% ----------------------------------------------------------------------

\chapter{Trigonometry}
\label{trig}
\index{trigonometry}

\emph{Trigonometry} is that branch of mathematics which relates angles to
lengths.  This chapter introduces the functions of trigonometry and
derives several of these functions' properties.

% ----------------------------------------------------------------------

\section{Definitions}
\label{trig:226}

\index{right triangle}
\index{triangle!right}
\index{circle}
Consider the circle-inscribed right triangle of Fig.~\ref{trig:226:f1}.

\index{angle}
\index{radian}
\index{radius}
\index{circle!unit}
\index{unit circle}
\index{arc}
\index{unit}
\index{unity}
\index{one}
\index{$1$ (one)}
\index{length!curved}
\index{dimensionlessness}
\index{revolution}
\index{angle!square}
\index{angle!right}
\index{$2\pi$}
In considering the triangle and its circle in the figure, we shall find
some terminology useful.  The \emph{angle}~$\phi$ in the figure is
measured in \emph{radians,} where a radian is that angle which, when
centered in a unit circle, describes or intercepts an arc of unit length
as measured---not in a straight line---but along the curve of the
circle's perimeter.  A \emph{unit circle} is a circle whose radius is
$\rho=1$\@.  Similarly, a \emph{unit length} is a length of~1 (not one
centimeter or one mile, or anything like that, but just an
abstract~1)\@.  In the more general circle of radius~$\rho$ (where the
\emph{radius} is the distance from the circle's center to its perimeter)
as in the figure, an angle~$\phi$ describes or intercepts an arc of
length~$\rho\phi$ along the curve.

An angle in radians is a dimensionless number, so one need not write,
``$\phi=2\pi/4\ \mbox{radians}$''; but it suffices to write,
``$\phi=2\pi/4$.'' In mathematical theory, we express angles in radians.

The angle of full revolution is given the symbol~$2\pi$---which thus is
the unit circle's circumference.%
\footnote{
  Section~\ref{taylor:355} computes the numerical value
  of~$2\pi$.
}
A quarter revolution,
$2\pi/4$, is then the \emph{right angle} or \emph{square angle.}

\index{sine}
\index{cosine}
\index{tangent}
\index{slope}
\index{diagonal}
\index{quadrant}
\index{trigonometric function}
\index{trigonometric function!inverse}
\index{arcsine}
\index{arccosine}
\index{arctangent}
\index{rise}\index{run}
\index{coordinates!rectangular}
\index{rectangular coordinates}
\index{north}\index{south}\index{east}\index{west}\index{up}\index{down}
\index{origin}
The trigonometric functions $\sin\phi$ and $\cos\phi$, the \emph{sine}
and \emph{cosine} of~$\phi$, relate the angle~$\phi$ to the lengths in
the figure.
\begin{figure}
  \caption[The sine and the cosine.]{The sine and the cosine (shown on a
  circle-inscribed right triangle, with the circle centered at the
  triangle's point).}
  \label{trig:226:f1}
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\end{figure}
The tangent function is then defined to be
\bq{trig:226:10}
  \tan\phi \equiv \frac{\sin\phi}{\cos\phi},
\eq
which is the ``rise'' per unit ``run,'' or \emph{slope,} of the
triangle's diagonal.%
\footnote{
  Often seen in print is the additional notation
  $\sec\phi \equiv 1/\cos\phi$,
  $\csc\phi \equiv 1/\sin\phi$ and
  $\cot\phi \equiv 1/\tan\phi$;
  respectively the ``secant,'' ``cosecant'' and ``cotangent.''
  This book does not use that notation.
}
Inverses of the three trigonometric functions can also be defined:
\[
  \begin{split}
    \arcsin\left(\sin\phi\right) &= \phi; \\
    \arccos\left(\cos\phi\right) &= \phi; \\
    \arctan\left(\tan\phi\right) &= \phi.
  \end{split}
\]
When the last of these is written,
\[
  \arctan\left(\frac{y}{x}\right),
\]
it is normally implied that~$x$ and~$y$ are to be interpreted as
rectangular coordinates%
\footnote{
  \emph{Rectangular coordinates} are pairs of numbers $(x,y)$ which
  uniquely specify points in a plane.  Conventionally, the~$x$
  coordinate indicates distance eastward as it were; the~$y$
  coordinate, northward.  For instance, the coordinates $(3,-4)$ mean
  the point three units eastward and four units southward (that is,~$-4$
  units northward) of the \emph{origin} $(0,0)$.  When needed, a third
  rectangular coordinate can moreover be added---$(x,y,z)$---the~$z$
  indicating distance upward, above the plane of the~$x$ and~$y$.
  (Consistency with the book's general style should have suggested the
  spelling ``co\"ordinates'' except that no mathematics book this writer
  knows spells the word that way, nor even as ``co-ordinates'' which is
  how the word after all is pronounced.)%
}
and that the $\arctan$ function is to return~$\phi$ in the
correct quadrant $-\pi<\phi\le\pi$ (for example, $\arctan[1/(-1)] =
[+3/8][2\pi]$, whereas $\arctan[(-1)/1] = [-1/8][2\pi]$).  This
is similarly the usual interpretation when an equation like
\[
  \tan\phi = \frac{y}{x}
\]
is written.

\index{Pythagorean theorem!and the sine and cosine functions}
By the Pythagorean theorem (\S~\ref{alggeo:223} and the introduction to
chapter~\ref{intro}), it is seen
that%
\footnote{
  The notation $\cos^2\phi$ means $\left(\cos\phi\right)^2$.
}
\bq{trig:226:25}
  \cos^2\phi + \sin^2\phi = 1
\eq
for any~$\phi.$

\index{sinusoid}
Fig.~\ref{trig:226:sinusoid} plots the sine function.  The shape in the
plot is called a \emph{sinusoid.}%
\footnote{\label{trig:226:fn1}%
  This section completes the forward reference of \S~\ref{alggeo:225}.
  See chapter~\ref{alggeo}'s footnote~\ref{alggeo:225:fn1}.
}%
\begin{figure}
  \caption{The sine function.}
  \label{trig:226:sinusoid}
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\end{figure}

% ----------------------------------------------------------------------

\section{Simple properties}
\label{trig:228}

Inspecting Fig.~\ref{trig:226:f1} and observing~(\ref{trig:226:10})
and~(\ref{trig:226:25}), one discovers the trigonometric properties of
Table~\ref{trig:228:table}.  (Observe in the table as usually elsewhere
in the book that $n\in\mathbb Z$, \S~\ref{alggeo:227}.)
\begin{table}
  \caption{Simple properties of the trigonometric functions.}
  \index{trigonometry!properties of}
  \label{trig:228:table}
  \bc
    \[
      \br{rclcrcl}
        \sin(-\phi) &=& -\sin\phi &\ &
        \cos(-\phi) &=& +\cos\phi \\
        \sin(2\pi/4 - \phi) &=& +\cos\phi &\ &
        \cos(2\pi/4 - \phi) &=& +\sin\phi \\
        \sin(2\pi/2 - \phi) &=& + \sin\phi &\ &
        \cos(2\pi/2 - \phi) &=& -\cos\phi \\
        \sin(\phi\pm2\pi/4) &=& \pm\cos\phi &\ &
        \cos(\phi\pm2\pi/4) &=& \mp\sin\phi \\
        \sin(\phi\pm2\pi/2) &=&  -\sin\phi &\ &
        \cos(\phi\pm2\pi/2) &=&  -\cos\phi \\
        \sin(\phi+n2\pi) &=&  \sin\phi &\ &
        \cos(\phi+n2\pi) &=&  \cos\phi
      \er
    \]
    \[
      \br{rcl}
        \tan(-\phi) &=& -\tan\phi \\
        \tan(2\pi/4 - \phi) &=& +1/\tan\phi \\
        \tan(2\pi/2 - \phi) &=& -\tan\phi \\
        \tan(\phi\pm2\pi/4) &=& -1/\tan\phi \\
        \tan(\phi\pm2\pi/2) &=&  +\tan\phi \\
        \tan(\phi+n2\pi) &=&  \tan\phi
      \er
    \]
    \bqb
      \frac{\sin\phi}{\cos\phi} &=& \tan\phi \\
      \cos^2\phi + \sin^2\phi &=& 1 \\
      1 + \tan^2 \phi &=& \frac{1}{\cos^2\phi} \\
      1 + \frac{1}{\tan^2 \phi} &=& \frac{1}{\sin^2\phi}
    \eqb
  \ec
\end{table}

