Find period, domain, and range of y= 3 csc (3x + pi) -2
y=3 csc 3 (x+ pi/3) -2 but I'm stuck from here.
This is review from last year, but I've totally forgotten everything about trig functions. Not looking for just answers but also explanations!

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Find period, domain, and range of y= 3 csc (3x + pi) -2
y=3 csc 3 (x+ pi/3) -2 but I'm stuck from here.
This is review from last year, but I've totally forgotten everything about trig functions. Not looking for just answers but also explanations!

At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga.
Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus.
Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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freckles

The period of y=csc(x) is 2pi.
So the period of y=csc(bx) is 2pi/|b|.
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y=sin(x) on [-2pi,2pi] looks like:
|dw:1440611891389:dw|
We can see the period is 2pi.
The reciprocal function of y=sin(x) is y=csc(x).
So noticed the zeros of y=sin(x) on [-2pi,2pi], they were at x=-2pi,-pi,0,pi,2pi
y=csc(x) will be undefined there.
Notice where sin(x)=1 on [-2pi,2pi], they were at x=-3pi/2,x=pi/2
well this is where csc(x)=1 also since the reciprocal of 1 is still 1.
Notice where sin(x)=-1 on [-2pi,2pi], they were at x=-pi/2,3pi/2
well this where csc(x)=-1 since the reciprocal of -1 is still -1.
The graph of y=csc(x) looks like:
|dw:1440612297776:dw|
you can also from this graph that the range is (-inf,-1] union [1,inf)
and the domain is everything but where we had the zeros fro sin(x) which was
npi where n is integer
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Anyways...we have
\[\csc(x) \le -1 \text{ or } \csc(x) \ge 1 \text{ from the range } \\ \text{ this holds for any input } \\ \csc(3x+\pi) \le -1 \text{ or } \csc(3x+\pi) \ge 1 \\ \text{ multiply both sides by 3 and then subtract both sides by 2 } \\ \text{ \to find range of } y=3\csc(3x+\pi)-2\]
--
Now for the domain part...
You could solve the following equation to find out what numbers to exclude:
\[\sin(3x+\pi)=0\]

cassieforlife5

Thank you for the awesome details! I'm still a bit confused but I'll try and go over it a few times. Could you help with the equation for finding the domain as well?