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Problems PreCal 1508 PLTL Workshop, October 25, 2011\\
PLs: Alex Knaust, Edith Mejia. Lecturer: Yi-Yu Liao\\

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These problems represent a review of the content that will be covered by the third exam. The answers
are available as a pdf at \url{http://alex.knaust.info/pltlfall2011/}
\begin{enumerate}

\item Verify the following trigonometric identities. \textbf{Make sure to practice writing these very formally, and mention which identity is used in each step (if any)}.
\begin{enumerate}
\item $\cos{x}(\tan^2{x} + 1) = \sec{x}$
\item $\sin^5{x}\cos^2{x} = (\cos^2{x} - 2\cos^4{x} + \cos^6{x})\sin{x}$
\item $\tan{\left(\frac{\pi}{4} - \theta \right)} = \frac{1 - \tan{\theta}}{1 + \tan{\theta}}$
\item $1 + \cos{10y} = 2\cos^2{5y}$
\item $\frac{\cos{3\beta}}{\cos{\beta}} = 1 - 4\sin^2{\beta}$
\item $\frac{\tan{x} + \tan{y}}{1 - \tan{x}\tan{y}} = \frac{\cot{x} + \cot{y}}{\cot{x}\cot{y} - 1}$
\end{enumerate}

\item Solve the following equations (all solutions)
\begin{enumerate}
\item $\sin^2{x} = 3\cos^2{x}$
\item $\cos{x} + \sin{x}\tan{x} = 2$
\item $\tan{(x + \pi)} + 2\sin{(x + \pi)} = 0$
\item $\cos{2x} + \sin{x} = 0$
\item $\sin{ \left( x + \frac{\pi}{2}\right) } - \cos^2{x} = 0$
\end{enumerate}

\item Find the following exact values
\begin{enumerate}
\item $\arcsin{(\sin{\left(\frac{5}{7}\right)})}$
\item $\arccos{(\sin{\left(\frac{\pi}{3}\right)})}$
\item $\tan{(\arcsin{\left(\frac{3}{5}\right)})}$
\end{enumerate}

\item Find the amplitude, phase shift, and period etc. and graph the following equations
\begin{enumerate}
\item $f(x) = -2\cos{\left( x + \frac{\pi}{4} \right)}$
\item $g(x) = \sin{\left(\frac{x}{3} - \pi \right)} + 1$
\end{enumerate}

\item A triangular parcel of land has sides of lengths 5m, 7m and 9m. Find the measure of the largest angle (in the triangle)

\item Two ships leave a port at 9 a.m. one travels at an angle of $53^\circ$ at 12 miles per hour and the other travels at an angle of $67$ at 16 miles per hour. Approximate how far apart they are from each other at noon.
\end{enumerate}
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