\hfill \thepage}
%}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\input{tcilatex}
\begin{document}
\section{Ma 115 Lecture 9/22/99}
\section{Review of Trigonometry}
\vspace{0in}
Many applications of calculus use trigonometry. \ So we here review the
subject of trigonometry.
\vspace{0in}
Trigonometry is the study of angles and functions of angles and their
application to circles, polygons and to science. \ So we start with the
definition of angles and their measures.
\subsection{Angles and Circles \ --- \ Angular Measure}
\vspace{0in}
\subsubsection{Angles}
Roughly, an \emph{angle} is the space between two \emph{rays} or \emph{line
segments} with a common \emph{endpoint}. \ The rays or line segments are
called the \emph{sides} and the common endpoint is called the \emph{vertex}.
\ More precisely, if a \emph{ray} or \emph{line segment} rotates about an
\emph{endpoint} from some initial position, called the \emph{initial side},
to some final position, called the \emph{final side}, then the \emph{angle}
between the sides is the space swept out. \ Within a plane, we say the angle
is \emph{positive} if the the rotation is \emph{counterclockwise}, and is
\emph{negative} if the rotation is \emph{clockwise}. \ An angle is in \emph{%
standard position} if the vertex is at the origin and the initial side is
along the positive $x$-axis.
\begin{center}
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\end{center}
\subsubsection{Measures of Angle}
The size of an angle may be measured in \emph{revolutions} ({\small rev}),
in \emph{degrees} ($%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$) or in \emph{radians} ({\small rad}).
\vspace{0in}
An angle is called a \emph{full rotation} if the ray rotates from the
initial side all the way around so that the final side coincides with the
initial side.
A \emph{full rotation} is measured as $1${\small rev}$\,=360%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=2\pi ${\small rad}.
\vspace{0in}
An angle is called a \emph{half rotation} or a \emph{straight angle} if the
ray rotates from the initial side to a final side which is directly opposite
to the initial side.
A \emph{half rotation} is measured as $\dfrac{1}{2}${\small rev}$\,=180%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=\pi ${\small rad}.
\vspace{0in}
An angle is called a \emph{quarter rotation} or a \emph{right angle} if the
ray rotates from the initial side to a final side which is perpendicular to
the initial side.
A \emph{quarter rotation} is measured as $\dfrac{1}{4}${\small rev}$\,=90%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=\dfrac{\pi }{2}${\small rad}.
\vspace{0in}
An angle is called a \emph{null rotation} if the ray never rotates so that
the final side coincides with the initial side.
A \emph{null rotation} is measured as $0${\small rev}$\,=0%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=0${\small rad}.
\vspace{0in}
Several other important angles are:
{\small \qquad }$\dfrac{3}{4}${\small rev}$\,=270%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=\dfrac{3\pi }{2}${\small rad\qquad\ }\qquad $\dfrac{1}{6}${\small rev}$%
\,=60%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=\dfrac{\pi }{3}${\small rad\qquad\ }\qquad $\dfrac{1}{8}${\small rev}$%
\,=45%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=\dfrac{\pi }{4}$rad\qquad\ {\small \qquad }$\dfrac{1}{12}${\small rev}$%
\,=30%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=\dfrac{\pi }{6}${\small rad}
\vspace{0in}
\begin{example}
Find the angular measure of one time zone on the surface of the earth.
\end{example}
\emph{Solution.} \ The earth rotates once a day or by $1${\small rev} in $24$
{\small hours}. \ Thus a 1 hour time zone has an angular measure of $\dfrac{1%
}{24}\,${\small rev}$\,=15%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=\dfrac{\pi }{12}\,${\small rad}.
\vspace{0in}
An angle will be bigger than a full rotation if the ray rotates from the
initial side all the way around and past the initial side again. \ There is
no limit to the size of an angle either positive or negative. \ Two angles
with the same initial and final sides are called \emph{coterminal} and their
measures must differ by an integral multiple of $1\,${\small rev}$\,=360%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=2\pi \,${\small rad}.
\vspace{0in}
Below are some pictures of various angles. \ In each figure, the initial
sides of the angles coincide.
\begin{center}
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\ \ \
\vspace{0in}
Below are some pictures of various angles in standard position.
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\end{center}
\vspace{0in}
\paragraph{You must primarily learn to identify angles in {\protect\Large %
standard position} in {\protect\Large radians}, since that is most often
needed for trig functions.}
\subsection{Angles and Circles \ --- \ The Parts of a Circle}
\vspace{0in}
\subsubsection{Circle, Disk}
The \emph{circle} with \emph{center} $P$ and \emph{radius} $r$ is the set of
all points $X$ in the plane whose distance from $P$ is $r$. \ If the center
is $P=\left( a,b\right) $ and the general point on the circle is $X=\left(
x,y\right) $ then the equation of the circle is
\begin{equation*}
\left( x-a\right) ^{2}+\left( y-b\right) ^{2}=r^{2}
\end{equation*}
We often take the center to be the origin $O=\left( 0,0\right) .$ \ Then its
equation is
\begin{equation*}
x^{2}+y^{2}=r^{2}
\end{equation*}
The region inside of a circle is called a \emph{disk}.
