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### How the Trig Functions Got their Names

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Date: 12/14/97 at 03:42:57
From: Frank Becker
Subject: How the Trig Functions got their names

I can guess why three of the trig functions are called cosine,
cotangent, and cosecant. But why were the other three named the sine,
the tangent, and the secant?

Does the choice of the words tangent and secant have anything to do
with the ordinary geometric meaning of these words?
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Date: 12/14/97 at 05:59:08
From: Doctor Luis
Subject: Re: How the Trig Functions got their names

Have you tried a dictionary? (you'd be surprised to know the number of
things you can find out by using one). Webster's New Collegiate
Dictionary suggests the following etymologies:

sine   :  Medieval Latin "sinus" from the Latin word for "curve"

tangent:  Latin "tangent-, tangens" from present participle of
"tangere" (to touch)

secant :  New Latin "secant-, secans" from  Latin present participle
of "secare" (to cut)

Webster's II International has, for sine:

Latin sinus, a bend, gulf, bosom of a garment, used as translation of
Arabic jayb, bosom of a garment, sine (in the latter sense from
Sanskrit Jiva, bowstring, chord of an arc, sine).

For more etymological information, see Jeff Miller's "Earliest Known
Uses of Some of the Words of Mathematics:

(S) http://jeff560.tripod.com/s.html

SINE. Aryabhata the Elder (476-550) used the word jya for sine in
Aryabhatiya, which was finished in 499.

According to some sources, sinus first appears in Latin in a
translation of the Algebra of al-Khowarizmi by Gherard of Cremona
(1114-1187). For example, Eves (page 177) writes:

The origin of the word sine is curious. Aryabhata called in
ardha-jya ("half-chord") and also jya-ardha ("chord-half"), and
then abbreviated the term by simply using jya ("chord"). From jya
the Arabs phonetically derived jiba, which, following Arabian
practice of omitting vowels, was written as jb. Now jiba, aside
from its technical significance, is a meaningless word in Arabic.
Later writers, coming across jb as an abbreviation for the
meaningless jiba, substituted jaib instead, which contains the
same letters and is a good Arabic word meaning "cove" or "bay."
Still later, Gherardo of Cremona (ca. 1150), when he made his
translations from the Arabic, replaced the Arabian jaib by its
Latin equivalent, sinus, whence came our present word sine.

However, Boyer (page 278) places the first appearance of sinus in a
translation of 1145. He writes:

It was Robert of Chester's translation from the Arabic that
resulted in our word "sine." The Hindus had given the name jiva
to the half chord in trigonometry, and the Arabs had taken this
over as jiba. In the Arabic language there is also a word jaib
meaning "bay" or "inlet." When Robert of Chester came to translate
the technical word jiba, he seems to have confused this with the
word jaib (perhaps because vowels were omitted); hence he used
the word sinus, the Latin word for "bay" or "inlet." Sometimes
the more specific phrase sinus rectus, or "vertical sine," was
used; hence the phrase sinus versus, or our "versed sine,"
was applied to the "sagitta," or the "sine turned on its side."

The origins of the names tangent and secant are related to the circle.

A line tangent to a circle is defined as a line that "touches" the
circle at only one point. A secant line is a line that "cuts" the
circle (it intersects the circle at two points).

Now, the tangent and the secant trigonometric functions are related to
the tangent and secant of a circle in the following way.

Consider a UNIT circle centered at point O, and a point Q outside the
unit circle. Construct a line tangent to the circle from point Q and
call the intersection of the tangent line and the circle point P. Also
construct a secant line that goes through the center O of the circle
from point Q. The line segment OQ will intersect the circle at some
point A. Next draw a line segment from the center O to point P. You
should now have a right triangle OPQ.

A little thought will reveal that the length of line segment QP on the
tangent line is nothing more but the tangent (trig function) of angle
POQ (or POA, same thing). Also, the length of the line segment QO on
the secant line is, not surprisingly, the secant (trig function) of
angle POA.

Now construct a line perpendicular to the line segment OP from point
A. Call the intersection of these lines point B. You can readily see
that

BA = sin POA
OB = cos POA

Can you find the other trigonometric functions?

As for the origin of the names for the co-functions, the reason comes
from the following obvious fact:

If you have a trigonometric function trig(x), then the cofunction
of trig(x) is

cotrig(angle) = trig(co-angle)

where co-angle is the "angle complementary" to angle. (Remember that
two angles are complementary if their sum is 90 degrees)

So, the trig cofunction of an angle is the trig function of the
complementary angle.

It is easy to see that in right triangles, the co-angle of an angle is
the other angle (the third angle is already 90 degrees!). That's why
cofunctions have the following properties, (try to prove all of them
from a right triangle)

cos(x) = sin(90-x)    (of course, x is in degrees)
sin(x) = cos(90-x)

cosec(x) = sec(90-x)
sec(x)   = cosec(90-x)

cotan(x) = tan(90-x)
tan(x)   = cotan(90-x)

I hope this helped.

-Doctors Luis and Sarah,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Definitions
High School Trigonometry
Middle School Definitions

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