% ----------------------------------------------------------------------

\section{Scalars, vectors, and vector notation}
\label{trig:230}
\index{scalar}
\index{vector}
\index{vector!notation for}
\index{vector!geometrical}
\index{geometrical vector}

\index{amplitude}
\index{direction}
\index{vector!unit basis}
\index{unit basis vector}
In applied mathematics, a \emph{geometrical vector}---whose name the
narrative will hereinafter usually abbreviate as \emph{vector}---is an
amplitude of some kind coupled with a direction.%
\footnote{
  The same word \emph{vector} also is used to indicate an ordered set
  of~$N$ scalars (\S~\ref{taylor:370}) or an $N\times 1$ matrix
  (chapter~\ref{matrix}), but those are not the uses of the word meant here.
  See also the introduction to chapter~\ref{vector}.%
}\mbox{${}^{,}$}\footnote{%
  The word ``amplitude'' is sometimes used interchangeably with
  ``magnitude,'' even by this writer.  However, used more precisely, an
  \emph{amplitude} unlike a magnitude can be negative or positive, as
  for example the~$A$ in $f(t)\equiv A\cos\omega t$.  Indeed,
  this~$A$ makes a typical example of the usage, for the word
  ``amplitude'' is especially applied to measure the extent to which a
  wave deviates from its neutral position.

  \index{amplitudinity}
  \index{quantity!linear}
  \index{quantity!squared}
  \index{linear quantity}
  \index{squared quantity}
  \index{energy}
  \index{power}
  \index{displacement}
  \index{force}
  \index{electric current}
  \index{current, electric}
  \index{tension, electric}
  \index{electric tension or potential}
  \index{connotation}
  In practice, the last usage typically connotes that the quantity whose
  amplitude the amplitude is is a \emph{linear} quantity like
  displacement, force, electric current or electric tension (voltage).
  A \emph{squared} quantity---that is, a quantity developed as the
  product of two other, linear quantities---is not usually said to
  possess an amplitude.  Especially, energy and power are not usually
  said to possess amplitudes, whereas displacement and force (whose
  product is an energy) are each amplitudinous; and, likewise, current
  and tension (whose product is a power) are each amplitudinous.%
}
For example, ``55 miles per hour
northwestward'' is a vector, as is the entity~$\ve u$ depicted in
Fig.~\ref{trig:231}.
\begin{figure}
  \caption[A two-dimensional vector $\ve u = \vu x x + \vu y y$.]{A
  two-dimensional vector $\ve u = \vu x x + \vu y y$, shown with its
  rectangular components.}
  \label{trig:231}
  \index{vector!two-dimensional}
  \bc
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          \rput(-0.30,0.80){$\vu y y$}
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  \ec
\end{figure}
The entity~$\ve v$ depicted in Fig.~\ref{trig:231f3} is also a vector,
in this case a three-dimensional one.
\begin{figure}
  \caption{A three-dimensional vector $\ve v = \vu x x + \vu y y + \vu
  z z$.}
  \label{trig:231f3}
  \index{vector!three-dimensional}
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\end{figure}

\index{unity}
\index{unit}
\index{vector!unit}
\index{unit vector}
\index{amplitude}
Many readers will already find the basic vector concept familiar, but
for those who do not, a brief review:  Vectors such as the
\bqb
  \ve u &=& \vu x x + \vu y y, \\
  \ve v &=& \vu x x + \vu y y + \vu z z,
\eqb
of the figures are composed of multiples of the \emph{unit basis
vectors}~$\vu x$, $\vu y$ and~$\vu z$, which are themselves
vectors of unit length pointing in the cardinal directions their
respective symbols suggest.%
\footnote{
  Printing by hand, one customarily writes a general vector
  like~$\ve u$ as ``$\,\vec u$\,'' or just ``\,$\overline u$\,'', and a
  unit vector like~$\vu x$ as ``$\,\hat x$\,''.
}
Any vector~$\ve a$ can be factored into an \emph{amplitude}~$a$ and a
\emph{unit vector}~$\vu a$, as
\[
  \ve a = \vu a a = \vu a \left| \ve a \right|,
\]
where the~$\vu a$ represents direction only and has unit magnitude by
definition, and where the~$a$ or~$\left| \ve a \right|$ represents
amplitude only and carries the physical units if any.%
\footnote{%
  \index{physical unit}%
  \index{financial unit}%
  \index{measure, unit of}%
  \index{unit!of measure}%
  \index{unit!physical}%
  \index{unit!financial}%
  \index{height}%
  \index{yen, Japanese}%
  \index{Japanese yen}%
  The word ``unit'' here is unfortunately overloaded.  As an
  adjective in mathematics, or in its nounal form ``unity,'' it refers to
  the number one~(1)---not one mile per hour, one kilogram, one Japanese
  yen or anything like that; just an abstract~1.  The word ``unit''
  itself as a noun however usually signifies a physical or financial
  reference quantity of measure, like a mile per hour, a kilogram or
  even a Japanese yen.  There is no inherent logical unity to~1
  mile per hour (otherwise known as~0.447 meters per second, among other
  names).  By contrast, a ``unitless~1''---a~1 with no physical unit
  attached, also known as a ``dimensionless 1''---does represent
  a logical unity.

  Consider the ratio $r=h_1/h_o$ of your height~$h_1$ to my
  height~$h_o$.  Maybe you are taller than I am and so $r=1.05$ (not
  1.05~cm or 1.05~feet, just~1.05).  Now consider the ratio $h_1/h_1$
  of your height to your own height.  That ratio is of course unity,
  exactly~1.

  There is nothing ephemeral in the concept of mathematical unity, nor
  in the concept of unitless quantities in general.  The concept is
  quite straightforward and is entirely practical.  That $r > 1$ means
  neither more nor less than that you are taller than I am.  In
  applications, one often puts physical quantities in ratio precisely to
  strip the physical units from them, comparing the ratio to unity
  without regard to physical units.

