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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:14:52 1997}
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\begin{document}


\section{Trigonometry\label{Trigonometry}}

Trigonometry developed from the study of triangles, particularly right
triangles, and the relations between the lengths of their sides and the
sizes of their angles. The trigonometric functions that measure the
relationships between the sides of similar triangles have far-reaching
applications that extend far beyond their use in the study of triangles.

\strut

\subsection{\ Trigonometric Functions}

The symbols used for the six basic trigonometric functions---$\sin $, $\cos $%
, $\tan $, $\cot $, $\sec $, $\csc $---are abbreviations for the words
cosine, sine, tangent, cotangent, secant, and cosecant,
respectively.\medskip You can enter these trigonometric functions and many
other functions either from the keyboard in mathematics mode or from the
dialog box that drops down when you click \FRAME{itbpF}{0.3009in}{0.3009in}{%
0.0701in}{}{}{function.wmf}{\special{language "Scientific Word";type
"GRAPHIC";display "PICT";valid-file "F";width 0.3009in;height 0.3009in;depth
0.0701in;original-width 0pt;original-height 0pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'FUNCTION.WMF';file-properties
"XNPEU";}} or choose \textsf{Insert + Math Name}. When you enter one of
these functions from the keyboard in mathematics mode, the function name
automatrically turns gray when you type the final letter of the name.

\strut

\begin{quotation}
\emph{Note}\quad \label{Function argument}\label{Trigtype}Ordinary functions
require parentheses around the function argument, while trigonometric
functions commonly do not. \textsl{Scientific Notebook} allows trigonometric
functions without parentheses. If you believe parentheses should be required
for all functions, you can change this behavior in the \textsf{Maple Settings%
} dialog. Click the \textsf{Definition Options }tab and under \textsf{%
Function Argument Selection Method}, check \textsf{Convert Trigtype to
Ordinary}.
\end{quotation}

\strut

To find values of the trigonometric functions, use \textsf{Evaluate} or 
\textsf{Evaluate Numerically}.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\sin \frac{3\pi }{4}=\allowbreak \frac{1}{2}\sqrt{2}$

$\sin \left( 1\right) =\allowbreak \sin 1$

$\sin 60^{\circ }=\allowbreak \frac{1}{2}\sqrt{3}$
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate Numerically}
\end{quote}

\begin{quotation}
$\sin \frac{3\pi }{4}=\allowbreak .\,70711$

$\sin \left( 1\right) =\allowbreak .\,84147$

$\sin 60^{\circ }=\allowbreak .\,86603$
\end{quotation}

\strut

The notation you use determines whether the argument is interpreted as
radians or degrees: $\sin 30=-.98803$, and $\sin 30^{\circ }=\,.5$. The
symbol used for degrees is the small circle that appears on the \textsf{%
Common Symbols} toolbar and on the \textsf{Binary Operations} symbol panel.
It should be typed in a superscript. All operations will convert angle
measure to radians. Click \hyperref{radians to degrees}{}{}{Radians to
degrees} for a discussion of conversion. Click \hyperref{Units in Scientific
Notebook}{}{}{DM2-6.tex#Units} for a discussion of units that can be used
for plane angles.

\strut

Your choice for \textsf{Digits Used in Display} in the \textsf{Maple +
Settings} or \textsf{Mathematica + Settings} dialog box determines the
number of places displayed in the response to \textsf{Evaluate Numerically}.

\begin{center}
{\ 
\begin{tabular}{lll}
\textbf{Expression} & \textbf{Evaluate} & \textbf{Evaluate Numerically} \\ 
\hline
$\sin \frac{\pi }{4}\medskip $ & $\frac{1}{2}\sqrt{2}$ & $.70711$ \\ 
$\sin 14^{\circ }37^{\prime }$ & $\,\sin \dfrac{877}{10800}\pi \medskip \;$
& $\,.25235$ \\ 
$\log _{10}\sin x\medskip $ & $\dfrac{\ln \left( \sin x\right) }{\ln 2+\ln 5}
$ & $\,.43429\ln \left( \sin x\right) $ \\ 
$3^{\circ }54^{\prime }$ & $\,\dfrac{13}{600}\pi $ & $\,6.8068\times 10^{-2}$%
\end{tabular}
}
\end{center}

