Question

Finding Vertical Asymptotes In Exercises \(13-28\) , find the vertical asymptotes (if any) of the graph of the function. $$ s(t)=\frac{t}{\sin t} $$

Short answer

The vertical asymptotes for the function \(s(t)=\frac{t}{\sin t}\) are at \(t = nπ\) for \(n ≠ 0\), where \(n\) is an integer.

Step-by-step solution

Set the Denominator Equals to Zero

Set the denominator of the expression equals to zero: \(\sin t = 0\). Solve this equation for \(t\). The general solution for this equation is \(t = nπ\), where \(n\) is any integer, because sine function is periodic and equals to zero at every multiple of \(π\).

Check if Numerator Equals Zero Simultaneously

Check if the numerator, which is \(t\), equals to zero at the same points. In our case, that means \(t = nπ\) may not be zero. We know that \(t = 0\) when \(n = 0\). Therefore, any integer except zero can be substituted for \(n\). Then \(t = 0\) is not a point of vertical asymptote.

Identify the Vertical Asymptotes

The vertical asymptotes of \(s(t) = \frac{t}{\sin t}\) are at \(t = nπ\) for \(n ≠ 0\), where \(n\) is an integer. That means \(t = π, -π, 2π, -2π, 3π, -3π, 4π, ...\). These are the values of \(t\) at which the denominator of the fraction equals zero, while the numerator does not equal zero, thus the function approaches positive or negative infinity.

Key concepts

Understanding the Sine Function

The sine function is a fundamental trigonometric function that relates the angles of a right triangle to its side lengths. It's defined for all real numbers and has the general form given by the equation \( \sin \theta \).

Here are a few characteristics of the sine function to help understand its nature:

• **Periodicity:** The sine function is periodic with a period of \( 2\pi \). This means every \( 2\pi \) units, the function repeats its values.
• **Range:** The values of \( \sin t \) are always between -1 and 1. So, \( -1 \leq \sin t \leq 1 \) for all real numbers \( t \).
• **Zeros:** The sine function equals zero at integer multiples of \( \pi \), such as \( t = 0, \pi, -\pi, 2\pi, \ldots \). These points are important for identifying vertical asymptotes in rational functions that include sine in their denominator.

Understanding these characteristics helps us in solving problems involving vertical asymptotes in trigonometric functions, as they reveal where the function might be undefined.

The Role of Rational Functions

Rational functions are expressions of the form \( \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. An important aspect of rational functions is identifying their vertical asymptotes, which occur where the denominator, \( Q(x) \), equals zero.

Some key points about rational functions include:

• **Vertical Asymptotes:** These occur at values of \( x \) where \( Q(x) = 0 \) but \( P(x) eq 0 \). They represent values where the function becomes undefined and the graph heads towards positive or negative infinity.
• **Determining Asymptotes:** While zeros of the denominator suggest potential vertical asymptotes, zeros of both the numerator and denominator at the same point denote removable discontinuities, not vertical asymptotes.
• **Behavior Near Asymptotes:** As the input approaches the value of the asymptote, the function’s outputs approach infinity, creating lines that the graph seems to 'avoid' but get infinitely close to.

By understanding these concepts, you can properly determine the domains of rational functions and identify possible vertical asymptotes, as demonstrated in exercises involving trigonometric rational functions.

Solving Trigonometric Equations for Asymptotes

Trigonometric equations often appear in functions like \( s(t) = \frac{t}{\sin t} \), where finding the values that make the denominator zero is crucial for locating vertical asymptotes.

Here are steps to solve trigonometric equations for such asymptotes:

• **Set Denominator to Zero:** First, determine where \( \sin t = 0 \). The solution is at multiples of \( \pi \) (\( t = n\pi \), with \( n \) as an integer), where the sine value naturally equals zero.
• **Exclude Removable Discontinuities:** Check if the numerator, here \( t \), also equals zero at these points. If \( t eq 0 \), then these values of \( t \) are vertical asymptotes.
• **Generalizing:** For \( s(t) = \frac{t}{\sin t} \), vertical asymptotes appear at \( t = n\pi \) (except for \( t = 0 \)), meaning \( t = \pi, -\pi, 2\pi, \ldots \), where the function transitions from undefined behavior due to sine equating to zero.

Knowing these steps aids in solving similar trigonometric equations, allowing for accurate location of vertical asymptotes in such functions.