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%TCIDATA{<META NAME="Title" CONTENT="Exercises">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:15:31 1997}
%TCIDATA{Language=American English}
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\input{tcilatex}
\begin{document}


\section{Exercises\label{Exercises}}

\begin{enumerate}
\item[1.]  \label{q1}Define the functions $f(x)=x^{3}+x\sin x$ and $%
g(x)=\sin x^{2\text{ }}$. Evaluate $f(g(x))$, $g(f(x))$, $f(x)g(x)$, and $%
f(x)+g(x)$.\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{0.2171in}{0.2006in}{0in%
}}{}{}{a1}

\item[2.]  \label{q2}At Metropolis Airport, an airplane is required to be at
an altitude of at least $800$ $\unit{ft}$ above ground when it has attained
a horizontal distance of $1$ $\unit{mi}$ from takeoff. What must be the
(minimum) average angle of ascent?\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{%
0.2171in}{0.2006in}{0in}}{}{}{a2}

\item  \label{q3}Experiment with expansions of $\sin nx$ in terms of $\sin x$
and $\cos x$ for $n=1,2,3,4,5,6$ and make a conjecture about the form of the
general expansion of $\sin nx$.\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{%
0.2214in}{0.205in}{0in}}{}{}{a3}

\item  \label{q4}Experiment with parametric plots of $\left( \cos t,\sin
t\right) $ and $(t,\sin t)$. Attach the point $\left( \cos 1,\sin 1\right) $
to the first plot and $\left( 1,\sin 1\right) $ to the second. Explain how
the two graphs are related.\hfill \hyperref{\TCIIcon{BITMAPSETAnswer}{%
0.2214in}{0.205in}{0in}}{}{}{a4}

\item  \label{q5}Experiment with parametric plots of $\left( \cos t,\sin
t\right) $, $\left( \cos t,t\right) $, and $\left( t,\cos t\right) $,
together with the point $\left( \cos 1,\sin 1\right) $ on the first plot, $%
\left( \cos 1,1\right) $ on the second, and $\left( 1,\cos 1\right) $ on the
third. Explain how these plots are related.\hfill \hyperref{\TCIIcon{%
BITMAPSETAnswer}{0.2214in}{0.205in}{0in}}{}{}{a5}
\end{enumerate}

\strut

\section{Solutions\label{Solutions}}

\begin{enumerate}
\item[1.]  \label{a1}Defining functions $f(x)=x^{3}+x\sin x$ and $g(x)=\sin
x^{2}$ and evaluating gives 
\begin{eqnarray*}
f(g(x)) &=&\sin ^{3}x^{2}+\sin x^{2}\sin \left( \sin x^{2}\right)  \\
g(f(x)) &=&\sin \left( x^{3}+x\sin x\right) ^{2}\, \\
f(x)g(x) &=&\left( x^{3}+x\sin x\right) \sin x^{2}\, \\
f(x)+g(x) &=&\,x^{3}+x\sin x+\sin x^{2}
\end{eqnarray*}

\hfill \hyperref{\TCIIcon{BITMAPSETProbSolvHint}{0.2006in}{0.243in}{0in}}{}{%
}{q1}

\item[2.]  \label{a2}You can find the minimum average angle of ascent by
considering the right triangle with legs of length $800\unit{ft}$ and $5280%
\unit{ft}$. The angle in question is the acute angle with sine equal to $%
\frac{800}{\sqrt{800^{2}+5280^{2}}}$. Find the answer in radians with 
\textsf{Evaluate Numerically}: 
\[
\arcsin \frac{800}{\sqrt{800^{2}+5280^{2}}}=\,.15037
\]
You can express this angle in degrees by using the following steps: 
\begin{eqnarray*}
360\times \dfrac{.15037}{2\pi } &=&8.6157 \\
0.6157\times 60 &=&36.942 \\
\theta  &=&8^{\circ }37^{\prime }
\end{eqnarray*}

\hfill \hyperref{\TCIIcon{BITMAPSETProbSolvHint}{0.1989in}{0.2413in}{0in}}{}{%
}{q2}