\vspace{0in}
\subsubsection{Radius, Diameter}
A \emph{radial line} (or a \emph{radius}) is any line segment from the
center of the circle to a point on the circle. \ The word radius can refer
to either a radial line or its length $r$.
\vspace{0in}
A \emph{diameter line} (or a \emph{diameter}) is any line segment between
two points on the circle which passes through the center. \ The word
diameter can refer to either a diameter line or its length $d=2r$.
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\end{center}
\subsubsection{Circumference, Area}
The \emph{circumference} of the circle is the distance around the circle. \
By the definition of $\pi $ the circumference is $C=\pi d=2\pi r$.
\vspace{0in}
The \emph{area} of the circle (actually of the disk) is $A=\pi r^{2}$.
\vspace{0in}
\subsubsection{Arc, Chord, Sector}
An \emph{arc} is any piece of the circle between two points on the circle. \
A \emph{chord} is any line segment between two points on the circle. \ A
\emph{sector} is any piece of the disk between two radial lines.
\begin{tabular}{l}
Two radial lines are shown in black.. \\
An \emph{arc} is shown in \emph{blue}. \\
A \hyperref{chord}{}{}{} is shown in \hyperref{green}{}{}{}. \\
A sector is shaded in cyan.%
\end{tabular}
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\subsubsection{Central angle, Arc length, Sector area}
An angle whose vertex is at the center of a circle is called a \emph{central
angle}. \ The sides of a central angle are radial lines which intersect the
circle at two points. \ The arc between these two points, the chord between
these two points and the sector between the two sides of the angle are
called the \emph{arc, chord and sector subtended by the central angle}. \ We
use the following notations:
\begin{center}
\vspace{0in}
\begin{tabular}{|lll|}
\hline
$C$ & $=$ & Circumference of the circle \\ \hline
$L$ & $=$ & Length of a sector of the circle \\ \hline
$A_{\bigcirc }$ & $=$ & Area of the whole circle \\ \hline
$A_{\sphericalangle }$ & $=$ & Area of a sector of the circle \\ \hline
\end{tabular}
\end{center}
In the previous figure the arc, chord and sector are subtended by the
central angle $\theta $.
{\Large T}he \emph{fraction} of the circle or disk \emph{subtended} by a
central angle $\theta $\ is $\dfrac{\theta }{360%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
}$ for $\theta $ in degrees, or $\dfrac{\theta }{2\pi \text{{\small rad}}}$
for $\theta $ in radians. \ So the length of an arc (the \emph{arc length})
is this fraction of the circumference and the area of a sector (the \emph{%
sector area}) is this fraction of the area of disk. \ Hence the arclength is
\begin{equation*}
L=\left\{
\begin{array}{ll}
\dfrac{\theta }{360%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
}C=\dfrac{2\pi r\theta }{360} & \text{for }\theta \text{ in degrees} \\
\dfrac{\theta }{2\pi \text{{\small rad}}}C=\dfrac{2\pi r\theta }{2\pi }%
=r\theta & \text{for }\theta \text{ in radians}%
\end{array}
\right.
\end{equation*}
and the sector area is
\begin{equation*}
A_{\sphericalangle }=\left\{
\begin{array}{ll}
\dfrac{\theta }{360%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
}A_{\bigcirc }=\dfrac{\pi r^{2}\theta }{360} & \text{for }\theta \text{ in
degrees} \\
\dfrac{\theta }{2\pi \text{{\small rad}}}A_{\bigcirc }=\dfrac{\pi
r^{2}\theta }{2\pi }=\dfrac{1}{2}r^{2}\theta & \text{for }\theta \text{ in
radians}%
\end{array}
\right.
\end{equation*}
\vspace{0in}
\begin{example}
In the figure above, suppose the radius is $r=6${\small cm} and the central
angle is $\theta =105%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
=\dfrac{7\pi }{12}${\small rad}. \ Find the circumference and area of the
circle. \ Find the fraction subtended as well as the arc length and sector
area of the arc and sector subtended by the central angle $\theta $.
\end{example}
\vspace{0in}
The circumference is $C=2\pi \left( 6\text{{\small cm}}\right) =12\pi $%
{\small cm}. \ The area is $A=\pi \left( 6\text{{\small cm}}\right)
^{2}=36\pi ${\small cm}$^{2}$.
Using degrees: \ The fraction subtended is $\dfrac{105%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
}{360%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
}=\dfrac{7}{24}$. \ The arc length is $L=\dfrac{2\pi \left( 6\text{{\small cm%
}}\right) 105}{360}=\dfrac{7\pi }{2}${\small cm}. \ And the sector area is $%
A=\dfrac{\pi \left( 6\text{{\small cm}}\right) ^{2}105}{360}=\dfrac{21\pi }{2%
}${\small cm}$^{2}$.
Using radians: \ The fraction subtended is $\dfrac{7\pi \text{{\small rad}}}{%
12\cdot 2\pi \text{{\small rad}}}=\dfrac{7}{24}$. \ The arc length is $%
L=\left( 6\text{{\small cm}}\right) \dfrac{7\pi }{12}=\dfrac{7\pi }{2}$%
{\small cm}. \ And the sector area is $A=\dfrac{1}{2}\left( 6\text{{\small cm%
}}\right) ^{2}\dfrac{7\pi }{12}=\dfrac{21\pi }{2}${\small cm}$^{2}$.
