Answer
The value of $\frac{x}{y}$ is $-\frac{\sqrt{3}}{3}$ .
Work Step by Step
Consider the provided values of $x$ and $y$:
$x=-\frac{1}{2}$ and $y=\frac{\sqrt{3}}{2}$
Now, calculate $\frac{x}{y}$ as:
$\begin{align}
& \frac{x}{y}=\frac{-\frac{1}{2}}{\frac{\sqrt{3}}{2}} \\
& =\frac{-1}{2}\cdot \frac{2}{\sqrt{3}} \\
& =-\frac{1}{\sqrt{3}}
\end{align}$
Now, rationalize the denominator by multiplying the numerator and the denominator by $\sqrt{3}$ :
$\begin{align}
& \frac{x}{y}=-\frac{1}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}} \\
& =-\frac{\sqrt{3}}{3}
\end{align}$
Therefore, the value of $\frac{x}{y}$ is $-\frac{\sqrt{3}}{3}$ .
