Question

what's the domain of the function f(x)=csc(2x-������/4)

Answer 1

What is the csc equal to (in terms of sin, cos, or tan)?

Answer 2

if you mean what i think you mean, the rest of the function is in the picture

Answer 3

There is an identity that equates the csc to an expression with either sin, cos, or tan.
Do you know that identity?

Answer 4

like 1/sin?

Answer 5

Yes.
Now you have:
\(f(x) = \csc \left(2x -\dfrac{\pi}{4}\right) = \dfrac{1}{\sin \left(2x -\dfrac{\pi}{4}\right) }\)

Answer 6

how do you use that to find the domain?

Answer 7

bear in mind that \(\large \sin \left(2x -\dfrac{\pi}{4}\right)\)
is just sin(x) with a couple of transformations, but basically the same domain

Answer 8

but its not all real numbers like sin is

Answer 9

some sites were saying its all real numbers except odd multiples of π/2, but that was with just csc x

Answer 10

that was meant to be pi over 2

Answer 11

You can't have zero in the denominator.
Where is the sine equal to zero?
Set the denominator equal to zero and solve.
Those are the values you need to exclude from the domain.

Answer 12

Set the denominator equal to zero, and solve for x.

Answer 13

hmmm well... sin(x) is also 0 at \(\Large \pi\)
so you may also want to check for that \(\large {sin \left(2x -\cfrac{\pi}{4}\right)\implies
\begin{cases}
2x -\cfrac{\pi}{4}&=0\\
2x -\cfrac{\pi}{4}&=\pi
\end{cases}
}\)

Answer 14

is it not 0 at pi over 8?

Answer 15

yeap it's
now check the 2nd case, for \(\pi\)

Answer 16

it can't be at pi

Answer 17

hmmm

Answer 18

so it can be at pi?

Answer 19

well, recall that sin(x) is 0 at those 2 points, thus
if the angle is 0, as mathstudent55 already pointed out
or if the angle is at \(\pi\), the value for the sine function becomes 0
and thus
the fraction undefined

Answer 20

right, so all real numbers except multiples of pi?

Answer 21

yes

Answer 22

oh my jeezus thank god

Answer 23

thank you so much

Answer 24

hmmm
all real numbers except \(\pi\) is the domain for sin(x)
is not the domain for this function though

Answer 25

omg...

Answer 26

recall, this is csc()

Answer 27

\(\large {
sin \left(2x -\cfrac{\pi}{4}\right)
\\ \quad \\
\implies
\begin{cases}
2x -\cfrac{\pi}{4}=0\to &x=\cfrac{\pi}{8}\\
2x -\cfrac{\pi}{4}=\pi\to 2x=\pi+\cfrac{\pi}{4}\to 2x=\cfrac{5\pi}{4}\to &x=\cfrac{5\pi}{8}
\end{cases}
}\)

Answer 28

all real numbers except odd multiples of pi over 8?