Notice how much simplier the computations are in terms of radians.
\subsubsection{Secant line, Tangent line}
\vspace{0in}
A \emph{secant line} is a line which intersects the circle twice. \ The part
inside the circle is a \emph{chord}.
A \emph{tangent line} is a line which intersects the circle at exactly one
point called the \emph{point of tangency}.
\begin{tabular}{l}
A secant line is shown in purple. \\
Its \hyperref{chord}{}{}{} is shown in \hyperref{green}{}{}{}. \\
A \emph{tangent line} is shown in \emph{blue}. \\
Its point of tangency is in yellow.%
\end{tabular}
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We are now ready to introduce the basic trig functions. \ There are two
definitions. \ The first is given in terms of a right triangle.
\subsection{Trig Functions \ --- \ Triangle Definitions}
\begin{tabular}{l}
Consider a right triangle with one angle $\theta $. \\
The sides are: \\
\qquad the leg adjacent to $\theta $:\qquad \emph{adj} \\
\qquad the leg opposite to $\theta $:\negthinspace \negthinspace \qquad
\emph{opp} \\
\qquad and the hypotenuse:\ \qquad \emph{hyp}%
\end{tabular}
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In terms of these sides, the trig functions, \emph{sine}, \emph{cosine},
\emph{tangent}, \emph{cotangent}, \emph{secant} and \emph{cosecant}, of the
angle $\theta $ are given by
\begin{equation*}
\begin{array}{lll}
\sin \theta =\dfrac{\text{\emph{opp}}}{\text{\emph{hyp}}}\qquad & \tan
\theta =\dfrac{\text{\emph{opp}}}{\text{\emph{adj}}}\qquad & \sec \theta =%
\dfrac{\text{\emph{hyp}}}{\text{\emph{adj}}} \\
\cos \theta =\dfrac{\text{\emph{adj}}}{\text{\emph{hyp}}} & \cot \theta =%
\dfrac{\text{\emph{adj}}}{\text{\emph{opp}}} & \csc \theta =\dfrac{\text{%
\emph{hyp}}}{\text{\emph{opp}}}%
\end{array}%
\end{equation*}
In terms of $\sin \theta $ and $\cos \theta $, the other trig functions are
\begin{equation*}
\tan \theta =\dfrac{\sin \theta }{\cos \theta }\qquad \cot \theta =\dfrac{%
\cos \theta }{\sin \theta }=\dfrac{1}{\tan \theta }\qquad \sec \theta =%
\dfrac{1}{\cos \theta }\qquad \csc \theta =\dfrac{1}{\sin \theta }
\end{equation*}
\begin{center}
\vspace{0in}
\end{center}
We now give a definition of the trig functions which works for all angles,
not just acute angles.
\subsection{Trig Functions \ --- \ Circle Definitions}
\begin{tabular}{l}
Consider a circle of radius $r$ centered at the origin \\
in which a radial line has been drawn at an angle $\theta $ \\
measured counterclockwise from the positive $x$-axis. \\
The radial line intersects the circle at a point $\left( x,y\right) $. \\
\emph{Note:} $x$ and/or $y$ may be negative.%
\end{tabular}
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In terms of $x$, $y$ and $r$, the trig functions, \emph{sine}, \emph{cosine}%
, \emph{tangent}, \emph{cotangent}, \emph{secant} and \emph{cosecant}, of
the angle $\theta $ are given by.
\begin{equation*}
\begin{array}{lll}
\sin \theta =\dfrac{y}{r}\qquad & \tan \theta =\dfrac{y}{x}\qquad & \sec
\theta =\dfrac{r}{x} \\
\cos \theta =\dfrac{x}{r} & \cot \theta =\dfrac{x}{y} & \csc \theta =\dfrac{r%
}{y}%
\end{array}%
\end{equation*}
Notice that this says a point on the circle is $\left( x,y\right) =\left(
r\cos \theta ,r\sin \theta \right) $.
\vspace{0in}
In terms of $\sin \theta $ and $\cos \theta $, the other trig functions are
\begin{equation*}
\tan \theta =\dfrac{\sin \theta }{\cos \theta }\qquad \cot \theta =\dfrac{%
\cos \theta }{\sin \theta }=\dfrac{1}{\tan \theta }\qquad \sec \theta =%
\dfrac{1}{\cos \theta }\qquad \csc \theta =\dfrac{1}{\sin \theta }
\end{equation*}
\vspace{0in}
Notice that this circle definition of the trig functions works for any angle
$\theta $ not just acute angles. \ However, to make sense, the definitions
in terms of circles and triangles must agree for acute angles. \ They do!
\subsubsection{Trig Functions on a Unit Circle}
It is often convenient to use a unit circle. \ Then the trig functions are
given by
\begin{equation*}
\begin{array}{lll}
\sin \theta =y\qquad & \tan \theta =\dfrac{y}{x}\qquad & \sec \theta =\dfrac{%
1}{x} \\
\cos \theta =x & \cot \theta =\dfrac{x}{y} & \csc \theta =\dfrac{1}{y}%
\end{array}%
\end{equation*}
and a point on the circle is $\left( x,y\right) =\left( \cos \theta ,\sin
\theta \right) $.
