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 $\pi$.

Answer

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Answer

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

Steps

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 $\boxed{\frac{7\pi}{6}}$.