Find the Asymptotes f(x)=4csc(1/4pix+1/3pi)
The question is asking to identify the asymptotes of the function f(x) = 4csc(1/4πx + 1/3π). An asymptote is a line that a graph approaches but never actually reaches. Typically, vertical asymptotes occur at values of x for which the function is undefined — in this case, where the cosecant function (csc), which is the reciprocal of the sine function, is undefined. That happens when the sine function is equal to zero since division by zero is undefined. The question seeks to determine the vertical lines (if any) that the graph of this particular trigonometric function approaches.
$f \left(\right. x \left.\right) = 4 csc \left(\right. \frac{1}{4} \pi x + \frac{1}{3} \pi \left.\right)$
Answer
Solution:
Step 1
Eliminate the parentheses in the expression: $y = 4 \csc\left(\frac{1}{4}\pi x + \frac{1}{3}\pi\right)$.
Step 2
Simplify the expression inside the cosecant function.
Step 2.1
Multiply $\pi x$ by $\frac{1}{4}$ to get $\frac{\pi x}{4}$.
Step 2.2
Combine $\frac{1}{3}$ with $\pi$ to get $\frac{\pi}{3}$.
Step 3
The function now looks like this: $y = 4 \csc\left(\frac{\pi x}{4} + \frac{\pi}{3}\right)$.
Step 4
Identify the vertical asymptotes for the cosecant function by setting the argument equal to $k\pi$, where $k$ is an integer.
Step 5
Find the first vertical asymptote by solving $\frac{\pi x}{4} + \frac{\pi}{3} = 0$.
Step 5.1
Isolate $\frac{\pi x}{4}$ by subtracting $\frac{\pi}{3}$ from both sides.
Step 5.2
Multiply through by $\frac{4}{\pi}$ to solve for $x$.
Step 5.3
Simplify to find $x = -\frac{4}{3}$.
Step 6
Find the next vertical asymptote by setting $\frac{\pi x}{4} + \frac{\pi}{3} = 2\pi$.
Step 6.1
Isolate $\frac{\pi x}{4}$ by subtracting $\frac{\pi}{3}$ from both sides.
Step 6.2
Multiply through by $\frac{4}{\pi}$ to solve for $x$.
Step 6.3
Simplify to find $x = \frac{20}{3}$.
Step 7
Determine the period of the function by calculating $\frac{2\pi}{|\frac{\pi}{4}|}$.
Step 7.1
Simplify to find the period is $8$.
Step 8
The vertical asymptotes occur at $x = -\frac{4}{3} + 8n$, where $n$ is an integer.
Step 9
There are no horizontal or oblique asymptotes for the cosecant function.
Step 10
The final result for the vertical asymptotes is $x = -\frac{4}{3} + 8n$, where $n$ is an integer.
Knowledge Notes:
Cosecant Function: The cosecant function, denoted as $\csc(x)$, is the reciprocal of the sine function. It is undefined when the sine of $x$ is zero, which occurs at integer multiples of $\pi$. The vertical asymptotes of the cosecant function occur at these points.
Vertical Asymptotes: A vertical asymptote of a function is a line $x = a$ where the function approaches infinity or negative infinity as $x$ approaches $a$. For trigonometric functions like cosecant, vertical asymptotes occur where the function is undefined.
Period of Trigonometric Functions: The period of a trigonometric function is the length of the smallest interval over which the function repeats its values. For the sine and cosine functions, the period is $2\pi$. For the cosecant function, which is derived from the sine function, the period is also $2\pi$.
Solving Trigonometric Equations: To solve equations involving trigonometric functions, it's often necessary to isolate the trigonometric term and then use the properties of the function to find solutions within a given interval. For periodic functions, solutions can be expressed in terms of the period.
LaTeX Formatting: In the solution, LaTeX is used to format mathematical expressions. For example, $\frac{1}{4}\pi x$ is written in LaTeX as $\frac{1}{4}\pi x$. LaTeX is a typesetting system that is widely used for mathematical and scientific documents.
