Trigonometric Functions and the Unit Circle

Date: 05/22/2000 at 21:26:17
From: Allison 
Subject: Theories behind trigonometric functions

Why don't sine and cosine graphs have values greater than one or less 
than negative one? Why does the tangent graph have asymptotes, and 
look the way it does? What do trig functions represent, and how are 
they calculated?

This is a writing assignment for my Algebra II / Trigonometry class. 
So far, I know that trig functions represent the ratios of the lengths 
of two sides of a triangle and they are calculated by substituting 
values into corresponding positions in the ratios and solving for the 
desired function. 

I am really stuck on why the 
values only stay in the range of -1 to +1. I know that it has 
something to do with the unit circle, but it is not because of the 
unit circle. Please help - thank you!

Allison S.

Date: 05/23/2000 at 09:09:25
From: Doctor Peterson
Subject: Re: Theories behind trigonometric functions

Hi, Allison.

The unit circle is a good place to start. Let's look at the sine 
first:

                  *********** -------------------- 1
              ****           ****   P
           ***                   **+
         **                      / |**
        *                hyp=1 /   |  **
      **                     /     |opp *
      *                    /       |    *
     *                   / A       |     *
     *                 +-----------+-----+ ------- 0
     *                                   *
      *                                 *
      **                                *
        *                             **
         **                         **
           ***                   ***
              ****           ****
                  *********** -------------------- -1

The sine of angle A is the ratio of the opposite to the hypotenuse. 
Using a unit circle, the hypotenuse is always 1, so the sine of A is 
simply the length of the opposite side. In fact, this is the y 
coordinate of point P. Do you see that this must always be between -1 
(the bottom of the circle) and 1 (the top of the circle)?

Now look at the tangent:

                                         +P
                  ***********          / |
              ****           ****    /   |
           ***                   **/     |
         **                      /  **   |opp
        *                      /      ** |
      **                     /          *|
      *                    /            *|
     *                   / A             |
     *                 +-----------------+B
     *                         adj=1     *
      *                                 *
      **                                *
        *                             **
         **                         **
           ***                   ***
              ****           ****
                  ***********

Here in the unit circle the adjacent side will always be 1, so the 
tangent of A, the ratio of the opposite to the adjacent, will be the 
length of the opposite side, BP. What happens as A approaches 90 
degrees? The opposite side gets larger and larger. That forms an 
asymptote at A = 90, where the line AP is parallel to the tangent line 
BP, and never intersects, so that there is no tangent. When you go 
past 90 degrees, the picture flips over:

                  ***********
              ****           ****
           ***   \               **
         **       \                 **
        *          \                  **
      **            \                   *
      *              \                  *
     *                \ A      adj=1     *
     *                 +-----------------+B
     *                  \                *
      *                  \              *|
      **                  \             *|
        *                  \          ** |
         **                 \       **   |
           ***               \   ***     |
              ****           ****        |
                  ***********  \         |opp
                                \        |
                                 \       |
                                  \      |
                                   \     |
                                    \    |
                                     \   |
                                      \  |
                                       \ |
                                        \|
                                         +P

Now the tangent is a large negative number, because the line at angle 
A intersects the tangent line below the circle rather than above it. 
As A increases to 180 degrees, the tangent goes to zero again.

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