There is perhaps nothing which so occupies the middle
position of mathematics as trigonometry.
-J. F. HERBART (1890)

This book is neither a textbook of trigonometry - of which there are many -
nor a comprehensive history of the subject, of which there is almost
none. It is an attempt to present selected topics in trigonometry from a
historic point of view and to show their relevance to other sciences. It
grew out of my love affair with the subject, but also out of my frustration
at the way it is being taught in our colleges.

First, the love affair. In the junior year of my high school we were
fortunate to have an excellent teacher, a young, vigorous man who
taught us both mathematics and physics. He was a no-nonsense teacher,
and a very demanding one. He would not tolerate your arriving late to
class or missing an exam-and you better made sure you didn't, lest it was
reflected on your report card. Worse would come if you failed to do your
homework or did poorly on a test. We feared him, trembled when he
reprimanded us, and were scared that he would contact our parents. Yet
we revered him, and he became a role model to many of us. Above all, he
showed us the relevance of mathematics to the real world - especially to
physics. And that meant learning a good deal of trigonometry.

He and I have kept a lively correspondence for many years, and we
have met several times. He was very opinionated, and whatever you said
about any subject - mathematical or otherwise - he would argue with you,
and usually prevail. Years after I finished my university studies, he would
let me understand that he was still my teacher. Born in China to a family
that fled Europe before World War 11, he emigrated to Israel and began
his education at the Hebrew University of Jerusalem, only to be drafted
into the army during Israel's war of independence. Later he joined the
faculty of Tel Aviv University and was granted tenure despite not having
a Ph.D. - one of only two faculty members so honored. In 1989, while
giving his weekly lecture on the history of mathematics, he suddenly collapsed and died instantly. His name was Nathan Elioseph. I miss him dearly.

And now the frustration. In the late 1950s, following the early Soviet
successes in space (Sputnik I was launched on October 4, 1957; 1
remember the date-it was my twentieth birthday) there was a call for
revamping our entire educational system, especially science education.
New ideas and new programs suddenly proliferated, all designed to close
the perceived technological gap between us and the Soviets (some dared
to question whether the gap really existed, but their voices were swept
aside in the general frenzy). These were the golden years of American
science education. If you had some novel idea about how to teach a
subject-and often you didn't even need that muchyou were almost
guaranteed a grant to work on it. Thus was born the "New Math" - an
attempt to make students understand what they were doing, rather than
subject them to rote learning and memorization, as had been done for
generations. An enormous amount of time and money was spent on
developing new ways of teaching math, with emphasis on abstract
concepts such as set theory, functions (defined as sets of ordered pairs),
and formal logic. Seminars, workshops, new curricula, and new texts were
organized in haste, with hundreds of educators disseminating the new
ideas to thousands of bewildered teachers and parents. Others traveled
abroad to spread the new gospel in developing countries whose
populations could barely read and write.

Today, from a distance of four decades, most educators agree that the
New Math did more harm than good. Our students may have been taught
the language and symbols of set theory, but when it comes to the
simplest numerical calculations they stumble-with or without a calculator.
Consequently, many high school graduates are lacking basic algebraic
skills, and, not surprisingly, some 50 percent of them fail their first college-
level calculus course. Colleges and universities are spending vast
resources on remedial programs (usually made more palatable by giving
them some euphemistic title like "developmental program" or "math lab"),
with success rates that are moderate at best.

Two of the casualties of the New Math were geometry and
trigonometry. A subject of crucial importance in science and engineering,
trigonometry fell victim to the call for change. Formal definitions and
legalistic verbosity-all in the name of mathematical rigor-replaced a real
understanding of the subject. Instead of an angle, one now talks of the
measure of an angle; instead of defining the sine and cosine in a
geometric context as ratios of sides in a triangle or as projections of the unit circle on the x- and y-axes - one talks about the wrapping function from the reals to the
interval [-1, 1]. Set notation and set language have pervaded all
discussion, with the result that a relatively simple subject became
obscured in meaningless formalism.

Worse, because so many high school graduates are lacking basic
algebraic skills, the level and depth of the typical trigonometry textbook
have steadily declined. Examples and exercises are often of the simplest
and most routine kind, requiring hardly anything more than the
memorization of a few basic formulas. Like the notorious "word
problems" of algebra, most of these exercises are dull and uninspiring,
leaving the student with a feeling of "so what?" Hardly ever are students
given a chance to cope with a really challenging identity, one that might
leave them with a sense of accomplishment. For example,

1. Prove that for any number x,

This formula was discovered by Euler. Substituting x = u12, using the
fact that cos 7T/4 = /2-/2 and repeatedly applying the half-angle formula
for the cosine, we get the beautiful formula

discovered in 1593 by François Viète in a purely geometric way.

2. Prove that in any triangle,

(The last formula has some unexpected consequences, which we will
discuss in chapter 12.) These formulas are remarkable for their symmetry;
one might even call them "beautiful" - a kind word for a subject that has
undeservedly gained a reputation of being dry and technical. In
Appendix 3, 1 have collected some additional beautiful formulas,
recognizing of course that "beauty" is an entirely subjective trait.

"Some students," said Edna Kramer in The Nature and Growth of
Modem Mathematics, consider trigonometry "a glorified geometry with
superimposed computational torture." The present book is an attempt to
dispel this view. I have adopted a historical approach, partly because I
believe it can go a long way to endear mathematics-and science in
general-to the students. However, I have avoided a strict chronological
presentation of topics, selecting them instead for their aesthetic appeal or
their relevance to other sciences. Naturally, my choice of subjects reflects
my own preferences; numerous other topics could have been selected.

The first nine chapters require only basic algebra and trigonometry;
the remaining chapters rely on some knowledge of calculus (no higher
than Calculus 11). Much of the material should thus be accessible to high
school and college students. Having this audience in mind, I limited the
discussion to plane trigonometry, avoiding spherical trigonometry
altogether (although historically it was the latter that dominated the
subject at first). Some additional historical material-often biographical in
nature-is included in eight "sidebars" that can be read independently of
the main chapters. If even a few readers will be inspired by these
chapters, I will consider myself rewarded.