The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Origin of the Terms Sine, Cosine, Tangent, etc.

Date: 10/27/1999 at 21:13:07
From: Natalie
Subject: Hypotenuse; SOH CAH TOA

I want to know how the term hypotenuse came about - from whom, from 
where, when, why, etc.  Also, I would like to know where the formulas 
for sine, cosine, and tangent, that is SOH CAH TOA, came from.

Date: 10/28/1999 at 08:57:19
From: Doctor Rick
Subject: Re: Hypotenuse; SOH CAH TOA

Hypotenuse is a Greek word that means "stretched under." I guess the 
early Greek mathematicians pictured the hypotenuse of a right triangle 
as being stretched across the ends of the two legs like a bowstring on 
a bow. By the way, another geometry word comes from exactly the same 
meaning, except that it is in Latin. The word is "subtend," as in "a 
chord subtends an arc" - there the bowstring image is very obvious.

As for SOH-CAH-TOA, that is a mnemonic for the sine, cosine, and 
tangent. The names sine, cosine, and tangent again date back to the 
Greeks; they originally referred to the lengths of various lines in 
the figure below. In the 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 - mine are badly congested right 
now - 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 

Here is a Web site that shows these relations with better figures.

  Dave's Short Trig Course - David Joyce   

Here is a good site tracing the first uses of many mathematical terms, 
including these. Look here to find dates and people who used the 

  Earliest Known Uses of Some of the Words of Mathematics
   - Jeff Miller   

- Doctor Rick, The Math Forum   
Associated Topics:
High School Definitions
High School History/Biography
High School Trigonometry
Middle School Definitions
Middle School History/Biography

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.