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://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/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.