Answer
The solution set is $$\{\frac{5\pi}{6}+\pi n, n\in Z\}$$
Work Step by Step
$$\cot x+\sqrt3=0$$
1) Solve the equation over the interval $[0,2\pi)$
$$\cot x+\sqrt3=0$$
$$\cot x=-\sqrt3$$
Over the interval $[0,2\pi)$, there are two values of $x$ where $\cot x=-\sqrt3$, which are $\frac{5\pi}{6}$ and $\frac{11\pi}{6}$.
2) Solve the equation for all solutions
- The integer multiples of the period of the cotangent function is $\pi$.
- We apply it to each solution found in step 1. The results are the solution set, which looks something like this
$$\{\frac{5\pi}{6}+\pi n, \frac{11\pi}{6}+\pi n, n\in Z\}$$
- However, a closer look shows that both $\frac{5\pi}{6}+\pi n$ and $\frac{11\pi}{6}+\pi n$ refer to the same set of points, meaning in the final solution set, only one set needs to be included.
Therefore, the final solution set is $$\{\frac{5\pi}{6}+\pi n, n\in Z\}$$
