Question

4 csc(2x)-1=9 csc(2x)+4 specified interval [0,2pi]

Answer 1

we can rewrite your function as follows:
\[\Large \begin{gathered}
4\csc \left( {2x} \right) - 1 = 9\csc \left( {2x} \right) + 4 \hfill \\
\hfill \\
5\csc \left( {2x} \right) = - 5 \hfill \\
\end{gathered} \]

Answer 2

yes, i got down to this:

csc(2x)=-1

Answer 3

that's right!

Answer 4

now, use this substitution:
\[\Large \csc \left( {2x} \right) = \frac{1}{{\sin \left( {2x} \right)}}\]

Answer 5

you should get this:
\[\Large \frac{1}{{\sin \left( {2x} \right)}} = - 1\]

Answer 6

how did you get 1/sin(2x) ?

Answer 7

it is a standard identity. By definition of csc(\alpha), we have:
\[\Large \csc \alpha \doteq \frac{1}{{\sin \alpha }}\]

Answer 8

oh, i found it tanks.

Answer 9

now we have to exclude those particular values for which sin(2x)=0, since we are not allowed to divide by zero

Answer 10

and we have:
\[\Large \sin \left( {2x} \right) = 0\]
when
\[\Large \begin{gathered}
2x = 0 \hfill \\
2x = \pi \hfill \\
\end{gathered} \]

Answer 11

or, when:
\[\Large \begin{gathered}
x = 0, \hfill \\
x = \pi /2 \hfill \\
\end{gathered} \]

Answer 12

so the acceptable solutions, have to be different from 0 and pi/2

Answer 13

now our last equation is equivalent to this one:
\[\Large \sin \left( {2x} \right) = - 1\]
please solve it, for x, what do you get? @quocski

Answer 14

the one i found so far is x=3pi/2

Answer 15

ok! since that value is different from x=0, and it is different from x=pi/2, then it is an acceptable solution

Answer 16

sorry, we have:
\[\Large 2x = \frac{{3\pi }}{2}\]
so, dividing by 2, we get:
\[\Large x = \frac{{3\pi }}{4}\]
am I right?

Answer 17

which is an acceptable solution

Answer 18

more precisely, the general solution is:
\[\Large 2x = \frac{{3\pi }}{2} + 2k\pi \]
or, equivalently:
\[\Large x = \frac{{3\pi }}{4} + k\pi \]

Answer 19

where k is an integer number, namely:
\[\Large k = 0, \pm 1, \pm 2,...\]

Answer 20

so, anothe acceptable solution is given adding pi, to the preceding solution, namely:
\[\Large x = \frac{{3\pi }}{4} + \pi = ...\]
please complete my computation

Answer 21

ok, one after that would be 3pi/4+2pi?

Answer 22

yes! nevertheless it is greater than 2*pi, and your problem asks for all solutions such that are greater or equal to zero and are less or equal to 2*pi

Answer 23

so we have these 2 solutions:
\[\Large \begin{gathered}
x = \frac{{3\pi }}{4}, \hfill \\
\\
x = \frac{{7\pi }}{4} \hfill \\
\end{gathered} \]
which are both acceptable, since they are different from 0 and from pi/2

Answer 24

thanks

Answer 25

:)