How do you graph $y=-csc3theta$ ?

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Answer 1 · 2018-02-18T02:01:03

See below.

$y=-csc3theta$

$=-1/sin(3theta)$

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.

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