Question

Find all solutions of the equation.
sinxcscx=-2sinx

I'm stuck and cannot figure out what to do with the cscx. Any help would be greatly appreciated!

Answer 1

ok.. thanks for sharing that

Answer 2

cscx is 1/sinx

Answer 3

Take the -2sinx to the LHS and factorise out the sinx. What do you get?

Answer 4

\[\Large\rm \sin x \csc x=-2\sin x\]Add 2sinx to each side,\[\Large\rm 2\sin x+\sin x \csc x=0\]Factor a sin x out of each term,\[\Large\rm \sin x(2+\csc x)=0\]Then apply the Zero-Factor Property, solve for x in each case,

Answer 5

\[\Large\rm \sin x=0, \qquad\qquad 2+\csc x=0\]

Answer 6

Thank you.

Answer 7

Were you given any specific interval to look for solutions?
If not, realize that you will have an infinite number of solutions, so you'll need to include a +2kpi or something like that..

Answer 8

Thank you @zepdrix! There was no interval.

Answer 9

pi, 7pi/6, 11pi/6...each one with +2kpi. Is that correct?

Answer 10

Should be `four` solutions.
So for your sine solutions don't forget about the 0 + rotations.\[\Large\rm \sin x=0\qquad\to \qquad x=\cases{0+2k \pi, \\\Large\rm \pi+2k \pi,}\quad k=0,~\pm1,~\pm2,~...\]

Answer 11

Those other two solution sets look good though! =)

Answer 12

Oh oh oh, but the cosecant kinda throws a wrench into everything doesn't it?
After we get our solution sets,
We should see if we have extraneous solutions.

So like...
For x=0,
Our csc x is undefined, so that is not a valid solution.
That becomes true for all of the solutions that we got from sinx.
(Because cscx is the reciprocal of sinx).
So everywhere we get 0 for our sinx, we're having a problem at our cscx.

Answer 13

So I guess we only care about the cscx solutions,\[\Large\rm x=\cases{7\pi/6~~+2k \pi,\\11\pi/6+2k \pi,},\qquad k=0,\pm1,\pm2,~...\]