Graph y=1/2csc(x)

For any ${\displaystyle y=\csc \left(x\right)}$, vertical asymptotes occur at ${\displaystyle x=n\pi }$, where ${\displaystyle n}$ is an integer. Use the basic period for ${\displaystyle y=csc\left(x\right)}$, ${\displaystyle (0,2\pi )}$, to find the vertical asymptotes for ${\displaystyle y=\frac{\csc \left(x\right)}{2}}$. Set the inside of the cosecant function, ${\displaystyle bx+c}$, for ${\displaystyle y=a\csc (bx+c)+d}$ equal to ${\displaystyle 0}$ to find where the vertical asymptote occurs for ${\displaystyle y=\frac{\csc \left(x\right)}{2}}$.

Set the inside of the cosecant function ${\displaystyle x}$ equal to ${\displaystyle 2\pi }$.

The basic period for ${\displaystyle y=\frac{\csc \left(x\right)}{2}}$ will occur at ${\displaystyle (0,2\pi )}$, where ${\displaystyle 0}$ and ${\displaystyle 2\pi }$ are vertical asymptotes.

Find the period ${\displaystyle \frac{2\pi }{\left|b\right|}}$ to find where the vertical asymptotes exist. Vertical asymptotes occur every half period.

The vertical asymptotes for ${\displaystyle y=\frac{\csc \left(x\right)}{2}}$ occur at ${\displaystyle 0}$, ${\displaystyle 2\pi }$, and every ${\displaystyle \pi n}$, where ${\displaystyle n}$ is an integer. This is half of the period.

There are only vertical asymptotes for secant and cosecant functions.

The period of the function can be calculated using ${\displaystyle \frac{2\pi }{\left|b\right|}}$.

Replace ${\displaystyle b}$ with ${\displaystyle 1}$ in the formula for period.

The absolute value is the distance between a number and zero. The distance between ${\displaystyle 0}$ and ${\displaystyle 1}$ is ${\displaystyle 1}$.

Divide ${\displaystyle 2\pi }$ by ${\displaystyle 1}$.

The phase shift of the function can be calculated from ${\displaystyle \frac{c}{b}}$.

Replace the values of ${\displaystyle c}$ and ${\displaystyle b}$ in the equation for phase shift.

Divide ${\displaystyle 0}$ by ${\displaystyle 1}$.