How the Trig Functions Got their Names

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?

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://members.aol.com/jeff570/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/