\vspace{0in}
\subsection{Trig Functions \ --- \ Special Angles}
You need to know (or be able to figure out) the values of the trig functions
for specific angles: those which are multiples of $30%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=\dfrac{\pi }{6}${\small rad} or $45%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=\dfrac{\pi }{4}${\small rad.}
\vspace{0in}
Here is a table of
\CustomNote[Trig Functions for Special Angles]{Margin Hint}{$\vspace{0in}%
\hspace{0.75in}$\emph{Trig Functions for Special Angles}
\par
$%
\begin{tabular}{|c|c||c|c|c|c|c|c|}
\hline
$\theta $ Deg & $\theta $ Rad & $\sin \theta $ & $\cos \theta $ & $\tan
\theta $ & $\cot \theta $ & $\sec \theta $ & $\csc \theta $ \\ \hline\hline
\multicolumn{1}{|r|}{$0%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$0$ {\small rad}} & $0$ & $1$ & $0$ & $\pm \infty
$ & $1$ & $\pm \infty $ \\ \hline
\multicolumn{1}{|r|}{$30%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{\pi }{6}$ {\small rad}} & $\dfrac{1}{2}$
& $\dfrac{\sqrt{3}}{2}$ & $\dfrac{1}{\sqrt{3}}$ & $\sqrt{3}$ & $\dfrac{2}{%
\sqrt{3}}$ & $2$ \\ \hline
\multicolumn{1}{|r|}{$45%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{\pi }{4}\,${\small rad}} & $\dfrac{1}{%
\sqrt{2}}$ & $\dfrac{1}{\sqrt{2}}$ & $1$ & $1$ & $\sqrt{2}$ & $\sqrt{2}$ \\
\hline
\multicolumn{1}{|r|}{$60%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{\pi }{3}$ {\small rad}} & $\dfrac{\sqrt{3}%
}{2}$ & $\dfrac{1}{2}$ & $\sqrt{3}$ & $\dfrac{1}{\sqrt{3}}$ & $2$ & $\dfrac{2%
}{\sqrt{3}}$ \\ \hline
\multicolumn{1}{|r|}{$90%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{\pi }{2}$ {\small rad}} & $1$ & $0$ & $%
\pm \infty $ & $0$ & $\pm \infty $ & $1$ \\ \hline\hline
\multicolumn{1}{|r|}{$120%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{2\pi }{3}$ {\small rad}} & $\dfrac{\sqrt{3%
}}{2}$ & $-\dfrac{1}{2}$ & $-\sqrt{3}$ & $-\dfrac{1}{\sqrt{3}}$ & $-2$ & $%
\dfrac{2}{\sqrt{3}}$ \\ \hline
\multicolumn{1}{|r|}{$135%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{3\pi }{4}\,${\small rad}} & $\dfrac{1}{%
\sqrt{2}}$ & $-\dfrac{1}{\sqrt{2}}$ & $-1$ & $-1$ & $-\sqrt{2}$ & $\sqrt{2}$
\\ \hline
\multicolumn{1}{|r|}{$150%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{5\pi }{6}$ {\small rad}} & $\dfrac{1}{2}$
& $-\dfrac{\sqrt{3}}{2}$ & $-\dfrac{1}{\sqrt{3}}$ & $-\sqrt{3}$ & $-\dfrac{2%
}{\sqrt{3}}$ & $2$ \\ \hline
\multicolumn{1}{|r|}{$180%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\pi $ {\small rad}} & $0$ & $-1$ & $0$ & $\pm
\infty $ & $-1$ & $\pm \infty $ \\ \hline\hline
\multicolumn{1}{|r|}{$210%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{7\pi }{6}$ {\small rad}} & $-\dfrac{1}{2}$
& $-\dfrac{\sqrt{3}}{2}$ & $\dfrac{1}{\sqrt{3}}$ & $\sqrt{3}$ & $-\dfrac{2}{%
\sqrt{3}}$ & $-2$ \\ \hline
\multicolumn{1}{|r|}{$225%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{5\pi }{4}\,${\small rad}} & $-\dfrac{1}{%
\sqrt{2}}$ & $-\dfrac{1}{\sqrt{2}}$ & $1$ & $1$ & $-\sqrt{2}$ & $-\sqrt{2}$
\\ \hline
\multicolumn{1}{|r|}{$240%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{4\pi }{3}$ {\small rad}} & $-\dfrac{\sqrt{%
3}}{2}$ & $-\dfrac{1}{2}$ & $\sqrt{3}$ & $\dfrac{1}{\sqrt{3}}$ & $-2$ & $-%
\dfrac{2}{\sqrt{3}}$ \\ \hline
\multicolumn{1}{|r|}{$270%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{3\pi }{2}$ {\small rad}} & $-1$ & $0$ & $%
\pm \infty $ & $0$ & $\pm \infty $ & $-1$ \\ \hline\hline
\multicolumn{1}{|r|}{$300%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{5\pi }{3}$ {\small rad}} & $-\dfrac{\sqrt{%
3}}{2}$ & $\dfrac{1}{2}$ & $-\sqrt{3}$ & $-\dfrac{1}{\sqrt{3}}$ & $2$ & $-%
\dfrac{2}{\sqrt{3}}$ \\ \hline
\multicolumn{1}{|r|}{$315%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{7\pi }{4}\,${\small rad}} & $-\dfrac{1}{%
\sqrt{2}}$ & $\dfrac{1}{\sqrt{2}}$ & $-1$ & $-1$ & $\sqrt{2}$ & $-\sqrt{2}$
\\ \hline
\multicolumn{1}{|r|}{$330%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{11\pi }{6}$ {\small rad}} & $-\dfrac{1}{2}
$ & $\dfrac{\sqrt{3}}{2}$ & $-\dfrac{1}{\sqrt{3}}$ & $-\sqrt{3}$ & $\dfrac{2%
}{\sqrt{3}}$ & $-2$ \\ \hline
\multicolumn{1}{|r|}{$360%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$2\pi $ {\small rad}} & $0$ & $1$ & $0$ & $\pm
\infty $ & $1$ & $\pm \infty $ \\ \hline
\end{tabular}%
\ $}. You should \emph{absolutely not memorize} this table.