  Incidentally, a more general term to comprise physical units,
  financial units and other such quantities is \emph{units of measure.}%
}
For example, $a=55\mbox{\ miles per hour}$, $\vu
a=\mbox{northwestward}$.  The unit vector~$\vu a$ itself can be
expressed in terms of the unit basis vectors; for example, if~$\vu x$
points east and~$\vu y$ points north, then $\vu a = -\vu x (1/\sqrt 2) +
\vu y (1/\sqrt 2)$ means ``northwestward,'' where per the Pythagorean
theorem $(-1/\sqrt 2)^2 + (1/\sqrt 2)^2 = 1^2.$

\index{scalar!complex} A single number which is not a vector or a matrix
(chapter~\ref{matrix}) is called a \emph{scalar.}  In the example,
$a=55\mbox{\ miles per hour}$ is a scalar.  Though the scalar~$a$ in the
example happens to be real, scalars can be complex, too---which might
surprise one, since scalars by definition lack direction and the Argand
phase~$\phi$ of Fig.~\ref{alggeo:225:fig} so strongly resembles a
direction.  However, phase is not an actual direction in the vector
sense (the real number line in the Argand plane cannot be said to run
west-to-east, or anything like that).  The~$x$, $y$ and~$z$ of
Fig.~\ref{trig:231f3} are each (possibly complex) scalars; $\ve v =
\vu x x + \vu y y + \vu z z$ is a vector.  If~$x$, $y$ and~$z$ are
complex, then%
\footnote{
  Some books print~$\left|\ve v\right|$ as~$\|\ve v\|$ or even~$\|\ve
  v\|_2$ to emphasize that it represents the real, scalar magnitude of a
  complex vector.  The reason the last notation subscripts a numeral~2
  is obscure, having to do with the professional mathematician's
  generalized definition of a thing he calls the ``norm.''  This book
  just renders it~$\left|\ve v\right|$.
}
\bqa
  \left|\ve v\right|^2 &=& \left|x\right|^2 + \left|y\right|^2 + \left|z\right|^2
    = x^{*}x + y^{*}y + z^{*}z  \xn\\
    &=&
        \left[\Re(x)\right]^2
      + \left[\Im(x)\right]^2
      + \left[\Re(y)\right]^2
      + \left[\Im(y)\right]^2
  \xn\\&&\mbox{\ \ }%
      + \left[\Re(z)\right]^2
      + \left[\Im(z)\right]^2.
  \label{trig:230:20}
\eqa

\index{vector!point}
\index{point!in vector notation}
\index{origin}
A point is often identified by the vector expressing its distance
from and direction relative to the origin $(0,0)$ of a coordinate
system.  That is, the point $(x,y)$ can be, and frequently is,
identified with the vector $\vu x x+\vu y y$.  [Likewise,
the origin $(0,0)$ itself can be identified with the null vector $\vu x
0+\vu y 0 = 0$.]
However, not every vector need be associated with an origin, for vectors
in the more general sense represent \emph{relative} distances and
directions, whether the thing to which they are relative happens to be
an origin or something else.

\index{mile per hour}
\index{kilogram}
\index{distance}
\index{speed}
Notice incidentally our mathematical use of the word ``distance.''  A
mathematical distance may or may not represent a distance in the
literal, physical sense, but it will at least represent a quantity one
might proportionably diagram as though it were a distance.  If such a
quantity also has direction, then it can be a vector.  This is why ``55
miles per hour northwestward'' is a vector despite that the mile per
hour is a unit of speed rather than of distance.  This is also why ``55
kilograms,'' which lacks direction, is not a vector.

\index{axes}
\index{right-hand rule}
\index{orientation}
\index{screw}
Observe the relative orientation of the axes of
Fig.~\ref{trig:231f3}.  The axes are oriented such that if you point
your flat right hand in the~$x$ direction, bend your fingers in the~$y$
direction, and extend your thumb, then the thumb points in the~$z$
direction.  This is orientation by the \emph{right-hand rule.}  A
left-handed orientation is equally possible, of course, but since neither
orientation owns a natural advantage over the other, mathematicians
arbitrarily but conventionally accept the right-handed one as standard.%
\footnote{
  The writer does not know the etymology for certain, but verbal lore
  in American engineering has it that the name ``right-handed'' comes
  from experience with a standard right-handed wood screw or machine
  screw.  If you hold the screwdriver in your right hand and turn the
  screw in the natural manner clockwise, turning the screw slot from the~$x$
  orientation toward the~$y$, then the screw advances away from you in
  the~$z$ direction into the bore.  If somehow you came across a
  left-handed screw, you'd probably find it easier to drive that screw
  with the screwdriver in your left hand.
}

Sections~\ref{trig:240} and~\ref{trig:277} and chapters~\ref{vector}
and~\ref{vcalc} will speak further of the geometrical vector.

% ----------------------------------------------------------------------

\section{Rotation}
\label{trig:240}
\index{rotation}
\index{vector!rotation of}
\index{axes!rotation of}
\index{axes!changing}

\index{symmetry!appeal to}
\index{dimensionality}
\index{space}
\index{plane}
\index{line}
\index{point}
\index{prime mark ($'$)}
\index{$'$}
A fundamental problem in trigonometry arises when a vector
\bq{trig:240:08}
  \ve u = \vu x x + \vu y y
\eq
must be expressed in terms of alternate unit vectors~$\vu x'$ and~$\vu
y'$, where~$\vu x'$ and~$\vu y'$ stand at right angles to one
another and lie in the plane%
\footnote{
  A \emph{plane,} as the reader on this tier undoubtedly knows, is a
  flat (but not necessarily level) surface, infinite in extent unless
  otherwise specified.  Space is three-dimensional.  A plane is
  two-dimensional.  A line is one-dimensional.  A point is
  zero-dimensional.  The plane belongs to this geometrical hierarchy.
}
of~$\vu x$ and~$\vu y$, but are rotated from the latter pair by an
angle~$\phi$ as depicted in Fig.~\ref{trig:240f}.%
\footnote{
  The ``$\,'\,$'' mark is pronounced ``prime'' or ``primed'' (for no
  especially good reason of which the author is aware, but anyway,
  that's how it's pronounced).  Mathematical writing employs the mark
  for a variety of purposes.  Here, the mark merely distinguishes the
  new unit vector~$\vu x'$ from the old~$\vu x$.
}
\begin{figure}
  \caption{Vector basis rotation.}
  \label{trig:240f}
  \bc
    \begin{pspicture}(-5,-3)(5,3)
      {
        \small
        \setlength\tla{1.6cm}
        \nc\xa{2.6}
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        \nc\xe{0.57358\tla}
        \nc\xg{35}
        \nc\xgg{17}
        \nc\xr{0.9}
        \nc\xrr{1.1}
        \nc\xh{
          \psarc{->}(0,0){\xr}{0}{\xg}
          \rput{\xgg}(0,0){
            \rput{*0}(\xrr,0){$\phi$}
          }
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        {
          \psset{linewidth=0.5pt}
          \psline(-\xa,0)(\xa,0)
          \psline(0,-\xa)(0,\xa)
          \rput(\xb,0){$x$}
          \rput(0,\xb){$y$}
          \rput{ 0}(0,0){\xh}
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        {
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          \psline{c->}(0,0)(-\xe,\xd)
          \psline{c->}(0,0)(2.2,2.6)
        }
        \rput(1.55,-0.25){$\vu x$}
        \rput(0.25,1.55){$\vu y$}
        \rput(1.55,1.10){$\vu x'$}
        \rput(-1.05,1.50){$\vu y'$}
        \rput(2.10,2.10){$\ve u$}
      }
    \end{pspicture}
  \ec
\end{figure}
In terms of the trigonometric functions of \S~\ref{trig:226}, evidently,
\bq{trig:240:10}
  \begin{split}
    \vu x' &= +\vu x \cos\phi + \vu y \sin\phi, \\
    \vu y' &= -\vu x \sin\phi + \vu y \cos\phi;
  \end{split}
\eq
and by appeal to symmetry it stands to reason that
\bq{trig:240:12}
  \begin{split}
    \vu x &= +\vu x' \cos\phi - \vu y' \sin\phi, \\
    \vu y &= +\vu x' \sin\phi + \vu y' \cos\phi.
  \end{split}
\eq
Substituting~(\ref{trig:240:12})
into~(\ref{trig:240:08}) yields that
\bq{trig:240:rot}
  \ve u = \vu x' (x\cos\phi+y\sin\phi) + \vu y'
  (-x\sin\phi+y\cos\phi),
\eq
which was to be derived.

