# How do you graph y=-csc3theta?

Feb 18, 2018

See below.

#### Explanation:

$y = - \csc 3 \theta$

$= - \frac{1}{\sin} \left(3 \theta\right)$

Let $x = 3 \theta \to y = - \frac{1}{\sin} x$

Consider: ${\lim}_{x \to {0}^{+}} - \frac{1}{\sin} x = - \infty$

and, ${\lim}_{x \to {0}^{-}} - \frac{1}{\sin} x = \infty$

This cycle is repeated for ${\lim}_{x \to n {\pi}^{+}} y$ and ${\lim}_{x \to n {\pi}^{-}} y$ $\forall n \in \mathbb{Z}$

Note that $y$ has local maxima of $- 1$ at $x = \frac{\left(2 n + 1\right) \pi}{2} \forall n \in \mathbb{Z}$ and $y$ has local minima of $+ 1$ at $x = \frac{\left(4 n - 1\right) \pi}{2} \forall n \in \mathbb{Z}$

This can be represented graphically below.

graph{-1/sinx [-10, 10, -5, 5]}

Replacing $x$ by $3 \theta$ has the effect of decreasing the period by a factor of 3. As shown below.

graph{-1/sin(3x) [-10, 10, -5, 5]}

Hence, the graph above is the graph of $y = - \csc \left(3 \theta\right)$
where $\theta$ is shown on the vertical axis.