\vspace{0in}
Rather, in each case, you should figure out the values of the trig functions
using the circle definition and your knowledge about $30%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
-60%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
-90%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$ and $45%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$ right triangles. \ These are typically remembered as
\quad\ \
\begin{tabular}{c}
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\end{tabular}
\begin{tabular}{l}
In the $30%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
-60%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
-90%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$\ triangle, remember the sides as ``$1,2,\sqrt{3}.$'' \\
Notice that the shortest side $1$ is opposite the smallest angle $30%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=\dfrac{\pi }{6}${\small rad}, \\
the middle length side $\sqrt{3}\approx 1.7$ is opposite the middle angle $60%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=\dfrac{\pi }{3}${\small rad} \\
and the longest side $2$ is opposite the biggest angle $90%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
\,=\dfrac{\pi }{2}${\small rad}.%
\end{tabular}
\FRAME{itbpF}{1.8386in}{1.6691in}{0.8769in}{}{}{g0204.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
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"0";croptop "1";cropright "1";cropbottom "0";filename
'/document/graphics/g0204.wmf';file-properties "XNPEU";}}\quad In the $45%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$ right triangle, remember the sides as ``$1,1,\sqrt{2}.$''
\begin{example}
Find the trig functions at $\dfrac{5\pi }{3}$ {\small rad.}
\end{example}
Using our knowledge of $30%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
-60%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
-90%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$ triangles, we draw a circle of radius $2$, draw the radial line at $\dfrac{%
5\pi }{3}$ {\small rad} and drop (raise) a perpendicular to the $x$-axis. \
The hypotenuse of the resulting right triangle is $2$ since this is the
radius of the circle. \ We label the two legs of the triangle with $x=1$ and
$y=-\sqrt{3}$, . \ Since this is the IV$^{\text{th}}$ quadrant, the $x$
coordinate is positive and the $y$ coordinate is negative.
\
\begin{tabular}{c}
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"1";cropright "1";cropbottom "0";filename
'/document/graphics/g0205.wmf';file-properties "XNPEU";}}%
\end{tabular}
$
\begin{array}{ll}
\text{Then the trig functions are\bigskip } & \\
\sin \dfrac{5\pi }{3}=\dfrac{y}{r}=\dfrac{-\sqrt{3}}{2}=-\dfrac{\sqrt{3}}{2}%
\qquad & \cos \dfrac{5\pi }{3}=\dfrac{x}{r}=\dfrac{1}{2} \\
\tan \dfrac{5\pi }{3}=\dfrac{y}{x}=\dfrac{-\sqrt{3}}{1}=-\sqrt{3} & \cot
\dfrac{5\pi }{3}=\dfrac{x}{y}=\dfrac{1}{-\sqrt{3}}=-\dfrac{1}{\sqrt{3}} \\
\sec \dfrac{5\pi }{3}=\dfrac{r}{x}=\dfrac{2}{1}=2 & \csc \dfrac{5\pi }{3}=%
\dfrac{r}{y}=\dfrac{2}{-\sqrt{3}}=-\dfrac{2}{\sqrt{3}}%
\end{array}
$
\begin{example}
Find the trig functions at $\dfrac{3\pi }{4}$ {\small rad.}
\end{example}
We draw a circle of radius $\sqrt{2}$, draw the radial line at $\dfrac{3\pi
}{4}$ {\small rad} and drop a perpendicular to the $x$-axis. \ The
hypotenuse of the resulting right triangle is $\sqrt{2}$ since this is the
radius of the circle. \ We label the two legs of the triangle with $x=-1$
and $y=1$, using our knowledge of $45%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$ right triangles. \ Since this is the II$^{\text{nd}}$ quadrant, the $x$
coordinate is negative and the $y$ coordinate is positive.