Equation~(\ref{trig:240:rot}) finds general application where rotations
in rectangular coordinates are involved.  If the question is asked,
``what happens if I rotate not the unit basis vectors but rather the
vector~$\ve u$ instead?''\ the answer is that it amounts to the same
thing, except that the sense of the rotation is reversed:
\bq{trig:240:rot2}
  \ve u' = \vu x (x\cos\phi-y\sin\phi) + \vu y (x\sin\phi+y\cos\phi).
\eq
Whether it is the basis or the vector which rotates thus depends on your
point of view.%
\footnote{
  This is only true, of course, with respect to the vectors themselves.
  When one actually rotates a physical body, the body experiences forces
  during rotation which might or might not change the body internally in
  some relevant way.
}

Much later in the book, \S~\ref{vector:210} will extend rotation in two
dimensions to reorientation in three dimensions.

% ----------------------------------------------------------------------

\section[Trigonometric sums and differences]{Trigonometric functions of
sums and differences of angles}
\label{trig:250}
\index{trigonometric function!of a sum or difference of angles}

\index{rotation!angle of}
\index{angle!of rotation}
\index{equation!solving a set of simultaneously}
With the results of \S~\ref{trig:240} in hand, we now stand in a
position to consider trigonometric functions of sums and differences of
angles.  Let
\[
  \begin{split}
    \vu a &\equiv \vu x \cos \alpha + \vu y \sin \alpha, \\
    \vu b &\equiv \vu x \cos  \beta + \vu y \sin \beta,
  \end{split}
\]
be vectors of unit length in the~$xy$ plane, respectively at angles~$\alpha$
and~$\beta$ from the~$x$ axis.  If we wanted~$\vu b$ to
co\"incide with~$\vu a$, we would have to rotate it by $\phi=\alpha-\beta$.
According to~(\ref{trig:240:rot2}) and the definition of~$\vu b$, if we
did this we would obtain that
\[
  \br{l}
    \vu b' =
    \vu x [\cos\beta\cos(\alpha-\beta)-\sin\beta\sin(\alpha-\beta)]
    \\
    \sh{2.0} + \vu y [\cos\beta\sin(\alpha-\beta)+\sin\beta\cos(\alpha-\beta)].
  \er
\]
Since we have deliberately chosen the angle of rotation such that $\vu
b' = \vu a$, we can separately equate the~$\vu x$ and~$\vu y$ terms
in the expressions for~$\vu a$ and~$\vu b'$ to obtain the pair of
equations
\[
  \begin{split}
    \cos \alpha &=
    \cos\beta\cos(\alpha-\beta)-\sin\beta\sin(\alpha-\beta), \\
    \sin \alpha &=
    \cos\beta\sin(\alpha-\beta)+\sin\beta\cos(\alpha-\beta).
  \end{split}
\]
Solving the last pair simultaneously%
\footnote{
  \label{trig:footnote10}%
  The easy way to do this is
  \bi
    \item to subtract $\sin\beta$ times the first equation from
      $\cos\beta$ times the second and then to solve the result for
      $\sin(\alpha-\beta)$;
    \item to add $\cos\beta$ times the first equation to
      $\sin\beta$ times the second and then to solve the result for
      $\cos(\alpha-\beta)$.
  \ei
  This shortcut technique for solving a pair of equations simultaneously
  for a pair of variables is well worth mastering.  In this book alone,
  it proves useful many times.
}
for $\sin(\alpha-\beta)$ and $\cos(\alpha-\beta)$ and observing that
$\sin^2(\cdot)+\cos^2(\cdot)=1$ yields that
\bq{trig:250:20}
  \begin{split}
    \sin(\alpha-\beta) &= \sin\alpha\cos\beta - \cos\alpha\sin\beta, \\
    \cos(\alpha-\beta) &= \cos\alpha\cos\beta + \sin\alpha\sin\beta.
  \end{split}
\eq
With the change of variable $\beta\la -\beta$ and the observations
from Table~\ref{trig:228:table} that $\sin(-\phi)=-\sin\phi$ and
$\cos(-\phi)=+\cos(\phi)$, eqns.~(\ref{trig:250:20})
become
\bq{trig:250:22}
  \begin{split}
    \sin(\alpha+\beta) &= \sin\alpha\cos\beta + \cos\alpha\sin\beta, \\
    \cos(\alpha+\beta) &= \cos\alpha\cos\beta - \sin\alpha\sin\beta.
  \end{split}
\eq
Equations~(\ref{trig:250:20}) and~(\ref{trig:250:22}) are the basic
formulas for trigonometric functions of sums and differences of angles.

\subsection{Variations on the sums and differences}
\label{trig:250.11}

{
  \nc\al{\left(\frac{\gamma+\delta}{2}\right)}
  \nc\bt{\left(\frac{\gamma-\delta}{2}\right)}
  Several useful variations on~(\ref{trig:250:20})
  and~(\ref{trig:250:22})
  are achieved by combining the equations in various
  straightforward ways.%
  \footnote{
    Refer to footnote~\ref{trig:footnote10} above for the technique.
  }
  These include that
  \bq{trig:250:30}
    \begin{split}
      \sin\alpha\sin\beta &= \frac{ \cos(\alpha-\beta) - \cos(\alpha+\beta) }{2},\\
      \sin\alpha\cos\beta &= \frac{ \sin(\alpha-\beta) + \sin(\alpha+\beta) }{2},\\
      \cos\alpha\cos\beta &= \frac{ \cos(\alpha-\beta) + \cos(\alpha+\beta) }{2}.
    \end{split}
  \eq
  With the change of variables $\delta \la \alpha-\beta$ and
  $\gamma \la \alpha+\beta$, (\ref{trig:250:20})
  and~(\ref{trig:250:22}) become
  \[
    \begin{split}
      \sin\delta &= \sin\al\cos\bt - \cos\al\sin\bt, \\
      \cos\delta &= \cos\al\cos\bt + \sin\al\sin\bt, \\
      \sin\gamma &= \sin\al\cos\bt + \cos\al\sin\bt, \\
      \cos\gamma &= \cos\al\cos\bt - \sin\al\sin\bt.
    \end{split}
  \]
  Combining these in various ways, we have that
  \bq{trig:250:50}
    \begin{split}
      \sin\gamma + \sin\delta &= 2\sin\al\cos\bt, \\
      \sin\gamma - \sin\delta &= 2\cos\al\sin\bt, \\
      \cos\delta + \cos\gamma &= 2\cos\al\cos\bt, \\
      \cos\delta - \cos\gamma &= 2\sin\al\sin\bt.
    \end{split}
  \eq
}

\subsection{Trigonometric functions of double and half angles}
\label{trig:250.12}
\index{trigonometric function!of a double or half angle}
\index{double angle}
\index{half angle}
\index{angle!double}
\index{angle!half}