\item  \label{a3}Note that 
\begin{eqnarray*}
\sin 2x &=&2\sin x\cos x \\
\sin 3x &=&\allowbreak 4\sin x\cos ^{2}x-\sin x \\
\sin 4x &=&\allowbreak 8\sin x\cos ^{3}x-4\sin x\cos x \\
\sin 5x &=&\allowbreak 16\sin x\cos ^{4}x-12\sin x\cos ^{2}x+\sin x \\
\sin 6x &=&\allowbreak 32\sin x\cos ^{5}x-32\sin x\cos ^{3}x+6\sin x\cos x
\end{eqnarray*}
It appears that in general, $\sin nx=2^{n-1}\sin x\cos ^{n-1}x\ -\cdots $
where the remaining terms are of the form $\sin x\cos ^{n-(2k+1)}x$.\hfill 
\hyperref{\TCIIcon{BITMAPSETProbSolvHint}{0.2006in}{0.2439in}{0in}}{}{}{q3}

\item  \label{a4}The first figure shows a circle of radius $1$ with center
at the origin. The graph is drawn by starting at the point $\left(
1,0\right) $ and is traced in a counter-clockwise direction. The second
figure shows the $y$-coordinates of the first figure as the angle varies
from $0$ to $2\pi $. The point $\left( \cos 1,\sin 1\right) $ is marked with
a small circle in the first figure. The corresponding point $\left( 1,\sin
1\right) $ is marked with a small circle in the second figure.

\begin{center}
\hfill \FRAME{itbpFU}{1.8957in}{2.0003in}{0in}{\Qcb{$\left( \cos t,\sin
t\right) $}}{}{Plot }{\special{language "Scientific Word";type
"MAPLEPLOT";width 1.8957in;height 2.0003in;depth 0in;display "PICT";function
\TEXUX{$\left( \cos t,\sin t\right) $};linecolor "red";linestyle
1;linethickness 1;pointstyle "circle";function \TEXUX{$\left( \cos 1,\sin
1\right) $};linecolor "black";linestyle 1;linethickness 1;pointstyle
"circle";pointplot TRUE;xmin "0";xmax "6.29";xviewmin "-1.04";xviewmax
"1.041";yviewmin "-1.04";yviewmax "1.04";rangeset"X";phi 45;theta
45;plottype 19;constrained TRUE;numpoints 10;axesstyle "normal";xis
\TEXUX{t};var1name \TEXUX{$x$};}}\FRAME{itbpFU}{2.0081in}{2.0081in}{0in}{%
\Qcb{$\left( t,\sin t\right) $}}{}{Plot }{\special{language "Scientific
Word";type "MAPLEPLOT";width 2.0081in;height 2.0081in;depth 0in;display
"PICT";function \TEXUX{$\left( t,\sin t\right) $};linecolor "blue";linestyle
1;linethickness 1;pointstyle "circle";function \TEXUX{$\left( 1,\sin
1\right) $};linecolor "black";linestyle 1;linethickness 1;pointstyle
"circle";pointplot TRUE;xmin "0";xmax "6.29";xviewmin "-0.1258";xviewmax
"6.418";yviewmin "-1.04";yviewmax "1.041";rangeset"X";phi 45;theta
45;plottype 19;numpoints 49;axesstyle "normal";xis \TEXUX{t};var1name
\TEXUX{$x$};}}\hfill \hyperref{\TCIIcon{BITMAPSETProbSolvHint}{0.2006in}{%
0.2439in}{0in}}{}{}{q4}
\end{center}

\item  \label{a5}The first figure shows a circle of radius $1$ with center
at the origin. The graph is drawn by starting at the point $\left(
1,0\right) $ and is traced in a counter-clockwise direction. The second
figure shows the $x$-coordinates of the first figure as the angle varies
from $0$ to $2\pi $. The point $\left( \cos 1,\sin 1\right) $ is marked with
a small circle in the first figure. The corresponding point $\left( \cos
1,1\right) $ is marked with a small circle in the second figure. The third
figure shows the graph in the second figure with the horizontal and vertical
axes interchanged. The third figure shows the usual view of $y=\cos x$.

