Cotangent is one of the six fundamental trigonometric functions. It is defined as the reciprocal of the tangent function. In a right triangle, cotangent of an angle theta equals the adjacent side divided by the opposite side. This makes it particularly useful in various mathematical and engineering applications.
There are three equivalent ways to define cotangent. First, in a right triangle, cotangent equals the adjacent side divided by the opposite side. Second, cotangent is the reciprocal of tangent, meaning one divided by tangent. Third, cotangent equals cosine divided by sine. These three definitions are mathematically equivalent and provide different perspectives for understanding this function.
On the unit circle, we can visualize cotangent geometrically. The x-coordinate represents cosine theta, and the y-coordinate represents sine theta. Therefore, cotangent equals x divided by y, or cosine divided by sine. Notice that cotangent becomes undefined when sine equals zero, which occurs at zero, one hundred eighty, and three hundred sixty degrees, where the point lies on the x-axis.
The cotangent function has a distinctive graph with vertical asymptotes. Unlike sine and cosine which are continuous, cotangent has breaks at multiples of pi where sine equals zero. Between these asymptotes, the function decreases from positive infinity to negative infinity. The period of cotangent is pi, meaning it repeats every pi units. Notice the vertical asymptotes at zero, pi, and two pi, where the function is undefined.
Cotangent has many practical applications in engineering, physics, navigation, and computer graphics. It's particularly useful when you know the opposite and adjacent sides of a triangle and need to find the angle. The function has a period of pi and is undefined at integer multiples of pi. For example, cotangent of thirty degrees equals square root of three. Understanding cotangent completes your knowledge of the fundamental trigonometric functions and their relationships.