Answer
The exact value of the expression is $\frac{1}{2}$.
Work Step by Step
$\sin \left( -\frac{35\pi }{6} \right)$ lies in quadrant I.
Add $6\pi $ to $-\frac{35\pi }{6}$ , to find a positive co-terminal angle less than $2\pi $. Consider $\alpha $ to be the positive co-terminal angle.
$\begin{align}
& \alpha =\left( -\frac{35\pi }{6} \right)+6\pi \\
& =\frac{-35\pi +36\pi }{6} \\
& =\frac{\pi }{6}
\end{align}$
The reference angle ${\theta }'$ is the positive acute angle formed by the x-axis and the terminal side of $\theta $.
The reference angle of $-\frac{35\pi }{6}$ is $\frac{\pi }{6}$.
The function value of the reference angle is
$\sin \frac{\pi }{6}=\frac{1}{2}$
The angle $-\frac{35\pi }{6}$ is in quadrant I and the sine function is positive in quadrant I.
Therefore,
$\sin \left( -\frac{35\pi }{6} \right)=\sin \frac{\pi }{6}$
Substitute $\frac{1}{2}$ for $\sin \frac{\pi }{6}$.
$\sin \left( -\frac{35\pi }{6} \right)=\frac{1}{2}$
The exact value of the expression is $\frac{1}{2}$.