\begin{tabular}{c}
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"1";cropright "1";cropbottom "0";filename
'/document/graphics/g0206.wmf';file-properties "XNPEU";}}%
\end{tabular}
\ $
\begin{array}{ll}
\text{Then the trig functions are\bigskip } & \\
\sin \dfrac{3\pi }{4}=\dfrac{y}{r}=\dfrac{1}{\sqrt{2}} & \cos \dfrac{3\pi }{4%
}=\dfrac{x}{r}=\dfrac{-1}{\sqrt{2}} \\
\tan \dfrac{3\pi }{4}=\dfrac{y}{x}=\dfrac{1}{-1}=-1 & \cot \dfrac{3\pi }{4}=%
\dfrac{x}{y}=\dfrac{-1}{1}=-1 \\
\sec \dfrac{3\pi }{4}=\dfrac{r}{x}=\dfrac{\sqrt{2}}{-1}=-\sqrt{2}\qquad &
\csc \dfrac{3\pi }{4}=\dfrac{r}{y}=\dfrac{\sqrt{2}}{1}=\sqrt{2}%
\end{array}
$
\begin{example}
Find the trig functions at $\dfrac{3\pi }{2}$ {\small rad.}
\end{example}
We draw a circle of radius $1$ and draw the radial line at $\dfrac{3\pi }{2}$
{\small rad}. \ The coordinates at the endpoint of the radial line are $x=0$
and $y=-1$.
\begin{tabular}{c}
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'/document/graphics/g0207.wmf';file-properties "XNPEU";}}%
\end{tabular}
\ $
\begin{array}{ll}
\text{So the trig functions are\bigskip } & \\
\sin \dfrac{3\pi }{2}=\dfrac{y}{r}=\dfrac{-1}{1}=-1 & \cos \dfrac{3\pi }{2}=%
\dfrac{x}{r}=\dfrac{0}{1}=0 \\
\tan \dfrac{3\pi }{2}=\dfrac{y}{x}=\dfrac{-1}{0}=\pm \infty & \cot \dfrac{%
3\pi }{2}=\dfrac{x}{y}=\dfrac{0}{-1}=0 \\
\sec \dfrac{3\pi }{2}=\dfrac{r}{x}=\dfrac{1}{0}=\pm \infty \qquad & \csc
\dfrac{3\pi }{2}=\dfrac{r}{y}=\dfrac{1}{-1}=-1%
\end{array}
$
\vspace{0in}
\subsection{Trig Functions \ --- \ Constructing Graphs}
If we graph the data from the table of
\CustomNote[Trig Functions for Special Angles]{Margin Hint}{$\hspace{0.75in}$%
\emph{Trig Functions for Special Angles}
\par
$%
\begin{tabular}{|c|c||c|c|c|c|c|c|}
\hline
$\theta $ Deg & $\theta $ Rad & $\sin \theta $ & $\cos \theta $ & $\tan
\theta $ & $\cot \theta $ & $\sec \theta $ & $\csc \theta $ \\ \hline\hline
\multicolumn{1}{|r|}{$0%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$0$ {\small rad}} & $0$ & $1$ & $0$ & $\pm \infty
$ & $1$ & $\pm \infty $ \\ \hline
\multicolumn{1}{|r|}{$30%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{\pi }{6}$ {\small rad}} & $\dfrac{1}{2}$
& $\dfrac{\sqrt{3}}{2}$ & $\dfrac{1}{\sqrt{3}}$ & $\sqrt{3}$ & $\dfrac{2}{%
\sqrt{3}}$ & $2$ \\ \hline
\multicolumn{1}{|r|}{$45%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{\pi }{4}\,${\small rad}} & $\dfrac{1}{%
\sqrt{2}}$ & $\dfrac{1}{\sqrt{2}}$ & $1$ & $1$ & $\sqrt{2}$ & $\sqrt{2}$ \\
\hline
\multicolumn{1}{|r|}{$60%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{\pi }{3}$ {\small rad}} & $\dfrac{\sqrt{3}%
}{2}$ & $\dfrac{1}{2}$ & $\sqrt{3}$ & $\dfrac{1}{\sqrt{3}}$ & $2$ & $\dfrac{2%
}{\sqrt{3}}$ \\ \hline
\multicolumn{1}{|r|}{$90%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{\pi }{2}$ {\small rad}} & $1$ & $0$ & $%
\pm \infty $ & $0$ & $\pm \infty $ & $1$ \\ \hline\hline
\multicolumn{1}{|r|}{$120%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{2\pi }{3}$ {\small rad}} & $\dfrac{\sqrt{3%
}}{2}$ & $-\dfrac{1}{2}$ & $-\sqrt{3}$ & $-\dfrac{1}{\sqrt{3}}$ & $-2$ & $%
\dfrac{2}{\sqrt{3}}$ \\ \hline
\multicolumn{1}{|r|}{$135%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{3\pi }{4}\,${\small rad}} & $\dfrac{1}{%
\sqrt{2}}$ & $-\dfrac{1}{\sqrt{2}}$ & $-1$ & $-1$ & $-\sqrt{2}$ & $\sqrt{2}$
\\ \hline
\multicolumn{1}{|r|}{$150%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{5\pi }{6}$ {\small rad}} & $\dfrac{1}{2}$
& $-\dfrac{\sqrt{3}}{2}$ & $-\dfrac{1}{\sqrt{3}}$ & $-\sqrt{3}$ & $-\dfrac{2%
}{\sqrt{3}}$ & $2$ \\ \hline
\multicolumn{1}{|r|}{$180%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\pi $ {\small rad}} & $0$ & $-1$ & $0$ & $\pm
\infty $ & $-1$ & $\pm \infty $ \\ \hline\hline
\multicolumn{1}{|r|}{$210%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{7\pi }{6}$ {\small rad}} & $-\dfrac{1}{2}$
& $-\dfrac{\sqrt{3}}{2}$ & $\dfrac{1}{\sqrt{3}}$ & $\sqrt{3}$ & $-\dfrac{2}{%
\sqrt{3}}$ & $-2$ \\ \hline
\multicolumn{1}{|r|}{$225%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{5\pi }{4}\,${\small rad}} & $-\dfrac{1}{%