\begin{center}
\FRAME{itbpFU}{2.0081in}{2.0081in}{0in}{\Qcb{$(\cos t,\sin t)$}}{}{Plot }{%
\special{language "Scientific Word";type "MAPLEPLOT";width 2.0081in;height
2.0081in;depth 0in;display "PICT";function \TEXUX{$\left( \cos t,\sin
t\right) $};linecolor "red";linestyle 1;linethickness 1;pointstyle
"point";function \TEXUX{$(\cos 1,\sin 1)$};linecolor "black";linestyle
1;linethickness 1;pointstyle "circle";pointplot TRUE;xmin "0";xmax
"6.29";xviewmin "-1.04";xviewmax "1.041";yviewmin "-1.04";yviewmax
"1.041";rangeset"X";recompute TRUE;phi 45;theta 45;plottype 19;constrained
TRUE;numpoints 49;axesstyle "normal";xis \TEXUX{t};var1name \TEXUX{$x$};}}%
\FRAME{itbpFU}{2.0081in}{2.0081in}{0in}{\Qcb{$(\cos t,t)$}}{}{Plot }{%
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$};linecolor "green";linestyle 1;linethickness 1;pointstyle "point";function
\TEXUX{$\left( \cos 1,1\right) $};linecolor "black";linestyle
1;linethickness 1;pointstyle "circle";pointplot TRUE;xmin "0";xmax
"6.29";xviewmin "-1.04";xviewmax "1.041";yviewmin "-0.1258";yviewmax
"6.418";rangeset"X";recompute TRUE;phi 45;theta 45;plottype 19;numpoints
49;axesstyle "normal";xis \TEXUX{t};var1name \TEXUX{$x$};}}\FRAME{itbpFU}{%
2.0081in}{2.0081in}{0in}{\Qcb{$(t,\cos t)$}}{}{Plot }{\special{language
"Scientific Word";type "MAPLEPLOT";width 2.0081in;height 2.0081in;depth
0in;display "PICT";function \TEXUX{$\left( t,\cos t\right) $};linecolor
"blue";linestyle 1;linethickness 1;pointstyle "point";function
\TEXUX{$(1,\cos 1)$};linecolor "black";linestyle 1;linethickness
1;pointstyle "circle";pointplot TRUE;xmin "0";xmax "6.29";xviewmin
"-0.1258";xviewmax "6.418";yviewmin "-1.04";yviewmax
"1.041";rangeset"X";recompute TRUE;phi 45;theta 45;plottype 19;numpoints
49;axesstyle "normal";xis \TEXUX{t};var1name \TEXUX{$x$};}}\hfill \bigskip 
\hyperref{\TCIIcon{BITMAPSETProbSolvHint}{0.2006in}{0.2439in}{0in}}{}{}{q5}
\end{center}
\end{enumerate}

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Trigonometric Functions}{}{}{DM4-1.tex#Trigonometry}

\item  \hyperref{Solving Trigonometric Equations}{}{}{DM4-1.tex#Solve,
trigonometry}

\item  \hyperref{Trigonometric Identities}{}{}{DM4-1.tex#Identities,
trigonometry}

\item  \hyperref{Simplifying Trigonometric Expressions}{}{}{%
DM4-1.tex#Simplify, trigonometry}

\item  \hyperref{Solution of Triangles}{}{}{DM4-1.tex#Solution of triangles}

\item  \hyperref{Inverse Trigonometric Functions and Trigonometric Equations%
}{}{}{DM4-1.tex#Inverse trig functions}

\item  \hyperref{DeMoivre's Theorem}{}{}{DM4-2.tex#DeMoivre's Theorem}
\end{itemize}

\rule{0.67in}{0.01in}

{\hyperref{\FRAME{itbpF}{38.875pt}{24.5pt}{6pt}{}{}{index.wmf}{\special%
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"XNPEU";}}}{}{}{DMIndex.tex#Math Index}\quad {\small Index entries: 
\hyperref{Trigonometry}{}{}{DMIndex.tex#Trigonometry}} }

\end{document}