\strut

\subsection{\ Solving \label{Solve, trigonometry}Trigonometric Equations}

You can use both \textsf{Exact} and \textsf{Numeric} from the \textsf{Solve}
submenu to find solutions to trigonometric equations. These operations also
convert degrees to radians. Use of decimal notation in the equation gives
you a numerical solution, even with \textsf{Exact}.

\begin{center}
{\ 
\begin{tabular}{lll}
\textbf{Equation} & \textbf{Solve + Exact} & \textbf{Solve + Numeric} \\ 
\hline
$x=\sin \frac \pi 4$ & $x=\frac 12\sqrt{2}$ & $x=.70711$ \\ 
$\sin 22^{\circ }=\dfrac{14}c$ & $\stackrel{\vspace{1pt}}{c=\dfrac{14}{\sin 
\frac{11}{90}\pi }}$ & $c=37.373$ \\ 
$\log _{10}\sin x=-1.1679$ & $x=6.7988\times 10^{-2}$ & $x=3.0736$ \\ 
$x=3^{\circ }54^{\prime }$ & $x=\,\frac{13}{600}\pi $ & $x=\,6.8068\times
10^{-2}\vspace{6pt}$%
\end{tabular}
}
\end{center}

Note that the answers are different for the equation $\log _{10}\sin
x=-1.1679$. This difference occurs because there are multiple solutions and
the two commands are finding different solutions. The \textsf{Numeric}
command from the \textsf{Solve} submenu offers the advantage that you can
specify the range in which you wish the solution to lie. Enter the equation
and the range in different rows of a display or a one-column matrix.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Numeric}
\end{quote}

\begin{quotation}
$
\begin{array}{c}
x=10\sin x \\ 
x\in \left( 5,\infty \right)
\end{array}
$, Solution is: $\left\{ x=7.0682\right\} $
\end{quotation}

\strut

The interval $\left( 5,\infty \right) $ was specified for the solution in
the preceding example. By specifying other intervals, you can find all seven
solutions: $\left\{ x=0\right\} $, $\left\{ x=\pm 2.8523\right\} $, $\left\{
x=\pm 7.0682\right\} $, $\left\{ x=\pm 8.4232\right\} $, as depicted in the
following graph. The \textsf{Exact} command for solving equations gives only
the solution $x=0$ for this equation.\FRAME{dtbpF}{3in}{2.0003in}{0pt}{}{}{%
Figure }{\special{language "Scientific Word";type "MAPLEPLOT";width
3in;height 2.0003in;depth 0pt;display "PICT";maintain-aspect-ratio
TRUE;function \TEXUX{$10\sin x,x$};linecolor "black";linestyle
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"-10.4";xviewmax "10.41";yviewmin "-10.4";yviewmax "10.41";rangeset"X";phi
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\TEXUX{x};var1name \TEXUX{$x$};}}

\label{Radians to degrees}\textsl{Scientific Notebook} automatically
converts degrees to radians when any operation is applied. To go in the
other direction, replace $2\pi $ radians with $360^{\circ }$ and convert
other angles proportionately. You can also solve directly for the number of
degrees. Both methods follow.

\begin{quote}
$\blacktriangleright $ To convert radians to degrees
\end{quote}

\begin{enumerate}
\item[1.]  Write the equation $\dfrac{\theta }{360}=\dfrac{x}{2\pi }$, where 
$x$ represents radians.

\item[2.]  Leave the insertion point in this equation.

\item[3.]  From the \textsf{Solve} submenu, choose \textsf{Exact} or \textsf{%
Numeric.}

\item[4.]  Name $\theta $ as the \textsf{Variable to Solve for}.

\item[5.]  Choose \textsf{OK} to get $\theta =\dfrac{180}{\pi }x.$
\end{enumerate}

\begin{quote}
$\blacktriangleright $ To convert radians to degrees
\end{quote}

\begin{enumerate}
\item  Write an equation such as $2=\theta ^{\circ }$

\item  Leave the insertion point in this equation.

