# How do you solve  sqrt3cscx-2=0?

Mar 22, 2018

{x | x = pi/3 + 2kpi, " " x = (2pi)/3 + 2kpi}," " forall k in ZZ

#### Explanation:

First, let's isolate $\csc x$:

$\sqrt{3} \csc x - 2 = 0$

$\sqrt{3} \csc x = 2$

$\csc x = \frac{2}{\sqrt{3}}$

Now, since we know that $\csc x = \frac{1}{\sin} x$, we can take the reciprocal of both sides of the equation and then solve the equation in terms of $\sin x$:

$\frac{1}{\csc} x = \frac{1}{\frac{2}{\sqrt{3}}}$

$\sin x = \frac{\sqrt{3}}{2}$

Now, we can see that our solution set will be all points where $\sin x$ is equal to $\frac{\sqrt{3}}{2}$. If we remember our unit circle, we can see that:

The two points with a y-coordinate of $\frac{\sqrt{3}}{2}$ are $x = \frac{\pi}{3}$ and $x = \frac{2 \pi}{3}$.

Therefore, our solution is:

$\left\{x | x = \frac{\pi}{3} , x = \frac{2 \pi}{3}\right\}$

One last touch: remember that the values of all trig functions are the same if you add $2 \pi$ to the angle, so any multiple of $2 \pi$ added to either of these solutions will ALSO be a valid solution of the equation. Therefore, we can represent our final solution set as:

{x | x = pi/3 + 2kpi, " " x = (2pi)/3 + 2kpi}, " for any integer " k

Or if you REALLY want to translate the last part into fancy math symbols:

{x | x = pi/3 + 2kpi, " " x = (2pi)/3 + 2kpi}," " forall k in ZZ