\sqrt{2}}$ & $-\dfrac{1}{\sqrt{2}}$ & $1$ & $1$ & $-\sqrt{2}$ & $-\sqrt{2}$
\\ \hline
\multicolumn{1}{|r|}{$240%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{4\pi }{3}$ {\small rad}} & $-\dfrac{\sqrt{%
3}}{2}$ & $-\dfrac{1}{2}$ & $\sqrt{3}$ & $\dfrac{1}{\sqrt{3}}$ & $-2$ & $-%
\dfrac{2}{\sqrt{3}}$ \\ \hline
\multicolumn{1}{|r|}{$270%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{3\pi }{2}$ {\small rad}} & $-1$ & $0$ & $%
\pm \infty $ & $0$ & $\pm \infty $ & $-1$ \\ \hline\hline
\multicolumn{1}{|r|}{$300%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{5\pi }{3}$ {\small rad}} & $-\dfrac{\sqrt{%
3}}{2}$ & $\dfrac{1}{2}$ & $-\sqrt{3}$ & $-\dfrac{1}{\sqrt{3}}$ & $2$ & $-%
\dfrac{2}{\sqrt{3}}$ \\ \hline
\multicolumn{1}{|r|}{$315%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{7\pi }{4}\,${\small rad}} & $-\dfrac{1}{%
\sqrt{2}}$ & $\dfrac{1}{\sqrt{2}}$ & $-1$ & $-1$ & $\sqrt{2}$ & $-\sqrt{2}$
\\ \hline
\multicolumn{1}{|r|}{$330%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$\dfrac{11\pi }{6}$ {\small rad}} & $-\dfrac{1}{2}
$ & $\dfrac{\sqrt{3}}{2}$ & $-\dfrac{1}{\sqrt{3}}$ & $-\sqrt{3}$ & $\dfrac{2%
}{\sqrt{3}}$ & $-2$ \\ \hline
\multicolumn{1}{|r|}{$360%
%TCIMACRO{\U{b0}}%
%BeginExpansion
{{}^\circ}%
%EndExpansion
$} & \multicolumn{1}{|r||}{$2\pi $ {\small rad}} & $0$ & $1$ & $0$ & $\pm
\infty $ & $1$ & $\pm \infty $ \\ \hline
\end{tabular}%
\ $}, we get the following six plots. \ Here and hereafter, the trig
functions will be regarded as functions of the angle in radians.
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1.7158in}{1.6483in}{0in}{\Qcb{$\csc \protect\theta $}}{}{g0212.wmf}{\special%
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\vspace{0in}
If we add more points, we get the following six plots.
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\subsection{Trig Functions \ --- \ General Graphs}
\vspace{0in}
There are three ways we need to clarify our graphs of the trig functions.
\begin{itemize}
\item First, although calculators will compute trig functions using either
degrees or radians, mathematicians always define the trig functions using
radians. \ The reasons will become clear after you learn about the
derivatives of the trig functions. \ The trig functions were already graphed
using radians on the previous page.
\item Second, in the circle definition of the trig functions, there is no
reason that the angle is restricted to $1$ revolution. \ So the trig
functions are also defined for angles less than $0$rad or greater than $2\pi
${\small rad}$\,\approx 6.28${\small rad}. \ Further, since the coordinates
of the point on the circle repeat periodically each revolution, the graphs
of the trig functions repeat periodically every $2\pi ${\small rad}.
\item Third, mathematicians like to use the letter $x$ for the independent
variable and the letter $y$ for the dependent variable of a function. \ So
we often write $y=\sin x$, or $y=\cos x$, or $y=\tan x$, or etc. \ Here $x$
is the angle (in radians) and $y$ is the value of the particular trig
function.
\end{itemize}
\subsubsection{Caution: The $x$ and $y$ in a formula such as $y=\sin x$ have
absolutely nothing to do with the $x$ and $y$ in the circle definition of
the trig functions.}
\vspace{0in}
With these modifications, the graphs of the six trig functions become:
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Notice that $\sin $ and $\cos $ have \emph{period} $2\pi $, while $\tan $
and $\cot $\ (below) have \emph{period} $\pi .$ \ In the next four trig
functions note the vertical asymptotes. \ Those for $\tan \theta $ and $\sec
\theta $ are at odd multiples of $\dfrac{\pi }{2};$ \ those for $\cot \theta
$ and $\csc \theta $ are at multiples of $\pi .$
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\vspace{0in}
Notice that $\sin $, $\cos $, $\sec $ and $\csc $ all have \emph{period} $%
2\pi $, while $\tan $ and $\cot $ have \emph{period} $\pi $. \ Also recall
that $\tan $ and $\sec $ have vertical asymptotes at the odd multiples of $%
\dfrac{\pi }{2}$ while $\cot $ and $\csc $ have vertical asymptotes at the
multiples of $\pi $.
