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\problemgrouptitle{Trigonometric Equations}

\begin{problemblock}{Solve the following trigonometric equations.}

\item $\displaystyle 2\cos\theta=\sqrt{3}$
\answer{$\displaystyle\left\{\frac{\pi}{6}+2n\pi\ |\ n\ \mbox{an integer}\right\}\cup\left\{\frac{11\pi}{6}+2n\pi\ |\ n\ \mbox{an integer}\right\}$}

\item $\displaystyle \sec\theta=-2$
\answer{$\displaystyle\left\{\frac{2\pi}{3}+2n\pi\ |\ n\ \mbox{an integer}\right\}\cup\left\{\frac{4\pi}{3}+2n\pi\ |\ n\ \mbox{an integer}\right\}$}

\item $\cos(4\alpha)=-\frac{1}{2}$
\answer{$\displaystyle\left\{\frac{\pi}{6}+\frac{n\pi}{2}\ |\ n\ \mbox{an integer}\right\}\cup\left\{\frac{\pi}{3}+\frac{n\pi}{2}\ |\ n\ \mbox{an integer}\right\}$\\  Note that in $[0,2\pi)$ there are eight angle solutions, namely $\displaystyle\left\{\frac{\pi}{6},\frac{\pi}{3},\frac{2\pi}{3},\frac{5\pi}{6},\frac{7\pi}{6},\frac{4\pi}{3},\frac{5\pi}{3},\frac{11\pi}{6}\right\}$\,.}

\item $\tan^2\!x=\tan x$
\answer{$\displaystyle\left\{\frac{\pi}{4}+2n\pi\ |\ n \in \mathset{Z}\right\}\cup\left\{\frac{5\pi}{4}+2n\pi\ |\ n \in \mathset{Z}\right\}\cup\left\{0+2n\pi\ |\ n \in \mathset{Z}\right\}\cup\left\{\pi+2n\pi\ |\ n \in \mathset{Z}\right\}$\\ $\displaystyle=\left\{\frac{\pi}{4}+n\pi\ |\ n \in \mathset{Z}\right\}\cup\left\{n\pi\ |\ n \in \mathset{Z}\right\}$}
\end{problemblock}