If $\alpha=\beta$, then eqns.~(\ref{trig:250:22})
become the \emph{double-angle formulas}
\bq{trig:250:62}
  \begin{split}
    \sin2\alpha &= 2\sin\alpha\cos\alpha, \\
    \cos2\alpha &= 2\cos^2\alpha - 1 = \cos^2\alpha - \sin^2\alpha = 1 - 2\sin^2\alpha.
  \end{split}
\eq
Solving~(\ref{trig:250:62}) for $\sin^2\alpha$ and $\cos^2\alpha$
yields the \emph{half-angle formulas}
\bq{trig:250:64}
  \begin{split}
    \sin^2\alpha &= \frac{1-\cos 2\alpha}{2},\\
    \cos^2\alpha &= \frac{1+\cos 2\alpha}{2}.
  \end{split}
\eq

% ----------------------------------------------------------------------

\section[Trigonometrics of the hour angles]{Trigonometric functions
of the hour angles}
\label{trig:260}
\index{angle}
\index{angle!hour}
\index{hour angle}
\index{trigonometric function!of an hour angle}
\index{hour}
\index{radian}
\index{circle}

\index{Greenwich}
\index{Observatory, Old Royal}
\index{Royal Observatory, Old}
\index{Old Royal Observatory}
\index{clock}
\index{day}
In general, one uses the Taylor series of chapter~\ref{taylor} to calculate
trigonometric functions of specific angles.  We're not ready for that
yet.  However, for angles which
happen to be integral multiples of an \emph{hour}---there being
twenty-four or 0x18 hours in a circle, just as there are twenty-four or
0x18 hours in a day%
\footnote{\label{trig:260:fn1}%
  Hence an hour is~$15^\circ$, but you weren't going to write your
  angles in such inelegant conventional notation as ``$15^\circ$,'' were
  you?  Well, if you were, you're in good company.

  The author is fully aware of the barrier the unfamiliar notation poses
  for most first-time readers of the book.  The barrier is erected
  neither lightly nor disrespectfully.  Consider:
  \bi
    \item There are 0x18 hours in a circle.
    \item There are 360 degrees in a circle.
  \ei

  % diagn: The above blank line and the below \noindent... should not
  % be necessary, should they?  They are given to work around an
  % apparent page-break-related glitch of LaTeX.  When text is modified
  % earlier in the chapter so that the page break occurs at a different
  % spot, the need for the workaround is expected to disappear.
  \noindent\mbox{}\\
  Both sentences say the same thing, don't they?  But even though the
  ``0x'' hex prefix is a bit clumsy, the first sentence nevertheless says
  the thing rather better.  The reader is urged to invest the attention
  and effort to master the notation.

  There is a psychological trap regarding the hour.  The familiar,
  standard clock face shows only twelve hours not twenty-four, so the
  angle between eleven o'clock and twelve \emph{on the clock face} is
  not an hour of arc!  That angle is two hours of arc.  This is so because
  the clock face's geometry is artificial.  If you have ever been to the
  Old Royal Observatory at Greenwich, England, you may have seen the big
  clock face there with all twenty-four hours on it.  It'd be a bit hard
  to read the time from such a crowded clock face were it not so big,
  but anyway, the angle between hours on the Greenwich clock is indeed
  an honest hour of arc.~\cite{KRHB}

  The hex and hour notations are recommended mostly only for theoretical
  math work.  It is not claimed that they offered much benefit in most
  technical work of the less theoretical kinds.  If you wrote an
  engineering memorandum describing a survey angle as 0x1.80 hours instead of
  22.5 degrees, for example, you'd probably not like the reception the
  memo got.  Nonetheless, the improved notation fits a book of this kind
  so well that the author hazards it.  It is hoped that, after trying
  the notation a while, the reader will approve the choice.
}---%
for such angles simpler expressions exist.  Figure~\ref{trig260:fig1}
shows the angles.
\begin{figure}
  \caption{The 0x18 hours in a circle.}
  \label{trig260:fig1}
  \bc
    \begin{pspicture}(-3,-5)(3,5)
      {
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        \rput{  0}(0,0)\xdl
        \rput{ 15}(0,0)\xbl
        \rput{ 30}(0,0)\xbl
        \rput{ 45}(0,0)\xbl
        \rput{ 60}(0,0)\xbl
        \rput{ 75}(0,0)\xbl
        \rput{ 90}(0,0)\xdl
        \rput{105}(0,0)\xbl
        \rput{120}(0,0)\xbl
        \rput{135}(0,0)\xbl
        \rput{150}(0,0)\xbl
        \rput{165}(0,0)\xbl
      }
    \end{pspicture}
  \ec
\end{figure}
Since such angles arise very frequently in practice,
it seems worth our while to study them specially.

\index{triangle!equilateral}
\index{square}
Table~\ref{trig:hourtable} tabulates the trigonometric functions of these
\emph{hour angles.}
\begin{table}
  \caption{Trigonometric functions of the hour angles.}
  \label{trig:hourtable}
  \bc
    \nc\st{\rule[-3.00ex]{0.0em}{7.00ex}}
    \[
      \br{ccccc}
        \multicolumn{2}{c}{\mr{ANGLE}\ \phi}&&\\
        \mr{[radians]} & \mr{[hours]} & \sin\phi & \tan\phi & \cos\phi \\
        \st\ds 0 & 0 & 0 & 0 & 1 \\
        \st\ds\frac{2\pi}{\mr{0x18}} & 1 &
          \ds\frac{\sqrt 3 - 1}{2\sqrt 2} & \ds\frac{\sqrt 3 - 1}{\sqrt 3 + 1} & \ds\frac{\sqrt 3 + 1}{2\sqrt 2} \\
        \st\ds\frac{2\pi}{\mr{0x10}} & \ds\frac 3 2 &
          \ds\frac{\sqrt{2-\sqrt{2}}}{2} & \ds\sqrt{\frac{2-\sqrt{2}}{2+\sqrt{2}}}& \ds\frac{\sqrt{2+\sqrt{2}}}{2} \\
        \st\ds\frac{2\pi}{\mr{0xC}} & 2 &
          \ds\frac{1}{2} & \ds\frac{1}{\sqrt 3} & \ds\frac{\sqrt 3}{2} \\
        \st\ds\frac{2\pi}{\mr{8}} & 3 &
          \ds\frac{1}{\sqrt 2} & 1 & \ds\frac{1}{\sqrt 2} \\
        \st\ds\frac{2\pi}{\mr{6}} & 4 &
          \ds\frac{\sqrt 3}{2} & \ds\sqrt 3 & \ds\frac{1}{2} \\
        \st\ds\frac{(3)(2\pi)}{\mr{0x10}} & \ds\frac 9 2 &
          \ds\frac{\sqrt{2+\sqrt{2}}}{2} & \ds\sqrt{\frac{2+\sqrt{2}}{2-\sqrt{2}}}& \ds\frac{\sqrt{2-\sqrt{2}}}{2} \\
        \st\ds\frac{(5)(2\pi)}{\mr{0x18}} & 5 &
          \ds\frac{\sqrt 3 + 1}{2\sqrt 2} & \ds\frac{\sqrt 3 + 1}{\sqrt 3 - 1} & \ds\frac{\sqrt 3 - 1}{2\sqrt 2} \\
        \st\ds\frac{2\pi}{4} & 6 & 1 & \infty & 0
      \er
    \]
  \ec
\end{table}
To see how the values in the table are calculated, look at the square
and the equilateral triangle%
\footnote{
  An \emph{equilateral} triangle is, as the name and the figure suggest,
  a triangle whose three sides all have the same length.
}
of Fig.~\ref{trig:260:f1}.
\begin{figure}
 \caption[Calculating the hour trigonometrics.]{A
 square and an equilateral triangle for calculating trigonometric
 functions of the hour angles.}
  \label{trig:260:f1}
  \bc
    \begin{pspicture}(-5,-2)(5,2)
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        \rput( 0.85,-0.00){$1$}
        \rput(-0.60,-1.50){$1/2$}
        \rput( 0.60,-1.50){$1/2$}
        \rput( 0.30,-0.65){$\frac{\sqrt 3}{2}$}
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  \ec
\end{figure}
Each of the square's four angles naturally measures six hours; and, since
a triangle's angles always total twelve hours (\S~\ref{alggeo:323.30}),
by symmetry each of the angles of the equilateral triangle in the figure
measures four.  Also by symmetry, the perpendicular splits the
triangle's top angle into equal halves of two hours each and its
bottom leg into equal segments of length~$1/2$ each; and the diagonal
splits the square's corner into equal halves of three hours each.  The
Pythagorean theorem (\S~\ref{alggeo:223} and the introduction to
chapter~\ref{intro}) then supplies the various
other lengths in the figure, after which we observe from
Fig.~\ref{trig:226:f1} that
\bi
  \item the sine of a non-right angle in a right triangle is the
    opposite leg's length divided by the diagonal's,
  \item the tangent is the opposite leg's length divided by the
    adjacent leg's, and
  \item the cosine is the adjacent leg's length divided by the
    diagonal's.
\ei
With this observation and the lengths in the figure, one can calculate
the sine, tangent and cosine of angles of two, three and four hours.

