Problem 69 - Precalculus Essentials

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 the cotangent function can be found at \( x = k\pi \) and \( x = (k+1)\pi \), where \( k \) is any integer.

Step 1: Define the cotangent function

An important thing to understand is how the cotangent function is defined. It is the reciprocal of the tangent function, or equivalently the ratio of the cosine to the sine of an angle. The function is defined as \( \cot(x) = \frac{\cos(x)}{\sin(x)} \).

Step 2: Calculating Asymptotes

Vertical asymptotes occur when the function is undefined, which for the cotangent function is when the sine of the angle is zero. The sine of an angle is zero at every integer multiple of \(\pi\). That implies the cotangent function, \( \cot(x) \), is undefined at \( x = k\pi \) for each integer \( k \). These values are the vertical asymptotes of the cotangent function.

Step 3: Find a pair of consecutive asymptotes

A pair of consecutive asymptotes of the cotangent function are found at \( x = k\pi \) and \( x = (k+1)\pi \), where \( k \) is any integer. To find a specific pair, simply substitute a particular integer value for \( k \). For example, if \( k = 0 \), then the pair of consecutive asymptotes would be at \( x = 0 \) and \( x = \pi \).