Graph y=csc(pi-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 ${\displaystyle y=\csc (\pi -x)}$. 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 ${\displaystyle y=\csc (\pi -x)}$.

Solve for ${\displaystyle x}$.

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

Solve for ${\displaystyle x}$.

The basic period for ${\displaystyle y=\csc (\pi -x)}$ will occur at ${\displaystyle (\pi ,-\pi )}$, where ${\displaystyle \pi }$ and ${\displaystyle -\pi }$ 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 ${\displaystyle y=\csc (\pi -x)}$ occur at ${\displaystyle \pi }$, ${\displaystyle -\pi }$, and every ${\displaystyle x=\pi +\pi n}$, where ${\displaystyle n}$ is an integer. This is half of the period.

Cosecant only has vertical asymptotes.

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 ${\displaystyle -1}$ and ${\displaystyle 0}$ 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.

Dividing two negative values results in a positive value.

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