# What are the asymptotes of g(x)=0.5 csc x?

Apr 22, 2018

infinite

#### Explanation:

$\csc x = \frac{1}{\sin} x$

$0.5 \csc x = \frac{0.5}{\sin} x$

any number divided by $0$ gives an undefined result, so $0.5$ over $0$ is always undefined.

the function $g \left(x\right)$ will be undefined at any $x$-values for which $\sin x = 0$.

from ${0}^{\circ}$ to ${360}^{\circ}$, the $x$-values where $\sin x = 0$ are ${0}^{\circ} , {180}^{\circ} \mathmr{and} {360}^{\circ}$.

alternatively, in radians from $0$ to $2 \pi$, the $x$-values where $\sin x = 0$ are $0 , \pi \mathmr{and} 2 \pi$.

since the graph of $y = \sin x$ is periodic, the values for which $\sin x = 0$ repeat every ${180}^{\circ} , \mathmr{and} \pi$ radians.

therefore, the points for which $\frac{1}{\sin} x$ and therefore $\frac{0.5}{\sin} x$ are undefined are ${0}^{\circ} , {180}^{\circ} \mathmr{and} {360}^{\circ}$ ($0 , \pi \mathmr{and} 2 \pi$) in the restricted domain, but can repeat every ${180}^{\circ}$, or every $\pi$ radians, in either direction.

graph{0.5 csc x [-16.08, 23.92, -6.42, 13.58]}

here, you can see the repeating points at which the graph cannot continue due to undefined values. for example, the $y$-value steeply increases when approaching closer to $x = 0$ from the right, but never reaches $0$. the $y$-value steeply decreases when approaching closer to $x = 0$ from the left, but never reaches $0$.

in summary, there are an infinite number of asymptotes for the graph $g \left(x\right) = 0.5 \csc x$, unless the domain is restricted. the asymptotes have a period of ${180}^{\circ}$ or $\pi$ radians.