Find the principal root of this equation: \[ \sin (x)=\frac{-1}{2} \] \[ x= \] - Give your answer in radians. - Do not use decimal approximation. Use 'pi' to represent $\pi$.

Step 1 ：The principal root of the equation \(\sin(x) = -\frac{1}{2}\) is the value of \(x\) for which the sine function equals \(-\frac{1}{2}\).

Step 2 ：The sine function equals \(-\frac{1}{2}\) at two points within a period: \(x = \frac{7\pi}{6}\) and \(x = \frac{11\pi}{6}\).

Step 3 ：However, the principal root is the smallest positive root, which is \(x = \frac{7\pi}{6}\).

Step 4 ：Final Answer: The principal root of the equation \(\sin(x) = -\frac{1}{2}\) is \(\boxed{\frac{7\pi}{6}}\).