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Problems PreCal 1508 PLTL Workshop, November 9, 2011\\
PLs: Alex Knaust, Edith Mejia. Lecturer: Yi-Yu Liao\\

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Please do the problems that you feel will help your group the most first (you don't have to do them in order). All handouts are available at \url{http://alex.knaust.info/pltlfall2011/}

\begin{enumerate}

\item Jimmy is trying to do the second problem and he makes the following argument : 
\begin{itemize}
\item The problem tells me $\tan(\theta) = \frac{5}{12}$
\item I also know $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$
\item So $\frac{5}{12} = \frac{\sin(\theta)}{\cos(\theta)}$
\item Since $\frac{a}{b} = \frac{c}{d}$ means $a=c$, $b=d$
\item Then $\sin(\theta) = 5$, $\cos(\theta) = 12$
\end{itemize}
How can he tell from his results that he did something incorrectly?


\item Given the following trigonometric functions, find all others of the given angle. $0 < \theta < \pi$
\begin{inparaenum}
\item $\sin(\theta) = \frac{-3}{5}\quad\quad\quad$
\item $\tan(\theta) = \frac{5}{12}\quad\quad\quad$
\item $\sec(\theta) = \frac{-17}{15}$
\end{inparaenum}


\item For the following functions, determine their amplitude, period, and phase shift, and use those parameters to sketch a graph of the function. verify with a calculator.

\begin{enumerate}
\item $y = -\sin\left(\frac{2\pi x}{3}\right)$
\item $y = 4\cos\left(x + \frac{\pi}{4}\right)$
\end{enumerate}

\item Determine the domain and range of $\sin$ $\cos$ and $\tan$. (use $\tan\theta = \frac{\sin\theta}{\cos\theta}$)

\item Find reference angles for the following angles (in radians)
\begin{enumerate}
\item $135^\circ$
\item $-\frac{5\pi}{6}$
\item $\frac{\pi}{2}$
\end{enumerate}


\item I want to find out the height of a pole, but unfortunately it is surrounded by a fence and guarded by vicious rabbits. Luckily I have some tools to measure distance and angles. I measure the angle of elevation (angle from ground to top of the pole) from my starting position and discover it is $30^\circ$. Then I walk exactly 12m towards the pole and measure the angle... It is now $45^\circ$. Then I try to climb over the fence to measure the height and am eaten alive by the rabbits. Did I die in vain? If so, how tall is the pole?


\item My friend says he wants to find all the Pythagorean triples (integers $a, b, c$ such that $a^2 + b^2 = c^2$), so he starts with $(1, 1, 1)$ and tries every tuple. I tell him he is a fool, instead he could just pick two integers $m, n$ and say $a = m^2 - n^2$, $b = 2mn$, $c = m^2+n^2$ and that will be a triple.
\begin{enumerate}
\item (Regular) He doesn't believe me, please show him I am right
\item (Hardened) Will my formula find every Pythagorean triple?
\item (Veteran) Fermat conjectured (in 1637) that there are none for $a^3 + b^3 = c^3$ or any power greater than $2$. (This was first proven in 1994)
\end{enumerate}
\end{enumerate}
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