The values for one and five hours are found by
applying~(\ref{trig:250:20}) and~(\ref{trig:250:22}) against the
values for two and three hours just calculated.  The values for zero and
six hours are, of course, seen by inspection.%
\footnote{
  The creative reader may notice that he can extend the table to any
  angle by repeated application of the various sum, difference and
  half-angle formulas from the preceding sections to the values already
  in the table.  However, the Taylor series (\S~\ref{taylor:315})
  offers a cleaner, quicker way to calculate trigonometrics of
  non-hour angles.
}

Though quarters of a right angle are half-integral multiples of an hour,
quarters of a right angle arise often enough in engineering practice that the
table mentions them, too.  The values for the quarters are found by
applying~(\ref{trig:250:64}) against the values for three and nine hours.

% ----------------------------------------------------------------------

\section{The laws of sines and cosines}
\label{trig:270}

\index{law of sines}
\index{sine!law of}
\index{variable!utility}
\index{utility variable}
Refer to the triangle of Fig.~\ref{trig:270:f1}.
\begin{figure}
  \caption{The laws of sines and cosines.}
  \label{trig:270:f1}
  \index{triangle}
  \bc
    \begin{pspicture}(-5,-1.0)(5,2.0)
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      \rput(-0.82,1.05){$b$}
      \rput( 1.45,1.05){$c$}
      \rput(-1.18, 0.22){$\gamma$}
      \rput( 1.35, 0.22){$\beta$}
      \rput(-3.5,0){
        \psset{linewidth=0.5pt}
        \psline(0,\xx)(0,0)(\xx,0)
        \rput(0,\xxa){$y$}
        \rput(\xxa,0){$x$}
      }
    \end{pspicture}
  \ec
\end{figure}
By the definition of the sine function, one can write that
\[
  c\sin\beta = h = b\sin\gamma,
\]
or in other words that
\[
  \frac{\sin\beta}{b} = \frac{\sin\gamma}{c}.
\]
But there is nothing special about~$\beta$ and~$\gamma$; what is true
for them must be true for~$\alpha$, too.%
\footnote{
  ``But,'' it is objected, ``there \emph{is} something special about~$\alpha$.
  The perpendicular~$h$ drops from it.''

  True.  However, the~$h$ is just a utility variable to help us to
  manipulate the equation into the desired form; we're not interested in~$h$
  itself.  Nothing prevents us from dropping additional
  perpendiculars~$h_\beta$ and~$h_\gamma$ from the other two corners and
  using those as utility variables, too, if we like.  We can use any
  utility variables we want.
}
Hence,
\bq{trig:270:sin}
  \frac{\sin\alpha}{a} = \frac{\sin\beta}{b} = \frac{\sin\gamma}{c}.
\eq
This equation is known as \emph{the law of sines.}

\index{law of cosines}
\index{cosine!law of}
On the other hand, if one expresses~$a$ and~$b$ as vectors emanating
from the point~$\gamma$,%
\footnote{
  Here is another example of the book's judicious relaxation of formal
  rigor, or at any rate of formal nomenclature.  Of course there is no
  ``point~$\gamma$''; $\gamma$ is an angle
  not a point.  However, the writer suspects in light of
  Fig.~\ref{trig:270:f1} that few readers will be confused
  as to which point is meant.  The skillful applied mathematician does
  not multiply labels without need!
}
\[
  \begin{split}
    \ve a &= \vu x a, \\
    \ve b &= \vu x b \cos\gamma + \vu y b \sin\gamma,
  \end{split}
\]
then
\bqb
  c^2 &=& \left|\ve b - \ve a\right|^2 \\
    &=& ( b\cos\gamma - a )^2 + ( b\sin\gamma )^2 \\
    &=& a^2 + (b^2)(\cos^2\gamma + \sin^2\gamma) - 2ab\cos\gamma.
\eqb
Since $\cos^2(\cdot) + \sin^2(\cdot) = 1$, this is that
\bq{trig:270:cos}
  c^2 = a^2 + b^2 - 2ab\cos\gamma,
\eq
known as \emph{the law of cosines.}

% ----------------------------------------------------------------------

\section{Summary of properties}
\label{trig:275}

Table~\ref{trig:hourtable} on page~\pageref{trig:hourtable} has listed
the values of trigonometric functions of the hour angles.
Table~\ref{trig:228:table} on page~\pageref{trig:228:table} has
summarized simple properties of the trigonometric functions.
Table~\ref{trig:275:table} on page~\pageref{trig:275:table}
summarizes further properties, gathering them
from \S\S~\ref{trig:240}, \ref{trig:250} and~\ref{trig:270}.
\begin{table}
  \caption{Further properties of the trigonometric functions.}
  \index{trigonometry!properties of}
  \label{trig:275:table}
  \bqb
    \ve u &=& \vu x' (x\cos\phi+y\sin\phi) + \vu y' (-x\sin\phi+y\cos\phi) \\
    \sin(\alpha\pm\beta) &=& \sin\alpha\cos\beta \pm \cos\alpha\sin\beta \\
    \cos(\alpha\pm\beta) &=& \cos\alpha\cos\beta \mp \sin\alpha\sin\beta \\
    \sin\alpha\sin\beta &=& \frac{ \cos(\alpha-\beta) - \cos(\alpha+\beta) }{2} \\
    \sin\alpha\cos\beta &=& \frac{ \sin(\alpha-\beta) + \sin(\alpha+\beta) }{2} \\
    \cos\alpha\cos\beta &=& \frac{ \cos(\alpha-\beta) + \cos(\alpha+\beta) }{2} \\
    \sin\gamma + \sin\delta &=& 2\sin\left(\frac{\gamma+\delta}{2}\right)
      \cos\left(\frac{\gamma-\delta}{2}\right) \\
    \sin\gamma - \sin\delta &=& 2\cos\left(\frac{\gamma+\delta}{2}\right)
      \sin\left(\frac{\gamma-\delta}{2}\right) \\
    \cos\delta + \cos\gamma &=& 2\cos\left(\frac{\gamma+\delta}{2}\right)
      \cos\left(\frac{\gamma-\delta}{2}\right) \\
    \cos\delta - \cos\gamma &=& 2\sin\left(\frac{\gamma+\delta}{2}\right)
      \sin\left(\frac{\gamma-\delta}{2}\right) \\
    \sin2\alpha &=& 2\sin\alpha\cos\alpha \\
    \cos2\alpha &=& 2\cos^2\alpha - 1 = \cos^2\alpha - \sin^2\alpha = 1 - 2\sin^2\alpha \\
    \sin^2\alpha &=& \frac{1-\cos 2\alpha}{2} \\
    \cos^2\alpha &=& \frac{1+\cos 2\alpha}{2} \\
    \frac{\sin\gamma}{c} &=& \frac{\sin\alpha}{a}
      \makebox[\arraycolsep]{}=\makebox[\arraycolsep]{} \frac{\sin\beta}{b} \\
    c^2 &=& a^2 + b^2 - 2ab\cos\gamma
  \eqb
\end{table}

