Find the Asymptotes 5csc(1/2pix+1/6pi)

Solution:

Step 1:

Eliminate the parentheses in the expression: $y=5\csc (\frac{1}{2}\cdot \pi x+\frac{1}{6}\cdot \pi )$.

Step 2:

Repeat the elimination of parentheses: $y=5\csc (\frac{1}{2}\cdot \pi x+\frac{1}{6}\cdot \pi )$.

Step 3:

Simplify each component in the expression.

Step 3.1:

Multiply $\frac{1}{2}$ by $\pi x$ to simplify.

Step 3.1.1:

Combine $\pi$ and $\frac{1}{2}$ to get $y=5\csc (\frac{\pi }{2}x+\frac{1}{6}\cdot \pi )$.

Step 3.1.2:

Merge $\frac{\pi }{2}$ with $x$ to obtain $y=5\csc (\frac{\pi x}{2}+\frac{1}{6}\cdot \pi )$.

Step 3.2:

Combine $\frac{1}{6}$ and $\pi$ to form $y=5\csc (\frac{\pi x}{2}+\frac{\pi }{6})$.

Step 4:

Identify vertical asymptotes for the function $y=\csc (x)$, which occur at $x=n\pi$, where $n$ is an integer. To find the vertical asymptotes for $y=5\csc (\frac{\pi x}{2}+\frac{\pi }{6})$, set the argument of the cosecant function to zero: $\frac{\pi x}{2}+\frac{\pi }{6}=0$.

Step 5:

Solve for $x$.

Step 5.1:

Subtract $\frac{\pi }{6}$ from both sides: $\frac{\pi x}{2}=-\frac{\pi }{6}$.

Step 5.2:

Multiply both sides by $\frac{2}{\pi }$: $x=-\frac{1}{3}$.

Step 6:

Set the argument of the cosecant function equal to $2\pi$: $\frac{\pi x}{2}+\frac{\pi }{6}=2\pi$.

Step 7:

Solve for $x$.

Step 7.1:

Isolate terms containing $x$: $\frac{\pi x}{2}=2\pi -\frac{\pi }{6}$.

Step 7.2:

Multiply both sides by $\frac{2}{\pi }$ to find $x=\frac{11}{3}$.

Step 8:

The basic period for $y=5\csc (\frac{\pi x}{2}+\frac{\pi }{6})$ is between the vertical asymptotes at $x=-\frac{1}{3}$ and $x=\frac{11}{3}$.

Step 9:

Determine the period of the function to locate all vertical asymptotes, which occur every half period: $\frac{2\pi }{\left|\frac{\pi }{2}\right|}=4$.

Step 10:

The vertical asymptotes for $y=5\csc (\frac{\pi x}{2}+\frac{\pi }{6})$ occur at $x=-\frac{1}{3}+4n$, where $n$ is an integer.

Step 11:

The function $y=5\csc (\frac{\pi x}{2}+\frac{\pi }{6})$ has no horizontal or oblique asymptotes, only vertical asymptotes at $x=-\frac{1}{3}+4n$.

Step 12:

There is no further action required in this step.

Knowledge Notes:

The problem involves finding the asymptotes of the function $y=5\csc (\frac{1}{2}\pi x+\frac{1}{6}\pi )$. An asymptote is a line that the graph of a function approaches but never touches.

• Vertical Asymptotes: These occur in trigonometric functions like cosecant ($\csc$) and secant ($\sec$) where the function is undefined. For the cosecant function, vertical asymptotes occur at values of $x$ where $\sin (x)=0$, since $\csc (x)=\frac{1}{\sin (x)}$.

• Horizontal Asymptotes: These occur when the values of $y$ approach a constant as $x$ goes to infinity or negative infinity. The cosecant function does not have horizontal asymptotes because it has an infinite range.

• Oblique Asymptotes: These are diagonal lines that the function approaches as $x$ goes to infinity or negative infinity. The cosecant function does not have oblique asymptotes.

• Period of Trigonometric Functions: The period of a function is the length of the smallest interval over which the function repeats itself. For $y=\csc (x)$, the period is $2\pi$. For a transformed function like $y=a\csc (bx+c)$, the period is $\frac{2\pi }{|b|}$.

• Solving Equations: To find the specific values of $x$ for the vertical asymptotes, we set the argument of the cosecant function equal to the values where $\sin (x)=0$ and solve for $x$. This involves algebraic manipulation, such as isolating terms and multiplying by reciprocals.

In this problem, we use the properties of the cosecant function and algebraic techniques to determine the locations of the vertical asymptotes. The function $y=5\csc (\frac{\pi x}{2}+\frac{\pi }{6})$ has vertical asymptotes at $x=-\frac{1}{3}+4n$, where $n$ is an integer, and no horizontal or oblique asymptotes.