91Ó°ÊÓ

Cotangent, often abbreviated as cot, is a trigonometric function closely related to tangent. It is defined as the reciprocal of the tangent function.
In mathematical terms, cotangent is expressed as \(\text{cot}(x) = \frac{1}{\text{tan}(x)}\).
It essentially measures the ratio of the adjacent side to the opposite side in a right-angled triangle as opposed to tangent, which measures the opposite to adjacent side.
The cotangent function is periodic and repeats its values every \(Ï€\) units. This means if you graph \( \text{y} = \text{cot}(x) \), the shape of the graph will repeat itself every \(Ï€ \). It's important to understand this foundational period when dealing with cotangent transformations.

Function transformation involves changing the standard form of a function.
Transformations can include shifts, stretches, shrinks, and reflections.
In this problem, we encounter a specific transformation where the x-variable is multiplied by a constant factor.
Consider the function \( \text{y} = \text{cot}(\frac{\text{Ï€x}}{2}) \). To understand how this affects the function, it helps to focus on that coefficient inside the argument.
What’s happening here is a horizontal stretch or shrink. Multiplying x by \( \frac{\text{π}}{2}\) changes the rate at which the cotangent function repeats.

The period of a transformed trigonometric function can be found by understanding the effect of the transformation on the original period.
For the cotangent function, the original period is \(Ï€\). When the argument of the function \(x\) is transformed by a factor \(k\), such as in \(\text{y} = \text{cot}(\frac{\text{Ï€x}}{2})\), the new period is given by the formula \(\frac{\text{Ï€}}{|\text{k}|}\).
In this problem, \(k\) is \( \frac{\text{Ï€}}{2}\). Substituting \(k\) into the formula gives \( \frac{\text{Ï€}}{|\frac{\text{Ï€}}{2}|} = 2\).
Therefore, the period of \(\text{y} = \text{cot}(\frac{\text{Ï€x}}{2})\) is 2. This means the function will repeat its values every 2 units.