% ----------------------------------------------------------------------

\section{Cylindrical and spherical coordinates}
\label{trig:277}

\index{coordinates}
\index{coordinates!rectangular}
\index{rectangular coordinates}
\index{axis}
\index{point}
\index{orthonormal vectors}
\index{vector!orthonormal}
Section~\ref{trig:230} has introduced the concept of the vector
\[
  \ve v = \vu x x + \vu y y + \vu z z.
\]
The coefficients $(x,y,z)$ on the equation's right side are
\emph{coordinates}---specif\-i\-cally, \emph{rectangular
coordinates}---which given a specific, orthonormal%
\footnote{
  \emph{Orthonormal} in this context means ``of unit length and at right
  angles to the other vectors in the
  set.''\ \cite[``Orthonormality,'' 14:19, 7 May 2006]{wikip}
}
set of unit basis vectors $[\vu x\ \vu y\ \vu z]$ uniquely identify a
point (see Fig.~\ref{trig:231f3} on page~\pageref{trig:231f3}; and also,
much later in the book, refer to \S~\ref{vector:230}).  Such
rectangular coordinates are simple and general, and are convenient for
many purposes.  However, there are at least two broad classes of
conceptually simple problems for which rectangular coordinates tend to
be inconvenient: problems in which an axis or a point dominates.
Consider for example an electric wire's magnetic field, whose intensity
varies with distance from the wire (an axis); or the illumination a
lamp sheds on a printed page of this book, which depends on the
book's distance from the lamp (a point).

\index{coordinates!cylindrical}
\index{cylindrical coordinates}
\index{coordinates!spherical}
\index{spherical coordinates}
\index{coordinate rotation}
\index{axis}
\index{point}
To attack a problem dominated by an axis, the \emph{cylindrical
coordinates} $(\rho;\phi,z)$ can be used instead of the rectangular
coordinates $(x,y,z)$.  To attack a problem dominated by a point, the
\emph{spherical coordinates} $(r;\theta;\phi)$ can be used.%
\footnote{
  Notice that the~$\phi$ is conventionally written second in cylindrical
  $(\rho;\phi,z)$ but third in spherical $(r;\theta;\phi)$ coordinates.
  This odd-seeming convention is to maintain proper right-handed
  coordinate rotation.  (The explanation will seem clearer once
  chapters~\ref{vector} and~\ref{vcalc}
  are read.)
}
Refer to Fig.~\ref{trig:fig-sphere}.  Such coordinates are related to
one another and to the rectangular coordinates by the formulas of
Table~\ref{trig:277:20}.
\begin{figure}
  \spherecaption
  \label{trig:fig-sphere}
  \index{sphere}
  \sphere
\end{figure}
\begin{table}
  \caption[Rectangular, cylindrical and spherical coordinate
    relations.]{Relations among the rectangular, cylindrical and spherical
    coordinates.}
  \label{trig:277:20}
  \index{coordinates!relations among}
  \settowidth\tla{$\ds \rho\cos\phi$}
  \bqb
    \rho^2 &=& x^2 + y^2 \\
    r^2 &=& \rho^2 + z^2 = x^2 + y^2 + z^2 \\
    \tan\theta &=& \frac \rho z \\
    \tan\phi &=& \frac y x \\
    z &=& r\cos\theta \\
    \rho &=& r\sin\theta \\
    x &=& \makebox[\tla][l]{$\ds \rho\cos\phi$} = r\sin\theta\cos\phi \\
    y &=& \makebox[\tla][l]{$\ds \rho\sin\phi$} = r\sin\theta\sin\phi
  \eqb
\end{table}

Cylindrical and spherical coordinates can greatly simplify the analyses
of the kinds of problems they respectively fit, but they come at a
price.  There are no constant unit basis vectors to match them.  That
is,
\[
  \ve v = \vu x x + \vu y y + \vu z z
  \neq \wu \rho \rho + \wu \phi \phi + \vu z z
  \neq \vu r r + \wu \theta \theta + \wu \phi \phi.
\]
It doesn't work that way.  Nevertheless, \emph{variable} unit basis vectors
are defined:
\bq{trig:277:30}
  \index{vector!unit basis, variable}
  \index{vector!unit basis, cylindrical}
  \index{vector!unit basis, spherical}
  \index{unit basis vector!variable}
  \index{unit basis vector!cylindrical}
  \index{unit basis vector!spherical}
  \settowidth\tla{$\ds + \vu x \cos\phi$}
  \begin{split}
    \wu \rho   &\equiv \makebox[\tla][l]{$\ds + \vu x \cos\phi$} + \vu y \sin\phi, \\
    \wu \phi   &\equiv \makebox[\tla][l]{$\ds - \vu x \sin\phi$} + \vu y \cos\phi, \\
    \vu r      &\equiv \makebox[\tla][l]{$\ds + \vu z \cos\theta$} + \wu \rho \sin\theta, \\
    \wu \theta &\equiv \makebox[\tla][l]{$\ds - \vu z \sin\theta$} + \wu \rho \cos\theta;
  \end{split}
\eq
or, substituting identities from the table,
\bq{trig:277:31}
  \settowidth\tla{$\ds + \vu x \cos\phi$}
  \begin{split}
    \wu \rho   &= \frac{\vu x x + \vu y y}{\rho}, \\
    \wu \phi   &= \frac{-\vu x y + \vu y x}{\rho}, \\
    \vu r      &= \frac{\vu z z + \wu \rho \rho}{r}
      = \frac{\vu x x + \vu y y + \vu z z}{r}, \\
    \wu \theta &= \frac{-\vu z \rho + \wu \rho z}{r}.
  \end{split}
\eq
Such variable unit basis vectors point locally in the directions in
which their respective coordinates advance.

