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Chapter 2: Problem 70

Find the equations for all vertical asymptotes for each function. $$ y=\csc (4 x+\pi) $$

Short Answer

Expert verified
The vertical asymptotes are at \( x = \frac{(k - 1)\pi}{4} \) for any integer \( k \).

Step by step solution

01

Understand the Function

The given function is given by the equation \( y = \csc(4x + \pi) \). Recall that the cosecant function, \csc(\theta) = \frac{1}{\sin(\theta)}, is undefined where \sin(\theta) = 0. These undefined points are the vertical asymptotes.
02

Identify the Zeros of Sine Function

To locate the vertical asymptotes, find the values of \(\theta = 4x + \pi \) where \( \sin(\theta) = 0 \). The sine function is zero at \( \theta = k\pi \) for any integer \( k \).
03

Solve for x

Set the argument of the sine function equal to the identified zeros: \( 4x + \pi = k\pi \). Solve for \( x \), giving: \( 4x = k\pi - \pi \), \( 4x = (k - 1)\pi \), then \( x = \frac{(k - 1)\pi}{4} \).
04

Write the Asymptote Equations

The equations for the vertical asymptotes are the values of \( x \) that make the cosecant function undefined. From \( x = \frac{(k - 1)\pi}{4} \), the vertical asymptotes are given by: \( x = \frac{(k - 1)\pi}{4} \) where \( k \) is any integer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosecant Function
The cosecant function, denoted as \(\text{csc}(\theta)\), is the reciprocal of the sine function. This means \( \text{csc}(\theta) = \frac{1}{\text{sin}(\theta)} \). Because it's the reciprocal, wherever the sine function is zero, the cosecant function becomes undefined.
In simpler terms, the cosecant function 'blows up' or goes to infinity at these points.
This is particularly important when graphing as it leads to vertical asymptotes.
Undefined Points of Trigonometric Functions
Trigonometric functions can be tricky because they aren't defined everywhere. For the cosecant function, it's not defined where the sine function equals zero. This happens because we can't divide by zero. When you're dealing with trigonometric functions, it's crucial to identify these undefined points.
These undefined points often become vertical asymptotes when you graph these functions. In the case of \(y = \text{csc}(4x + \pi)\), the function is undefined where \( \text{sin}(4x + \pi) = 0 \).
Solving Trigonometric Equations
Solving trigonometric equations involves finding all possible values of the variable that satisfy the equation. For \( \text{csc}(4x + \pi) \), you need to solve \(4x + \pi = k\pi\) for \(x\). Here鈥檚 a quick guide:
  • Start with the given equation: \( 4x + \pi = k\pi \).

  • Isolate \(x\) by rearranging terms: \( 4x = k\pi - \pi \).

  • Simplify further: \( 4x = (k - 1)\pi \).

  • Divide both sides by 4: \( x = \frac{(k - 1)\pi}{4} \).

Keep in mind that \(k\) is any integer. This shows how the trigonometric equation helps us find the vertical asymptotes.
Sine Function Zeros
The zeros of the sine function are the points where \( \text{sin}(\theta) = 0 \). For the standard sine function, these points occur at \( \theta = k\pi \), where \( k \) is an integer.
In essence, these zeros are evenly spaced along the x-axis at intervals of \( \pi \). Knowing this helps you find where other trigonometric functions, like the cosecant function, are undefined. For instance, in \( \text{csc}(4x + \pi) \), you need to locate these zeros within the transformed argument \( 4x + \pi \).
Thus, understanding these basic properties of the sine function is critical for solving more complex trigonometric functions.

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Most popular questions from this chapter

WRITING/DISCUSSION Even or Odd An even function is one for which \(f(-x)=f(x)\) and an odd function is one for which \(f(-x)=-f(x)\). Determine whether \(f(x)=\sin (x)\) and \(f(x)=\cos (x)\) are even or odd functions and explain your answers.

Determine the amplitude, phase shift, and range for each function. Sketch at least one cycle of the graph and label the five key points on one cycle as done in the examples. $$ y=\frac{1}{2} \cos x $$

Find the equations of all asymptotes to the graph of $$ y=-6 \tan (2 x)+1 $$

Determine the amplitude, phase shift, and range for each function. Sketch at least one cycle of the graph and label the five key points on one cycle as done in the examples. $$ y=4 \sin x $$

A person owns two businesses. For one the profit is growing exponentially at a rate of \(1 \%\) per month. The other business is cyclical, with a higher profit in the summer than in the winter. The function \(P(x)=1000(1.01)^{x}+500 \sin \left(\frac{\pi}{6}(x-4)\right)+2000\) gives the total profit as a function of the month with \(x=1\) corresponding to January of 2018 . a. Graph the function for 60 months. b. What does the graph look like if the domain is 600 months (50 years)?
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