\item  From the \textsf{Solve} submenu, choose \textsf{Exact} or \textsf{%
Numeric}.
\end{enumerate}

\strut

\begin{example}
Convert $x=\dfrac{13}{600}\pi \,$ radians to degrees as follows.

\begin{enumerate}
\item  Write the equation $\dfrac{\theta }{360}=\dfrac{\left( \frac{13}{600}%
\pi \right) }{2\pi }$.

\item  Leave the insertion point in this equation.

\item  From the \textsf{Solve} submenu, choose \textsf{Exact} to get $\theta
=\frac{39}{10}$ degrees, or choose \textsf{Numeric} to get $\theta =3.9$
degrees$.$
\end{enumerate}

\begin{itemize}
\item  - or -\medskip
\end{itemize}

\begin{enumerate}
\item  Write the equation $\dfrac{13}{600}\pi =\theta ^{\circ }$

\item  Leave the insertion point in this equation

\item  From the \textsf{Solve} submenu, choose \textsf{Exact} to get $\theta
=\frac{39}{10}$, or choose \textsf{Numeric} to get $\theta =3.9$ .

\item[4.]  Change $0.9$ degrees to minutes by computing $0.9\times 60=\,54.0$
to get 
\[
\theta =3^{\circ }54^{\prime } 
\]
\end{enumerate}

\begin{itemize}
\item  To check these results, apply \textsf{Evaluate} to $3^{\circ
}54^{\prime }$ to get 
\[
3^{\circ }54^{\prime }=\,\frac{13}{600}\pi 
\]
\end{itemize}
\end{example}

\strut

\subsection{\ Trigonometric Identities \label{Identities, trigonometry}}

This section illustrates the effects of some operations on trigonometric
functions. First, simplifications and expansions of various trigonometric
expressions illustrate many of the familiar trigonometric identities.

\subsubsection{Definitions in Terms of Basic Trigonometric Functions}

Apply \textsf{Simplify} to the tangent, cotangent, secant, and cosecant to
find their definition in terms of the sine and cosine functions.

\begin{quote}
$\blacktriangleright $ \textsf{Simplify}
\end{quote}

\begin{quotation}
$\tan x=\,\dfrac{\sin x}{\cos x}$

\strut

$\cot x=\,\,\dfrac{\cos x}{\sin x}$

\strut

$\sec x=\,\dfrac{1}{\cos x}$

\strut

$\csc x=\,\dfrac{1}{\sin x}$
\end{quotation}

\subsubsection{Pythagorean Identities}

\begin{quote}
$\blacktriangleright $ \textsf{Simplify}
\end{quote}

\begin{quotation}
$\sin ^{2}x+\cos ^{2}x=\,1\vspace{4pt}$

$\tan ^{2}x-\sec ^{2}x=\,-1\vspace{4pt}$

$\cot ^{2}x-\csc ^{2}x=\,-1\vspace{6pt}$
\end{quotation}

\subsubsection{Addition Formulas\label{Addition formulas}\label%
{Combine, trigonometric functions}}

\begin{quote}
\textsf{Expand}
\end{quote}

\begin{quotation}
$\sin \left( x+y\right) =\,\sin x\cos y+\cos x\sin y$

$\cos \left( x+y\right) =\,\cos x\cos y-\sin x\sin y$

$\tan \left( x+y\right) =\,\dfrac{\sin x\cos y+\cos x\sin y}{\cos x\cos
y-\sin x\sin y}$
\end{quotation}

Apply \textsf{Combine + Trig Functions } 
\index{Combine@Combine!trigonometric functions@trigonometric functions}to
the expansions of $\sin \left( x+y\right) $ and $\cos \left( x+y\right) $ to
return them to their original form and to change the expansion of $\tan
\left( x+y\right) $ to the form $\,%
\frac{\sin \left( x+y\right) }{\cos \left( x+y\right) }$.

