Problem 69 - Algebra and Trigonometry

Question

If you are given the equation of a cotangent function, how do you find a pair of consecutive asymptotes?

Step-by-Step Solution

Answer

A pair of consecutive asymptotes for a cotangent function can be found at \(x = n\pi\) and \(x = (n + 1)\pi\) where \(n\) is any integer.

Step 1: Understand the Cotangent Function

Cotangent function, represented as \(cot(x)\), is a periodic function with a period of \(\pi\) radians (or 180 degrees). There are vertical asymptotes at the endpoints of every period.

Step 2: Identify Key Points and Period

For any given cotangent function, you start with the period at zero. Since cotangent function has a period of \(\pi\), the consecutive periods will be at \(n\pi\) where \(n\) is any integer (both positive and negative). The endpoints of these periods, \(n\pi\), correspond to vertical asymptotes.

Step 3: Locate the Asymptotes

With the critical points identified, the asymptotes for a cotangent function will thus be at \(x = n\pi\) for all integer \(n\). Pick successive integers to find a pair of consecutive asymptotes. For example, for consecutive integers 1 and 2, the consecutive asymptotes will be at \(\pi\) and \(2\pi\) respectively.