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://members.aol.com/jeff570/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/