# How do you find all (cos2x)/(sin3x-sinx) in the interval [0,2pi)?

Jul 28, 2018

Graph reads y values, $x \in \left[0 , 2 \pi\right]$
sans indeterminate holes at $x = \frac{\pi}{4} , \frac{3}{4} \pi , \frac{5}{4} \pi \mathmr{and} \frac{7}{4} \pi$, and
asymptotic $x = 0 , \pi \mathmr{and} 2 \pi$

#### Explanation:

$y = \frac{\cos 2 x}{\sin 3 x - \sin x}$

$= \frac{\cos 2 x}{2 \cos \left(\frac{1}{2} \left(3 x + x\right)\right) \sin \left(\frac{1}{2} \left(3 x - x\right)\right)}$

$= \frac{1}{2} \frac{\cos 2 x}{\cos 2 x} \left(\frac{1}{\sin} x\right) ,$

$= \frac{1}{2} \csc x , \cos 2 x \ne 0$
$\Rightarrow 2 x \ne \left(2 k + 1\right) \left(\frac{\pi}{2}\right) , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

$x \ne \left(2 k + 1\right) \frac{\pi}{4} \mathmr{and}$ asymptotic $x \ne k \pi$

Note that

$y = \frac{1}{2} \csc x \notin \frac{1}{2} \left(- 1 , 1\right) = \left(- \frac{1}{2} , \frac{1}{2}\right)$

The graph reads y. sans y at duly marked $x = 0 , \frac{\pi}{4} , \frac{3}{4} \pi ,$

$\pi , \frac{5}{4} \pi , \frac{7}{4} \pi , 2 \pi$. .
graph{(2y sin x +1)(x-pi/4+0.001y)(x+0.001y)(x-pi+0.001y) (x-3pi/4+0.001y)(x-5pi/4+0.001y)(x-2pi+0.001y)(x-7pi/4+0.001y)= 0[-0.2 8 -2 2]}