[tex] \bf sin(\theta )=\cfrac{1}{2}\implies \measuredangle \theta =sin^{-1}\left( \cfrac{1}{2} \right)\implies \theta \measuredangle =
\begin{cases}
\frac{\pi }{6}\\\\
\frac{5\pi }{6}
\end{cases}\\\\
-------------------------------\\\\
cot(\theta )=-1\implies \measuredangle \theta =cot^{-1}(-1)\implies \measuredangle \theta =
\begin{cases}
\frac{3\pi }{4}\\\\
\frac{7\pi }{4}
\end{cases} [/tex]
[tex] \bf -------------------------------\\\\
cos(3\theta )=\cfrac{1}{2}\implies 3\theta =cos^{-1}\left( \cfrac{1}{2} \right)\implies 3\theta =
\begin{cases}
\frac{\pi }{3}\\\\
\frac{5\pi }{3}
\end{cases}
\\\\\\
3\theta =\cfrac{\pi }{3}\implies \theta =\cfrac{\pi }{9}\qquad \qquad ,\qquad \qquad 3\theta =\cfrac{5\pi }{3}\implies \theta =\cfrac{5\pi }{9} [/tex]
for the second one, cot(θ) = -1, bear in mind that cotangent is cosine/sine, so where both of them are the same values BUT different signs, you'd get a -1 from their fraction, and that happens at halfway in II and IV Quadrants, since on those quadrants both of them are of different signs.
