Why Sine, Cosine, and Tangent?
Date: 11/29/2001 at 10:44:11 From: Sara Allen Subject: Sine, cosine, and tangent Why is sine=opp/hyp, cos=adj/hyp, and tan=adj/opp? I know these things, I just want to know why! Thanks, Sara
Date: 11/29/2001 at 12:01:08 From: Doctor Rob Subject: Re: Sine, cosine, and tangent Thanks for writing to Ask Dr. Math, Sara. Of these, sine is rooted in antiquity. For historical questions about math words, I use Jeff Miller's: Earliest Known Uses of Some of the Words of Mathematics http://jeff560.tripod.com/mathword.html . Click on "S" for the page containing the discussion of "sine." Tangent is a lot easier to explain. Start with a unit circle with center O. Draw two radii OA and OB making angle x. Draw the tangent line to the circle at point A. Extend the radius OB in both directions until it meets that tangent line at point C. | o C /| / | / | / | sec(x)/ | / | _,---._B/ |tan(x) _,-' o-._ | ,' / `. | / 1/ \ | / /x \| : O o-----------o A \ / 1 /| \ / / | `. / ,' | `-./ _,-' | / `-...-' | / | Then the distance AC along the tangent line will be tan(x). (It is the ratio of opposite over adjacent in the right triangle OCA.) It is the length on the tangent line cut out by the angle x, hence the "tangent of angle x." The word "tangent" for the line touching the circle at just one point is derived from the Latin verb "tangere," to touch, and means "touching." There is a similar explanation for using "secant" for the ratio hypotenuse/adjacent. In this case, the distance OC, measured along the extended radius OB, is sec(x). (It is the ratio of hypotenuse over adjacent in the right triangle OCA.) The extended radius is a secant line, that is, a line cutting the circle in two points. "Secant," too, is derived from a Latin verb, "secare," to cut, and means "cutting." All the "co-" trigonometric functions are related to the complement of the angle. For any angle x, the complement is 90o - x. Thus the cosine of an angle is the sine of the complementary angle: cos(x) = sin(90o-x), cot(x) = tan(90o-x), csc(x) = sec(90o-x). Thus "cosine" is a contraction of "complementary sine," and likewise for cotangent and cosecant. Feel free to write again if I can help further. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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