Question

How do you graph `y=-csc3theta`?

Answer 1

See below.

Explanation:

`y=-csc3theta`

`=-1/sin(3theta)`

Let `x=3theta -> y=-1/sinx`

Consider: `lim_(x->0^+) -1/sinx = -oo`

and, `lim_(x->0^-) -1/sinx = oo`

This cycle is repeated for `lim_(x->npi^+) y` and `lim_(x->npi^-) y` `forall n in ZZ`

Note that `y` has local maxima of `-1` at `x=((2n+1)pi)/2 forall n in ZZ` and `y` has local minima of `+1` at `x=((4n-1)pi)/2 forall n in ZZ`

This can be represented graphically below.

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

Replacing `x` by `3theta` 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(3theta)`
where `theta` is shown on the vertical axis.