\subsubsection{Double-Angle Formulas\label{Double-angle formulas}}

\begin{quote}
$\blacktriangleright $ \textsf{Expand}
\end{quote}

\begin{quotation}
$\sin 2\theta =\,2\sin \theta \cos \theta \vspace{4pt}$

$\cos 2\theta =\,2\cos ^{2}\theta -1$

$\tan 2\theta =\,2\left( \sin \theta \right) \dfrac{\cos \theta }{2\cos
^{2}\theta -1}\vspace{8pt}$
\end{quotation}

You can uncover other multiple-angle formulas with \textsf{Expand}.
Following are some examples.\vspace{4pt}

\begin{quote}
$\blacktriangleright $ \textsf{Expand}
\end{quote}

\begin{quotation}
$\sin 6\theta =\,32\sin \theta \cos ^{5}\theta -32\sin \theta \cos
^{3}\theta +6\sin \theta \cos \theta $

\ 

$\sin 24\theta =8388608\sin \theta \cos ^{23}\theta \,-\,46137344\sin \theta
\cos ^{21}\theta $

\qquad \qquad \qquad $+\,110100480\sin \theta \cos ^{19}\theta
\,-\,149422080\sin \theta \cos ^{17}\theta $

\qquad \qquad \qquad $+\,127008768\sin \theta \cos ^{15}\theta
\,-\,70189056\sin \theta \cos ^{13}\theta $

\qquad \qquad \qquad $+\,25346048\sin \theta \cos ^{11}\theta
\,-\,5857280\sin \theta \cos ^{9}\theta $

\qquad \qquad \qquad $+\,823680\sin \theta \cos ^{7}\theta \,-\,64064\sin
\theta \cos ^{5}\theta $

\qquad \qquad \qquad $+\,2288\sin \theta \cos ^{3}\theta \,-\,24\sin \theta
\cos \theta $

\ 

$\sin (2a+3b)=\,8\sin a\cos a\cos ^{3}b\,-\,6\sin a\cos a\cos b$

\qquad \qquad \qquad \qquad $+\;\,8\cos ^{2}a\sin b\cos ^{2}b\,-\,2\cos
^{2}a\sin b$

\qquad \qquad \qquad \qquad $-\;\,4\sin b\cos ^{2}b\,+\,\sin b$\thinspace \
\thinspace $\,$
\end{quotation}

\strut

\subsection{\ Simplifying\label{Simplify, trigonometry} Trigonometric
Expressions}

The command \textsf{Simplify} combines and simplifies trigonometric
expressions, as in the following examples.

\begin{quote}
$\blacktriangleright $ \textsf{Simplify}
\end{quote}

\begin{quotation}
$\cos ^{2}x+\frac{1}{4}\sin ^{2}2x-\sin ^{2}x\cos ^{2}x+2\sin ^{2}x=\,-\cos
^{2}x+2$

$\left( \cos 3t+3\cos t\right) \sec t=\,4\cos ^{2}t$

$\sin 3a+4\sin ^{3}a=\,3\sin a$

$\tan \frac{\theta }{2}\sin \frac{\theta }{2}+\cos \frac{\theta }{2}-2\csc
\theta \sin \frac{\theta }{2}=\,0$
\end{quotation}

You may need to apply repeated operations to get the result you want. The
order in which you apply the operations is not necessarily critical. You
achieve the first of the following examples by applying \textsf{Simplify}
followed by \textsf{Expand}. For the second example, apply \textsf{Expand}
followed by \textsf{Simplify}.

\begin{quote}
$\blacktriangleright $ \textsf{Simplify}, \textsf{Expand}
\end{quote}

\begin{quotation}
$\left( \sec t\right) \left( 1+\cos 2t\right) =\,\dfrac{1}{\cos t}\left(
1+\cos 2t\right) =\,2\cos t$
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Expand}, \textsf{Simplify}
\end{quote}

\begin{quotation}
$\left( \sec t\right) \left( 1+\cos 2t\right) =\,2\sec t\cos ^{2}t=\,2\cos t%
\vspace{4pt}$
\end{quotation}

\strut

\subsection{\ Solution \label{Solution of triangles}of Triangles}

To \emph{solve} a triangle means to determine the lengths of the three sides
and the measures (in degrees or radians) of the three angles.