\subsection{Composition of Functions}
So far we have rules for differentiating sums, products, and quotients of
functions. There is another way to combining functions---compositions---for
which we also need a differentiation rule. \
\vspace{1pt}
\begin{definition}
Let $f$ and $g$ be two functions, and suppose for each $x$, the \emph{%
number\/} $g(x)$ is in the domain of $f$. Then we can evaluate the function $%
f$ at the \emph{number\/} $g(x)$. We write this as
\begin{equation*}
f(g(x))
\end{equation*}
and call it the \emph{composition\/} of the function $f$ with the function \
$g$. This can be visualized by the diagram below.
\end{definition}
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\begin{center}
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\begin{example}
Write the composition of $f$ \ with $g,$ where $g(x)=x^{2}$ and $%
f(x)=x^{3}+1 $.
\end{example}
\noindent \emph{Solution.}~~Since $f(x)=x^{3}+1$, \ it follows that
\begin{equation*}
f(g(x))=\lbrack g(x)\rbrack ^{3}+1
\end{equation*}
Thus,
\begin{equation*}
f(g(x))=f(x^{2})=(x^{2})^{3}+1=x^{6}+1
\end{equation*}
Note that $g(f(x))=g(x^{3}+1)=(x^{3}+1)^{2}=x^{6}+2x^{3}+1$. \ \ Thus, in
general $g(f(x))\neq f(g(x))$.$\hfill \blacksquare $
\begin{example}
Write the composition of $f$ \ with $g$, where $g(x)=\sqrt{x}$ and \ $%
f(x)=1/x^{2}$.
\end{example}
\noindent \emph{Solution.}~~Since $f(x)=1/x^{2}$, it follows that for $x\geq
0$
\begin{eqnarray*}
f(g(x)) &=&\frac{1}{[g(x)]^{2}} \\
f(g(x)) &=&f(\sqrt{x})=\frac{1}{(\sqrt{x})^{2}}=\frac{1}{x}
\end{eqnarray*}
\section{Formulation of the Chain Rule}
\vspace{1pt}Knowing what the composition of two functions is, we are now in
a position to formulate the rule for its derivative.
\begin{quotation}
\textbf{Chain Rule}~~~If $g(x)$ and $f(x)$ are two differentiable functions,
the derivative of the composition of $f$ \ with $g$ \ is given by
\begin{equation*}
\frac{d}{dx}\lbrack f(g(x))\rbrack =f^{\prime }(g(x))g^{\prime }(x)
\end{equation*}
for all $x$ \ in the domain of $f(g(x))$.
\end{quotation}
\vspace{1pt}
The function $f(x)$ \ is usually called the \textbf{outer function}, and the
function $g(x)$ \ is usually called the \textbf{inner function}.
\begin{center}
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\paragraph{\protect\vspace{1pt}Connection with the Extended Power Rule.}
The Extended Power Rule is a special case of the Chain Rule. Indeed, the
function
\begin{equation*}
\lbrack g(x)]^{n}
\end{equation*}%
could be replaced by
\begin{equation*}
f(g(x))
\end{equation*}%
where $f$ is the $n^{\text{th}}$ power function. So, for $f(x)=x^{n}$, $%
f\,^{\prime }(x)=nx^{n-1}$, and the Chain Rule gives us
\begin{equation*}
\frac{d}{dx}[f(g(x))]=n[g(x)]^{n-1}g^{\prime }(x)
\end{equation*}%
This result is exactly the Extended Power Rule.
\vspace{1pt}
\section{Derivative of the Trigonometric Functions}
\subsection{\protect\vspace{1pt}Two fundamental limits}
\vspace{1pt}
To compute the derivatives of the trigonometric functions, $\sin x$ in
particular, we apply the secant method. This means we must compute
\begin{equation*}
\dfrac{d}{dx}\sin t=\lim_{h\rightarrow 0}\dfrac{\sin \left( t+h\right) -\sin
t}{h}
\end{equation*}%
To compute this limit we use the double angle formula sin$\left( \alpha
+\beta \right) =\sin \alpha \cos \beta +\cos \alpha \sin \beta $ for the
sine function. \ Thus
\begin{eqnarray*}
\dfrac{d}{dx}\sin t &=&\lim_{h\rightarrow 0}\dfrac{\sin \left( t+h\right)
-\sin t}{h} \\
&=&\lim_{h\rightarrow 0}\dfrac{\sin t\cos h+\cos t\sin h-\sin t}{h} \\
&=&\lim_{h\rightarrow 0}\dfrac{\left( \cos h-1\right) \sin t+\cos t\sin h}{h}
\\
&=&\sin t\lim_{h\rightarrow 0}\dfrac{\left( \cos h-1\right) }{h}+\cos
t\lim_{h\rightarrow 0}\dfrac{\sin h}{h}
\end{eqnarray*}
This leaves the two undetermined limits to compute:
\begin{equation*}
\lim_{h\rightarrow 0}\dfrac{\left( \cos h-1\right) }{h}\text{ \ \ \ and \ \ }%
\lim_{h\rightarrow 0}\dfrac{\sin h}{h}
\end{equation*}%
We use a geometric diagram to help us see what to do.
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\begin{tabular}{l}
{\Large W}e have a relation between three areas: \\
Area$\,\triangle OAB<\,$Area$\sphericalangle OAB<\,$Area$\,\triangle OAC$ \\
{\Large C}omputing these areas gives the following \\
$\;\;\;\;\;\sin h