of common angles and a graph of the full period are included below.
What is the cotangent?
The cotangent (cot) is one of the six fundamental trigonometric functions. In a right triangle, cot(x) is the ratio of the adjacent side to the opposite side: cot(x) = adjacent / opposite. This is the reciprocal of the tangent, so cot(x) = 1 / tan(x). Using the unit circle definitions, cotangent can also be written as cot(x) = cos(x) / sin(x), which works for any angle, not just those in a right triangle. Because cot(x) depends on sin(x) in the denominator, it is undefined whenever sin(x) = 0, that is at 0 degrees, 180 degrees, 360 degrees, and any integer multiple of pi radians.
How to calculate cotangent
There are three equivalent formulas: (1) cot(x) = adjacent / opposite, from the right-triangle definition. (2) cot(x) = cos(x) / sin(x), using the unit-circle definitions of sine and cosine. (3) cot(x) = 1 / tan(x), the reciprocal identity. For a 45-degree angle, sin(45) = cos(45) = sqrt(2)/2, so cot(45) = 1. For a 30-degree angle, sin(30) = 1/2 and cos(30) = sqrt(3)/2, giving cot(30) = sqrt(3) approximately 1.7321. If your calculator does not have a cot key, compute tan(x) first and take its reciprocal.
Degrees, radians, and pi-multiples
Angles can be expressed in three ways. Degrees split the full circle into 360 equal parts, the most common choice in everyday geometry and navigation. Radians measure the angle as the length of the arc on a unit circle, with a full circle equal to 2*pi radians. Pi-multiples (or "turns in pi") express the angle as a fraction of pi, so pi/4 radians is written as 0.25 in this mode. To convert, use degrees = radians * 180 / pi, or radians = degrees * pi / 180. This calculator accepts all three forms and shows both the degree and radian equivalents in the output.
Properties and graph of cot(x)
The cotangent function has a period of pi (180 degrees), half the period of sine and cosine. It is positive in the first and third quadrants (0 to 90 and 180 to 270 degrees) and negative in the second and fourth quadrants. The graph looks like a decreasing curve with vertical asymptotes at every integer multiple of pi: as the angle approaches 0 from above, cot(x) rises to positive infinity; as it approaches pi from below, cot(x) falls to negative infinity. The function passes through zero at pi/2 (90 degrees) and 3*pi/2 (270 degrees). There is no maximum or minimum value: the range is all real numbers.
Inverse cotangent (arccotangent)
If you know the cotangent value and want the angle, use the inverse cotangent: x = arccot(value). Most calculators do not have a dedicated arccot key, but since cot(x) = 1/tan(x), the inverse is arccot(v) = arctan(1/v). For example, cot(x) = 2 means tan(x) = 0.5, so x = arctan(0.5) approximately 26.57 degrees. A small adjustment is needed for the correct quadrant when v is negative: add 180 degrees to bring the result into the 90-to-180-degree range if desired. The principal value of arccot is typically defined on the interval (0, pi), or equivalently 0 to 180 degrees.
Cotangent values at common angles
| Degrees | Radians | cot(x) | Exact form |
|---|---|---|---|
| 0 | 0 | undefined | - |
| 15 | pi/12 | 3.7321 | 2 + sqrt(3) |
| 30 | pi/6 | 1.7321 | sqrt(3) |
| 45 | pi/4 | 1 | 1 |
| 60 | pi/3 | 0.5774 | 1/sqrt(3) |
| 75 | 5pi/12 | 0.2679 | 2 - sqrt(3) |
| 90 | pi/2 | 0 | 0 |
| 120 | 2pi/3 | -0.5774 | -1/sqrt(3) |
| 135 | 3pi/4 | -1 | -1 |
| 150 | 5pi/6 | -1.7321 | -sqrt(3) |
| 180 | pi | undefined | - |
| 210 | 7pi/6 | 1.7321 | sqrt(3) |
| 225 | 5pi/4 | 1 | 1 |
| 270 | 3pi/2 | 0 | 0 |
| 315 | 7pi/4 | -1 | -1 |
| 330 | 11pi/6 | -1.7321 | -sqrt(3) |
| 360 | 2pi | undefined | - |
Key angles from 0 to 360 degrees with their radian equivalents and exact cotangent values. Cotangent is undefined at 0, 180, and 360 degrees (integer multiples of pi) where sine equals zero.
Frequently asked questions
What is cot(45 degrees)?
cot(45 degrees) = 1. At 45 degrees, sin and cos are both equal to sqrt(2)/2, so their ratio is exactly 1. This is one of the simplest exact values of the cotangent function.
Why is cotangent undefined at 0 and 180 degrees?
Cotangent is defined as cos(x) / sin(x). At 0 degrees and 180 degrees (and every multiple of 180 degrees), sin(x) = 0. Division by zero is undefined in standard arithmetic, so cot(0) and cot(180) do not exist. The graph of cot(x) shows vertical asymptotes at each of these angles.
How do I convert from degrees to radians?
Multiply the degree value by pi / 180. For example, 90 degrees = 90 * pi / 180 = pi/2 approximately 1.5708 radians. To go the other way, multiply radians by 180 / pi. This calculator accepts both units so you do not need to convert manually.
Is cotangent the same as 1 divided by tangent?
Yes. By definition, cot(x) = 1 / tan(x) = cos(x) / sin(x). This reciprocal identity is the easiest way to compute cotangent on a standard scientific calculator: find tan(x), then press the reciprocal key (1/x). The only exception is when tan(x) = 0, meaning the angle is 90 degrees or 270 degrees, where cotangent equals zero but tangent is zero (not undefined), so the reciprocal of zero is still undefined. Wait - tangent equals zero at 0, 180, 360 degrees, and cotangent is undefined there. Tangent is undefined at 90 and 270 degrees, where cotangent equals zero.
What is the period of the cotangent function?
The period of cot(x) is pi radians, or 180 degrees. This means cot(x + pi) = cot(x) for all x where the function is defined. Because the period is pi rather than 2*pi, the cotangent curve repeats twice as often as sine or cosine within a full 360-degree rotation.
How do I find the angle if I know the cotangent value?
Use the arccotangent function: x = arccot(v). Since most calculators lack an arccot key, you can compute arctan(1/v) instead. For a positive value v, this gives an angle in the first quadrant (0 to 90 degrees). For a negative value, arctan(1/v) gives an angle in the fourth quadrant; add 180 degrees to get the principal arccotangent value in the range (90 to 180 degrees).