Answer
a) The value of $\theta '$ is $15{}^\circ $.
b) The value of $\theta '$ is $\frac{\pi }{6}$.
Work Step by Step
(a)
The angle formed by a full rotation of the terminal side in degrees is $360{}^\circ $.
The angle $\theta '$ is the difference between $360{}^\circ $ and $\theta $.
$\theta '=360{}^\circ -\theta $
Put $345{}^\circ $ for $\theta $.
$\begin{align}
& \theta '=360{}^\circ -345{}^\circ \\
& =15{}^\circ
\end{align}$
(b)
And the angle formed by a half rotation of the terminal side in radians is $\pi $.
The angle $\theta '$ is the difference between $\pi $ and $\theta $.
$\theta '=\pi -\theta $
Put $\frac{5\pi }{6}$ for $\theta $.
$\begin{align}
& \theta '=\pi -\frac{5\pi }{6} \\
& =\frac{6\pi -5\pi }{6} \\
& =\frac{\pi }{6}
\end{align}$
