How do you sketch one cycle of y=cscx?

Feb 27, 2017

Like a $\sin$ graph, but opposite.

Explanation:

$\csc \left(x\right) = \frac{1}{\sin} \left(x\right)$

Sketch a $\sin$ graph, then draw loops down to meet each maximum or minimum point.

graph{csc(x) [-10, 10, -5, 5]}

Notice that at each stationary point on this graph, there would usually be a stationary point for $\sin \left(x\right)$.

Each time the $\sin$ graph would approach $0$, the $\csc$ graph will approach $\infty$ or $- \infty$. Each time the $\sin$ graph approaches $\pm 1$, the $\csc$ graph approaches $\pm 1$.

These are because

$\csc \left(x\right) = \frac{1}{\sin} \left(x\right)$

$\sin \left(x\right) = 0 \to \csc \left(x\right) = \frac{1}{0} = \infty$
$\sin \left(x\right) = 1 \to \csc \left(x\right) = \frac{1}{1} = 1$