Question

True or False The graphs of y = tan x, y = cot x, y = sec x, and y = csc x each have infinitely many vertical asymptotes.

Short answer

True.

Step-by-step solution

Understand the Given Functions

The functions to evaluate are: - \(y = \tan x\) - \(y = \cot x\) - \(y = \sec x\) - \(y = \csc x\)

Analyze \(y = \tan x\)

The tangent function, \(\tan x\), has vertical asymptotes where \(\cos x = 0\). This occurs at \(x = \frac{\pi}{2} + k\pi\) for all integers \(k\). Hence, there are infinitely many vertical asymptotes.

Analyze \(y = \cot x\)

The cotangent function, \(\cot x\), has vertical asymptotes where \(\sin x = 0\). This occurs at \(x = k\pi\) for all integers \(k\). Hence, there are infinitely many vertical asymptotes.

Analyze \(y = \sec x\)

The secant function, \(\sec x\), has vertical asymptotes where \(\cos x = 0\). This occurs at \(x = \frac{\pi}{2} + k\pi\) for all integers \(k\). Hence, there are infinitely many vertical asymptotes.

Analyze \(y = \csc x\)

The cosecant function, \(\csc x\), has vertical asymptotes where \(\sin x = 0\). This occurs at \(x = k\pi\) for all integers \(k\). Hence, there are infinitely many vertical asymptotes.

Conclude

Since \(y = \tan x\), \(y = \cot x\), \(y = \sec x\), and \(y = \csc x\) each have vertical asymptotes at infinitely many points, the statement is True.

Key concepts

tangent function asymptotes

Understanding the vertical asymptotes in the tangent function, denoted as \(y = \tan x\), is key. A vertical asymptote represents a line where the function approaches infinity or negative infinity.
For the tangent function, these asymptotes occur where \(\tan x\) is undefined. This happens when \(\frac{1}{\textrm{cos} x}\) goes to infinity, meaning \(\textrm{cos} x = 0\). Mathematically, it occurs at:

• \(x = \frac{\frac{\text{π}}{2}}{π} + kπ\) for any integer \(k\)

So, there are indeed infinitely many vertical asymptotes.
For example, the values \(k=0, 1,\) and \(-1\) correspond to asymptotes at \(\frac{π}{2}, \frac{3π}{2},\) and \(-\frac{\text{π}}{2}\) respectively.

cotangent function asymptotes

Next, let's explore the vertical asymptotes in the cotangent function, expressed as \(y = \text{\text{cot}} x\). The same idea applies: vertical asymptotes occur where \(\text{cot} x\) is undefined.
In this case, cotangent is the reciprocal of sine, so the function becomes undefined where \(\textrm{sin} x = 0\). These values are:

• \(x = kπ\) for any integer \(k\)

This shows that cotangent also has infinitely many vertical asymptotes.
Examples include \(k=0\) and \(k=1\), corresponding to asymptotes at 0 and \(\text{π}\). A crucial takeaway is to remember this fundamental property of cotangent.

secant function asymptotes

For the secant function, noted as \(y = \text{\text{sec}} x\), vertical asymptotes show up where \(\text{\text{cos}} x = 0\).
The secant function is the reciprocal of the cosine function. Whenever cosine hits zero, secant becomes undefined. The points where \(\text{cos} x = 0\) are:

• \(x = \frac{\text{π}}{2} + kπ\) for any integer \(k\)

Thus, the secant function also has infinitely many vertical asymptotes.
Some examples include \(k=0\) and \(k=1\), leading to asymptotes at \(\frac{π}{2}\) and \(\frac{3π}{2}\). Remember, secant and cosine work hand in hand in defining these asymptotes.

cosecant function asymptotes

Lastly, the cosecant function, represented by \(y = \text{\text{csc}} x\), has vertical asymptotes where \(\textrm{sin} x = 0\).
Cosecant is the reciprocal of the sine function. Vertical asymptotes appear where the sine of \(x\) is zero. These points can be listed as:

• \(x = kπ\) for any integer \(k\)

The cosecant function, therefore, features infinitely many vertical asymptotes.
For instance, when \(k=0\) or \(k=1\), the asymptotes occur at 0 and \(π\). Understanding the functions of sine and cosecant helps illustrate how these vertical asymptotes function.
All these instances clearly point out that each function rains vertical asymptotes infinitely, affirming the statement.