Cotangent
Online calculator — enter the values and get the result instantly, with the formula and a worked example.
α = angle alpha
a = side "a"
b = side "b"
c = side "c"
The cotangent of an angle, written cot, is one of the six trigonometric functions. In a right triangle it is the ratio of the side adjacent to the angle to the side opposite it, which makes it the reciprocal of the tangent. It can also be viewed as the cosine of the angle divided by its sine, so it is undefined wherever the sine equals zero, such as at 0° and 180°. Its graph repeats every 180°, forming separate branches that fall steeply from very large positive values toward very large negative ones between consecutive vertical asymptotes. As the angle approaches a right angle the cotangent passes through zero, and unlike sine and cosine it is unbounded, taking every real value.
Cotangent appears throughout trigonometry, geometry, and physics, especially when a slope or an angle is more naturally described by the horizontal-to-vertical ratio than the other way round. Surveyors, engineers, and navigators rely on it to compute heights, distances, and inclinations, and it shows up in calculus, wave analysis, and computer graphics. Because it complements the tangent, knowing one immediately gives the other, which is why cotangent remains a handy tool for solving triangles and modelling periodic behaviour.
cot(θ)
–
Drag the point around the circle — or enter an angle
Worked example
Let's assume we have a right triangle ABC, where ∠C=90o. Side a is adjacent to angle α, side b is opposite to angle α, and c is the hypotenuse (the longest side of the triangle).
The definition of the cotangent of angle α is:
The cotangent of an angle (cot) is the ratio of the length of the adjacent side to the opposite side.
For this example:
cot α = a/b
Example:
Assume the side lengths are:
- a = 3
- b = 4
- c = 5
- Verify that the triangle is right-angled using the Pythagorean theorem:
a2 + b2 = c2
32 + 42 = 52
9 + 16 = 25
Since the equality holds, it is a right triangle.
2. Calculate the cotangent of angle α:
cot α = a/b = 3/4 = 0.75
Result:
cot α = 0.75
Thus, the cotangent of angle α in this right triangle is 0.75.