\subsubsection{Solving a Right Triangle\label{Solve, right triangle}}

You can solve a right triangle 
\index{Solve@Solve!right triangle@right triangle}with sides $a,b,c$ and
opposite angles $\alpha ,\beta ,\gamma $, respectively, if you know the
value of one side and one acute angle, or the value of any two sides.

\strut

\begin{example}
To solve the right triangle with one side of length $c=2$ and one angle $%
\alpha =%
\frac{\pi }{9}$,

\begin{enumerate}
\item  Choose \textsf{New Definition} from the \textsf{Define} menu for each
of the given values $\alpha =\frac{\pi }{9}$ and $c=2$.

\item  Apply \textsf{Evaluate} to $\beta =\frac{\pi }{2}-\alpha $ to get $%
\beta =\,\frac{7}{18}\pi $.

\item  Apply \textsf{Evaluate\ }(or \textsf{Evaluate Numerically}) to $%
a=c\sin \alpha $ to get $a=2\sin \frac{1}{9}\pi $ $\left( =\,.68404\right) $.

\item  Apply \textsf{Evaluate\ }to $b=c\cos \alpha $ to get $b=\,2\cos \frac{%
1}{9}\pi $ $\left( =\,1.8794\right) $.
\end{enumerate}
\end{example}

\strut

\begin{example}
To solve a right triangle given two sides, say $a=19$ and $c=23$,

\begin{enumerate}
\item  Apply \textsf{Define}\ to each of the given values, $a=19$ and $c=23$.

\item  Place the insertion point in the equation $a^{2}+b^{2}=c^{2}$.

\item  From the \textsf{Solve} submenu, choose \textsf{Exact} to get $b=2%
\sqrt{42}$.

\item  Place the insertion point in each of the equations $\sin \alpha =%
\dfrac{a}{c}$, $\cos \beta =\dfrac{a}{c}$ in turn.

\item  From the \textsf{Solve} submenu, choose \textsf{Exact} to get $\alpha
=\arcsin \frac{19}{23}$, $\beta =\arccos \frac{19}{23}.$
\end{enumerate}
\end{example}

\strut

For a numerical result, evaluate these functions numerically, or choose 
\textsf{Numeric}\ rather than \textsf{\QTR{MenuDialog}{Exact}} from the 
\textsf{Solve} submenu in the final step. You need to specify intervals for $%
\alpha $ and $\beta $, such as $\left[ 
\begin{array}{c}
\sin \alpha =a/c \\ 
\alpha \in \left( 0,\pi /2\right)
\end{array}
\right] $ and $\left[ 
\begin{array}{c}
\cos \beta =a/c \\ 
\beta \in \left( 0,\pi /2\right)
\end{array}
\right] $, before applying \textsf{Solve + Numeric} or you may get a
solution greater than $\frac{\pi }{2}$. Specifying these intervals gives the
solution 
\[
b=\,12.96,\alpha =\,.9721\,,\beta =.5987 
\]

\subsubsection{Solving General Triangles\label{Law of sines}\label%
{Solve, general triangle}}

The \emph{law of sines} 
\[
\frac{a}{\sin \alpha }=\frac{b}{\sin \beta }=\frac{c}{\sin \gamma } 
\]
enables you to solve a triangle 
\index{Solve@Solve!general triangle@general triangle}if you are given one
side and two angles, or if you are given two sides and an angle opposite one
of these sides.%
\FRAME{dtbpF}{2.3298in}{1.5489in}{0pt}{}{}{triangle.wmf}{\special{language
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'TRIANGLE.WMF';file-properties "XNPEU";}}

\strut

\begin{example}
To solve a triangle given one side and two angles,

\begin{enumerate}
\item  Use \textsf{New Definition} on the \textsf{Define} submenu to define $%
\alpha =\frac{\pi }{9}$, $\beta =\frac{2\pi }{9}$, and $c=2$

\item  Evaluate $\gamma =\pi -\alpha -\beta $ to get $\gamma =\,\frac{2}{3}%
\pi $.

\item  Use \textsf{New Definition} on the \textsf{Define} submenu to define $%
\gamma =\,\frac{2}{3}\pi $.

