Sine, Cosine, Tangent

Date: 03/13/2001 at 00:49:35
From: kelsie gale
Subject: Sine, cosine, and tangent

What are sine, cosine, and tangent?

Date: 03/13/2001 at 11:25:42
From: Doctor Rob
Subject: Re: Sine, cosine, and tangent

Thanks for writing to Ask Dr. Math, Kelsie!

Sine, cosine, and tangent are functions that take an angle and give 
you a corresponding real number.

Let A be an acute angle, and consider the following diagram:

                                 B"_,-'
                             B'_,o'
                         B _,o'  |
                       _,o'  |   |
                   _,-'  |   |   |
               _,-'      |   |   |
           _,-'          |   |   |
       _,-'              |   |   |
   A o-------------------o---o---o-----
                         C   C'  C"

Here <ACB, <AC'B', and <AC"B" are all right angles.  

The three triangles ABC, AB'C', and AB"C", are similar. (Why?) That 
means their corresponding sides are in proportion to each other. This 
tells you that, for example,

   CB/AB = C'B'/AB' = C"B"/AB"

That means that in a right triangle, no matter its size, the ratio of 
the leg opposite the angle A, to the hypotenuse, is a certain 
constant.  That ratio is called the sine of the angle A, and is 
written sin(A).

Similarly, the ratio of the leg adjacent to the angle, to the 
hypotenuse,

   AC/AB = AC'/AB' = AC"/AB"

is a constant, called the cosine of the angle A, and written cos(A).

Similarly, the ratio of the leg opposite to the angle, to the leg
adjacent to the angle,

   CB/AC = C'B'/AC' = C"B"/AC"

is a constant, called the tangent of the angle A, and written tan(A).

These three functions can also be defined for zero, right, obtuse,
and other angles using the diagram at the top of the following Web 
page from our Frequently Asked Questions (FAQ):

  Trigonometry Formulas
  http://mathforum.org/dr.math/faq/formulas/faq.trig.html   

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/