91Ó°ÊÓ

Understanding the concept of the period is crucial when working with trigonometric functions like the cotangent. Generally, the period of a function is the interval after which it repeats its values. For the basic cotangent function, the period is \(\pi\), meaning it repeats its pattern and behavior every \(\pi\) units.
However, when transformations are applied, such as \(\cot\left(\frac{\pi}{3} x\right)\), the period changes. In this modified function, the \(x\)-variable is multiplied by a fraction, \(\frac{\pi}{3}\), which affects the period length. The period becomes \(3\pi\).
To determine the new period, use the formula for transformed trigonometric functions:\[ \text{New period} = \frac{\text{Original period}}{|b|} \]where \(b\) is the coefficient of \(x\) in the argument. Thus, calculating for the modified cotangent:\[ \text{New period} = \frac{\pi}{|\frac{\pi}{3}|} = 3\pi \]
This means that every \(3\pi\) units, the function repeats its values, creating a new repeating pattern.

Vertical asymptotes are important features of the cotangent function. They are the \(x\)-values where the function does not exist and it tends to infinity. For the standard cotangent function \(\cot(x)\), vertical asymptotes occur at every integer multiple of \(\pi\), such as \(x = 0, \pi, 2\pi\), etc.
These asymptotes occur because the cotangent is the reciprocal of the tangent function, and wherever the tangent equals zero, the cotangent becomes undefined, shooting up to positive or negative infinity.
For the function \(\cot\left(\frac{\pi}{3} x\right)\), the asymptotes are moved by the transformation. Use this principle to locate them:

• Set the inner argument equal to each standard asymptote value: \(\frac{\pi}{3}x = \pi \times n\) (where \(n\) is an integer).
• Solve for \(x\): \(x = 3 \times n\).

This results in vertical asymptotes at \(x = 0, 3, 6, \ldots\)