\item  Apply \textsf{Solve + Exact} to $\dfrac{a}{\sin \alpha }=\dfrac{c}{%
\sin \gamma }$ and to $\dfrac{b}{\sin \beta }=\dfrac{c}{\sin \gamma }$ to
get 
\[
a=\frac{4}{3}\sqrt{3}\sin \frac{1}{9}\pi \text{ and }b=\frac{4}{3}\sqrt{3}%
\sin \frac{2}{9}\pi 
\]
\end{enumerate}
\end{example}

\strut

You can apply \textsf{Solve + Numeric\ }to get numerical solutions, or you
can evaluate the preceding solutions numerically.

Using both the law of sines and 
\index{Law of cosines@Law of cosines}\label{Law of cosines}the \emph{law of
cosines,} 
\[
a^{2}+b^{2}-2ab\cos \gamma =c^{2} 
\]
you can solve a triangle given two sides and the included angle, or given
three sides.

\strut

\begin{example}
To solve a triangle given two sides and the included angle,

\begin{enumerate}
\item  Define each of $a=2.34$, $b=3.57$, and $\gamma =\,%
\frac{29}{216}\pi $.

\item  Apply \textsf{Solve + Exact} to $a^{2}+b^{2}-2ab\cos \gamma =c^{2}$
to get $c=\,1.7255$.

\item  Define $c=\,1.7255$.

\item  Apply \textsf{Solve + Exact} to both $\dfrac{a}{\sin \alpha }=\dfrac{c%
}{\sin \gamma }$ and $\dfrac{b}{\sin \beta }=\dfrac{c}{\sin \gamma }$ to get 
$\alpha =.58859$ and $\beta =1.0104$.
\end{enumerate}
\end{example}

\strut

A triangle with three sides given is solved similarly: interchange the
actions on $\gamma $ and $c$ in the steps just described.

\strut

\begin{example}
To solve a triangle given three sides,

\begin{enumerate}
\item  Define $a=2.53$, $b=4.15$, and $c=6.19$.

\item  Apply \textsf{Solve + Exact} to $a^{2}+b^{2}-2ab\cos \gamma =c^{2}$
to get $\gamma =2.3458$.

\item  Define $\gamma =2.3458$.

\item  Apply \textsf{Solve + Exact} to $\dfrac{a}{\sin \alpha }=\dfrac{c}{%
\sin \gamma }$ to get $\alpha =.29632$, and to $\dfrac{b}{\sin \beta }=%
\dfrac{c}{\sin \gamma }$ to get $\beta =.49948$.
\end{enumerate}
\end{example}

\strut

\subsection{\ Inverse\label{Inverse trig functions} Trigonometric Functions
and Trigonometric Equations}

The following type of question arises frequently when working with the
trigonometric functions: for which angles $x$ is $\sin x=y$? There are many
correct answers to these questions, since the trigonometric functions are
periodic. The inverse trigonometric functions provide answers to such
questions that lie within a restricted domain. The inverse sine function,
for example, produces the angle $x$ between $-\frac \pi 2$ and $\frac \pi 2$
that satisfies $\sin x=y$. This solution is denoted by $\arcsin x$ or $\sin
^{-1}x$.

The inverse trigonometric functions and a number of other functions are
available in the dialog box that comes up when you click the \textsf{Math
Name} button on the \textsf{Math} toolbar. They can also be entered from the
keyboard in mathematics mode.

\strut

\begin{example}
To find 
\index{Inverse@Inverse!trigonometric functions@trigonometric functions}the
angle $x$ (between $-%
\frac{\pi }{2}$ and $\frac{\pi }{2}$) for which $\tan x=100$,

\begin{itemize}
\item  Leave the insertion point in the expression $\arctan 100.$

