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 angle. Here is a Web site that shows these relations with better figures. Dave's Short Trig Course - David Joyce http://aleph0.clarku.edu/~djoyce/java/trig/ 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 terms. Earliest Known Uses of Some of the Words of Mathematics - Jeff Miller http://jeff560.tripod.com/mathword.html - Doctor Rick, The Math Forum http://mathforum.org/dr.math/
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