Combining pairs of (\ref{trig:277:30})'s equations appropriately, we
have also that
\bq{trig:277:30a}
  \settowidth\tla{$\ds + \wu \phi \cos\phi$}
  \begin{split}
    \vu x      &= \makebox[\tla][l]{$\ds + \wu \rho \cos\phi$} - \wu \phi \sin\phi, \\
    \vu y      &= \makebox[\tla][l]{$\ds + \wu \rho \sin\phi$} + \wu \phi \cos\phi, \\
    \vu z      &= \makebox[\tla][l]{$\ds + \vu r \cos\theta$} - \wu \theta \sin\theta, \\
    \wu \rho   &= \makebox[\tla][l]{$\ds + \vu r \sin\theta$} + \wu \theta \cos\theta.
  \end{split}
\eq

Convention usually orients~$\vu z$ in the direction of a problem's
axis.  Occasionally however a problem arises in which it is more
convenient to orient~$\vu x$ or~$\vu y$ in the direction of the
problem's axis (usually because~$\vu z$ has already been established in
the direction of some other pertinent axis).  Changing the meanings of
known symbols like~$\rho$, $\theta$ and~$\phi$ would probably not be a good
idea, but you can use symbols like
\bq{trig:277:40}
  \renewcommand\arraystretch{2.0}
  \br{rclcrcl}
    \ds (\rho^x)^2 &=& \ds y^2 + z^2, &\sh{1.0}&
    \ds (\rho^y)^2 &=& \ds z^2 + x^2, \\
    \ds \tan \theta^x &=& \ds \frac{\rho^x}{x}, &\sh{1.0}&
    \ds \tan \theta^y &=& \ds \frac{\rho^y}{y}, \\
    \ds \tan \phi^x &=& \ds \frac{z}{y}, &\sh{1.0}&
    \ds \tan \phi^y &=& \ds \frac{x}{z},
  \er
\eq
instead if needed.%
\footnote{
  Symbols like~$\rho^x$ are logical but, as far as this writer is aware,
  not standard.  The writer is not aware of any conventionally
  established symbols for quantities like these, but
  \S~\ref{vector:270} at least will use the $\rho^x$-style symbology.
}

% ----------------------------------------------------------------------

\section{The complex triangle inequalities}
\label{trig:278}
\index{triangle inequalities!complex}
\index{triangle inequalities!vector}

If the real, two-dimensional vectors~$\ve a$, $\ve b$ and~$\ve c$
represent the three sides of a triangle such that $\ve a + \ve b + \ve
c = 0$, then per~(\ref{alggeo:323:20})
\[
  \left|\ve a\right|-\left|\ve b\right| \le \left|\ve a
  + \ve b\right| \le \left|\ve a\right| + \left|\ve b\right|.
\]
These are just the triangle inequalities of \S~\ref{alggeo:323.20} in vector
notation.%
\footnote{
  Reading closely, one might note that \S~\ref{alggeo:323.20} uses
  the~``$<$'' sign rather than the~``$\le$,'' but that's all right.  See
  \S~\ref{intro:284.1}.
}
But if the triangle inequalities hold for real vectors in a
plane, then why not equally for complex scalars?  Consider the geometric
interpretation of the Argand plane of Fig.~\ref{alggeo:225:fig} on
page~\pageref{alggeo:225:fig}.
Evidently,
\bq{trig:278:triangle2}
  \left|z_1\right|-\left|z_2\right| \le \left|z_1
  + z_2\right| \le \left|z_1\right| + \left|z_2\right|
\eq
for any two complex numbers~$z_1$ and~$z_2$.  Extending the sum
inequality, we have that
\bq{trig:278:triangle}
  \left|\sum_k z_k \right| \le \sum_k \left|z_k\right|.
\eq
(Naturally, the inequalities~\ref{trig:278:triangle2}
and~\ref{trig:278:triangle} hold as well for real numbers as for
complex.  One may find the latter inequality useful for sums of real
numbers, for example, when some of the numbers summed are positive and
others negative.)%
\footnote{
  Section~\ref{mtxinv:447} proves the triangle inequalities more generally.
}

\index{convergence}
\index{series!convergence of}
\index{summation!convergence of}
\index{magnitude}
An important consequence of~(\ref{trig:278:triangle}) is that if $\sum
\left|z_k\right|$ converges, then $\sum z_k$ also converges.  Such a consequence is
important because mathematical derivations sometimes need the
convergence of $\sum z_k$ established, which can be hard to do directly.
Convergence of $\sum \left|z_k\right|$, which per~(\ref{trig:278:triangle}) implies
convergence of $\sum z_k$, is often easier to establish.

See also~(\ref{inttx:250:triangle}).  Equation~(\ref{trig:278:triangle})
will find use among other places in \S~\ref{taylor:316.30}.

% ----------------------------------------------------------------------

\section{De~Moivre's theorem}
\label{trig:280}
\index{de Moivre, Abraham (1667--1754)}
\index{Moivre, Abraham de (1667--1754)}
\index{de Moivre's theorem}
\index{complex number}
\index{number!complex}
\index{complex number!multiplication and division}
\index{multiplication}
\index{division}

Compare the Argand-plotted complex number of Fig.~\ref{alggeo:225:fig}
(page~\pageref{alggeo:225:fig}) against the vector of
Fig.~\ref{trig:231} (page~\pageref{trig:231}).  Although complex numbers
are scalars not vectors, the figures do suggest an analogy between
complex phase and vector direction.
With reference to Fig.~\ref{alggeo:225:fig} we can write,
\bq{trig:280:10}
  z = (\rho) ( \cos\phi + i\sin\phi ) = \rho\cis\phi,
\eq
where
\bq{trig:280:cis}
  \cis\phi \equiv \cos\phi + i\sin\phi.
\eq
If $z=x+iy$, then evidently
\bq{trig:280:11}
  \begin{split}
    x&=\rho\cos\phi, \\
    y&=\rho\sin\phi.
  \end{split}
\eq

Per~(\ref{alggeo:225:10}),
\[
  z_1z_2 = (x_1x_2 - y_1y_2) + i(y_1x_2 + x_1y_2).
\]
Applying~(\ref{trig:280:11}) to this equation yields that
\[
  \frac{z_1z_2}{\rho_1\rho_2} =
    (\cos\phi_1\cos\phi_2 - \sin\phi_1\sin\phi_2)
    + i(\sin\phi_1\cos\phi_2 + \cos\phi_1\sin\phi_2).
\]
But according to~(\ref{trig:250:22}), this is just that
\[
  \frac{z_1z_2}{\rho_1\rho_2} =
    \cos(\phi_1 + \phi_2) + i\sin(\phi_1 + \phi_2),
\]
or in other words that
\bq{trig:280:20}
  z_1z_2 = \rho_1\rho_2 \cis(\phi_1 + \phi_2).
\eq
Equation~(\ref{trig:280:20}) is a significant result.  It says that if
you want to multiply complex numbers, it suffices
\bi
  \item to multiply their magnitudes and
  \item to add their phases.
\ei
It follows by parallel reasoning (or by extension) that
\bq{trig:280:22}
  \frac{z_1}{z_2} = \frac{\rho_1}{\rho_2} \cis(\phi_1 - \phi_2)
\eq
and by extension that
\bq{trig:280:24}
  z^a = \rho^a \cis a\phi.
\eq
Equations~(\ref{trig:280:20}), (\ref{trig:280:22})
and~(\ref{trig:280:24}) are known as \emph{de~Moivre's theorem.}%
\footnote{
  Also called \emph{de~Moivre's formula.}  Some authors apply the name
  of de~Moivre directly only to~(\ref{trig:280:24}), or to some variation
  thereof; but, since the three equations express essentially the same
  idea, if you refer to any of them as~\emph{de~Moivre's theorem} then
  you are unlikely to be misunderstood.
}$\mbox{}^,$\footnote{\cite{Spiegel}\cite{wikip}}

We have not shown yet, but will in \S~\ref{cexp:230}, that
\[
  \cis \phi = \exp i\phi = e^{i\phi},
\]
where $\exp(\cdot)$ is the natural exponential function and~$e$ is the
natural logarithmic base, both defined in chapter~\ref{cexp}\@.  De~Moivre's
theorem is most useful in this light.

Section~\ref{cexp:240} will revisit the derivation of de~Moivre's theorem.

