Latin Origins of Trig Functions

Date: 11/20/98 at 02:23:33
From: Kim Taing
Subject: Definitions in Latin

What are the origins of the words sine, cosine, tangent, etc.? 
Basically the six trigonometric functions: sin, cos, tan, cscin, 
cos, tan, csc, sec, cot, sec, cot.

Thanks.

Date: 11/20/98 at 08:39:06
From: Doctor Rick
Subject: Re: Definitions in Latin

Hi, Kim. I've wanted to put together this information for some time. 
Thanks for getting me to do it! The following is based on etymological 
information from Webster's Third New International Dictionary.

I will refer to the following figure. O is the center of the circle 
shown passing through A and D.

                       *          B
                          * D   /|
                            */   |
                          / |*   |
                       /    | *  |
                    /       |  * |
                 /          |   *|
             /              |   *|
          /                 |    *
       /                    |    *
    /_______________________|____*
   O                        C     A

SINE comes from the Latin SINUS, meaning a bend or gulf, or the bosom 
of a garment. (We know the word from its anatomical meaning: the 
cavities or bays in the facial bones and from the names of some "bays" 
on the moon.) The term was used as a translation for the Arabic word 
"jayb," the word for a sine that also meant the bosom of a garment, and 
which in turn comes from the Sanskrit word "jiva" meaning a bowstring.

The word was originally applied to the line segment CD in the figure: 
half the chord of twice the angle AOB. You can see how this could be 
called a "bowstring." The ratio of the sine CD to the radius of the 
circle, OA, is the SINE of angle AOB.

TANGENT comes from the Latin TANGENS, the present participle of TANGERE, 
"to touch." In other words, it means "touching." It was originally 
applied to the line segment AB in the figure: the segment of the 
tangent to the circle at A that is cut off by the extension of OB. 
The ratio of the tangent AB to the radius of the circle, OA, is the 
TANGENT of angle AOB.

SECANT comes from the Latin SECANS, the present participle of SECARE, "to 
cut." In other words, it means "cutting." It was originally applied to 
the line segment OB in the figure - the line that cuts off the tangent. 
The ratio of the secant OB to the radius OA is the SECANT of angle AOB.

COSINE was originally written "co.sine," short for COMPLEMENTI SINUS: 
the sine of the complement. The COSINE of angle AOB is the sine of the
complementary angle (ABO in the figure). Likewise COTANGENT and 
COSECANT are the tangent and secant respectively of the complementary 
angle.

David Joyce's "Short Course in Trigonometry" site shows these relations 
with better figures:

   http://aleph0.clarku.edu/~djoyce/java/trig/   

Jeff Miller maintains pages that trace the first uses of many 
mathematical terms, including these. Look here to find dates and people 
who used the terms:

   http://members.aol.com/jeff570/mathword.html   

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/