Find the Asymptotes f(x)=4csc(1/4pix+1/3pi)

Solution:

Step 1

Eliminate the parentheses in the expression: $y=4\csc (\frac{1}{4}\pi x+\frac{1}{3}\pi )$.

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: $y=4\csc (\frac{\pi x}{4}+\frac{\pi }{3})$.

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:

1. 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.

2. 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.

3. 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 $2\pi$.

4. 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.

5. 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.