\item  Apply \textsf{Evaluate Numerically}$\,$
\end{itemize}

This gives\textsf{\ }$\arctan 100=\,1.56079666$
\end{example}

\strut

You can also find an angle satisfying $\tan x=100$ by applying \textsf{Solve
+ Numeric} to the equation. This technique does not necessarily find the
solution between $-\frac{\pi }{2}$ and $\frac{\pi }{2}$. In this case, in
fact, it gives the solution $x=102.09176$, which is $1.56079666+32\pi $. You
can specify the interval for the solution, as follows.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Numeric}
\end{quote}

\begin{quotation}
$
\begin{array}{c}
\tan x=100 \\ 
x\in \left( -\frac{\pi }{2},\frac{\pi }{2}\right)
\end{array}
$, Solution is : $\left\{ x=1.56079666\right\} $
\end{quotation}

\strut

Using this technique, you can find solutions to a variety of trigonometric
equations in specified intervals. Following are some examples of equations
that you can solve with \textsf{Solve + Exact} and \textsf{Solve + Numeric.}%
\ 

\begin{center}
{\ 
\begin{tabular}{lll}
\textbf{Equation} & \textbf{Solve + Exact} & \textbf{Solve + Numeric} \\ 
\hline
$\sin t=\sin 2t$ & $\left\{ t=0\right\} ,\,\left\{ t=\frac{1}{3}\pi \right\} 
$ & $t=5.236$ \\ 
$8\tan x-13+5\tan ^{2}x=3$ & $x=\arctan \left( -\frac{4}{5}\pm \frac{4}{5}%
\sqrt{6}\right) $ & $x=-2.2824$ \\ 
$\tan ^{2}x-\cot ^{2}x=1$ & $x=\pm \arcsin \left( \frac{1}{2}\sqrt{-2\pm 2%
\sqrt{5}}\right) $ & $x=2.237$%
\end{tabular}
}
\end{center}

\strut

Note that \textsf{Solve + Exact}\ gives two or four solutions in these
examples, whereas \textsf{Solve + Numeric}\ returns only one solution. In
general, applying \textsf{Evaluate Numerically\ }to the exact solutions
gives numerical solutions different from those produced by \textsf{Solve +
Numeric. }This is a good place to experiment with a plot to visualize the
complete solution.\FRAME{dtbpFU}{3.0441in}{2.0384in}{0pt}{\Qcb{$y=\tan
^{2}x-\cot ^{2}x=1$, $y=1$}}{}{solvtrig.wmf}{\special{language "Scientific
Word";type "GRAPHIC";display "PICT";valid_file "F";width 3.0441in;height
2.0384in;depth 0pt;original-width 216.8125pt;original-height
144.5625pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'Solvtrig.wmf';file-properties "XNPEU";}}

\strut

Solutions can be found in a specified range with \textsf{Solve + Numeric},
as demonstrated in the following example. Enter the equation and the range
in different rows of a one-column matrix.

\begin{quote}
$\blacktriangleright $ \textsf{Solve + Numeric}
\end{quote}

\begin{quotation}
$
\begin{array}{l}
8\tan x-13+5\tan ^{2}x=3 \\ 
x\in \left( -\frac{\pi }{2},\frac{\pi }{2}\right)
\end{array}
$, Solution is : $\left\{ x=.85916\right\} $
\end{quotation}

\strut

\subsection{\ Related topics}

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

\item  \hyperref{2D Plots of Functions and Expressions}{}{}{DM6-4.tex#Plot
expression}

\item  \hyperref{Exercises and Solutions}{}{}{DM4-3.tex#Exercises}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{38.875pt}{24.5pt}{6pt}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 38.875pt;height 24.5pt;depth
6pt;original-width 36.875pt;original-height 22.5625pt;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'index.wmf';file-properties
"XNPEU";}}}{}{}{DMIndex.tex#Math Index}\quad {\small Index entries: 
\hyperref{Combine}{}{}{DMIndex.tex#Combine}, \hyperref{Function argument
selection method}{}{}{DMIndex.tex#Function argument}, \hyperref{Inverse
functions}{}{}{DMIndex.tex#Inverse functions}, \hyperref{Angular measure}{}{%
}{DMIndex.tex#Angular measure}, \hyperref{Solve}{}{}{DMIndex.tex#Solve}, 
\hyperref{Trigonometry}{}{}{DMIndex.tex#Trigonometry}, \hyperref{Trigtype
functions}{}{}{DMIndex.tex#Trigtype functions}}

\end{document}
