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\begin{document}

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\centerline{\textbf{\large Exam \#2 $\cdot$ Monday November 20, 2006}}
\vspace*{.2in}
MATH 111 $\cdot$ Section 7 $\cdot$ Fall 2006  \hspace*{0.5in} Name \hrulefill
\vspace*{0.2in}

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\textbf{Problem 1.}
Given the function $y=\frac{1}{3}\sin(4x+1)$, find the average value and period.
\vspace*{2.0in}

\textbf{Problem 2.}
Given the following graph, find the maximum value, minimum value, and
amplitude.

\includegraphics{e2f1.eps}

\vspace*{1.5in}

\textbf{Problem 3.}
Write an equation for a sine function which passes through all the points
in the following data table:
$$
\begin{tabular}{|r||r|r|r|r|r|r|}
\hline $x$ & $0$ & $1$ & $2$ & $3$ & $4$ & $5$ \\
\hline $y$ & $-1$ & $3$ & $7$ & $3$ & $-1$ & $3$ \\
\hline
\end{tabular}
$$

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\newpage
\textbf{Problem 4.}
Which of the following correctly represents $\cos(-\theta)+\sin(-\theta)$?

\begin{tabular}{llllllllll}
(A) & $ \cos(\theta)+\sin(\theta)$ &
(B) & $ \cos(\theta)-\sin(\theta)$ &
(C) & $-\cos(\theta)+\sin(\theta)$ &
(D) & $-\cos(\theta)-\sin(\theta)$ \\
(E) & None of these.
\end{tabular}
\vspace*{1.0in}

\textbf{Problem 5.}
Verify the following trigonometric identity.  (Hint:  use a sum identity and
the Pythagorean identity.)
$$
	\cos(2x) = 1 - 2\sin^2(x).
$$
\vspace*{2.0in}

\textbf{Problem 6.}
Determine the exact value of $\sin(7\pi/12)$ using sum and/or difference identities.
\vspace*{2.5in}

\textbf{Problem 7.}
If $\cos(\theta)=-4/5$ and $\sin(\theta)>0$, find $\tan(\theta)$.

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\newpage
\textbf{Problem 8.}
Which of the following is not an identity?

\begin{tabular}{llllllllll}
(A) & $\sin^2(x)+\cos^2(x)=1$ &
(B) & $\cos(x+y) = \cos(x)\cos(y) + \sin(x)\sin(y)$ \\
(C) & $\sin(2 x) = 2 \sin(x)\cos(x)$ &
(D) & $\cos(\pi/2-x) = \sin(x)$  \\
(E) & These are all identities.
\end{tabular}
\vspace*{1.5in}

\textbf{Problem 9.}
Find all solutions for $0 \le x < 2\pi$ (or, if you prefer, do it in degrees
for $0^\circ \le x < 360^\circ$):
$$
	(\sin(x) + 1/2)(\sin(x)-1/2) = 0.
$$
\vspace*{2.5in}

\textbf{Problem 10.}
Solve for $x$:  $$y=5\,\cos^{-1}(2x)+3.$$
\vspace*{2.5in}

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\newpage
\textbf{Problem 11.}
Find the exact value of $\cos(\sin^{-1}(-2/3))$.
\vspace*{2.5in}

\textbf{Problem 12.}
Find the exact value of $\arcsin(\sin(135^\circ))$.
\vspace*{2.5in}

\textbf{Problem 13.}
Writing question:  Why is it that $\tan^{-1}(2)$ is defined, while
$\sin^{-1}(2)$ and $\cos^{-1}(2)$ are undefined?

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\end{document}
