Cotangent

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.