How do you solve $4csc^2x-2=0$ and find all solutions in the interval $[0,2pi)$ ?

Question

4388 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-12-24T16:58:52

No solutions exist for the given equation.

Note that the $abs(csc(theta)) >= 1$ for all values of $theta$ .
[See image below or note that $csc=1/sin$ and $sin in [-1,+1]$

Therefore $csc^2(x) >= 1$
and $4csc^2(x) >= 4$

and finally $4csc^2(x) - 2 >= 2$

Therefore $4csc^2(x)-2 != 0$ for any value of $x$

enter image source here

How do you solve 4csc^2x-2=04csc^2x-2=0 and find all solutions in the interval [0,2pi)[0,2pi)?