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 $π$ .

Answer

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Answer

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

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: $x = \frac{7 π}{6}$ and $x = \frac{11 π}{6}$ .

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

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