Answer
The exact value of the provided expression is $\frac{\sqrt{3}}{2}$.
Work Step by Step
We know that $\theta =-\frac{35\pi }{6}$ lies in quadrant I, so the reference angle for $\theta =-\frac{35\pi }{6}$ is:
$\begin{align}
& \theta '=-\frac{35\pi }{6}+6\pi \\
& =\frac{-35\pi +36\pi }{6} \\
& =\frac{\pi }{6}
\end{align}$
Now, the function value for the reference angle is:
$\cos \frac{\pi }{6}=\frac{\sqrt{3}}{2}$
Thus, we see that the angle $\theta =\frac{22\pi }{3}$ lies in quadrant I, so the cos function is positive:
$\begin{align}
& \cos \left( -\frac{35\pi }{6} \right)=\cos \frac{\pi }{6} \\
& =\frac{\sqrt{3}}{2}
\end{align}$
