Explanation of Sine and Cosine

Date: 7/30/96 at 20:38:31
From: SunjayMishra
Subject: Explanation of Sine, Cosine

Hello,

I am having trouble with the topics of sine and cosine.

Thanks for the help!
Jay

Date: 7/31/96 at 4:2:44
From: Doctor Pete
Subject: Re: Explanation of Sine, Cosine

Hello,

I'm not sure what math courses you have taken so far, but it sounds 
like you're somewhere around junior high or high school geometry.  If 
not, please feel free to ask me to clarify anything I explain here.

Think of a right triangle.  Say you increase or decrease the length of 
one of its legs, while keeping the other leg the same (the hypotenuse 
will increase as well).  Then you'll notice that the *subtending* 
angle (the angle which is opposite to the increasing leg) gets larger, 
while the *adjacent* angle (the angle with the increasing leg as one 
of its rays) will decrease.  Here's a picture:

                .
               /| B'
              / |
             /  |
            /   |
           /    |
          /     | B
         /     /|
        /    /  |
       /   /    |
      /  /      |  <---- increasing leg
     / /        |
    //         _|
   /__________|_|    
  A              C

So ABC was our original triangle, and AB'C is the changed triangle, 
while leg BC increases in length to B'C.  Then angle BAC increases as 
well, to B'AC, where AB'C decreases accordingly from ABC.  This is 
quite clear.  But there's a question to be asked here:  what is the 
relation between the increase of length BC and the increase of 
angle BAC?  Is there some sort of relation connecting the two?  
Indeed, there is.  But we will find out exactly what this relationship 
is in a moment.

Now, think about another right triangle.  Only this time, place it in 
a circle, like this:

                y-axis
            ___|___
          .-   |    -.A
        .-     |    /|-.
       /       |   / |  \
      |        |  /  |   |
               | /   |    |
    ___________|/____|____|___
               |O    B        x-axis
               |
               |
               |
               |
               |

Here I have drawn in a Cartesian (x-y) coordinate system, and placed 
part of a circle with radius OA and center O at the origin.  Our right 
triangle is AOB with right angle at B.  Now, imagine what happens to 
AOB as we move A along the circumference of the circle, while point B 
is determined by always drawing a line parallel to the y-axis until it 
hits the x-axis.  The hypotenuse OA is the radius of the circle, so it 
remains constant.  If we move point A around the circle at a constant 
rate, angle AOB will also increase at a constant rate.  So what we're 
actually doing is increasing angle AOB while keeping the radius OA 
constant.  Now, we ask a similar question to the previous one:  what 
is the relationship between the increase in angle AOB and the 
increase/decrease of lengths AB and OB?

To answer this, mathematicians developed a pair of functions called 
sine and cosine.  We define the sine of angle AOB to be the ratio of 
AB to OA, or

                AB
     sin(AOB) = -- .
                OA

Since OA stays the same as we increase angle AOB, the sine of AOB 
depends only on the length of AB.  Similarly, we define the cosine of 
angle AOB to be the ratio of OB to OA, or

                OB
     cos(AOB) = -- ,
                OA

and again cos(AOB) depends only on OB.  So we take sines and cosines 
of angles, and the purpose is for expressing the relationship between 
the angles and the sides of a triangle.  Therefore, cos and sin are *
functions*, whose *domain* consists of angles, and the *range* is a 
real number.

Now, there is a third function, which we call the *tangent*, or tan.  
It is defined as simply

                sin(AOB)   AB/OA   AB
     tan(AOB) = -------- = ----- = -- .
                cos(AOB)   OB/OA   OB

So the tangent defines a relationship which is *independent* of the 
hypotenuse.  Notice that this answers our first question, which was 
how the angle BAC related to the increase in length of BC.  Since AC 
remains constant,
                BC
     tan(BAC) = --
                AC

and therefore BC and angle BAC are related through the tangent 
function.

One thing that students ask when first introduced to the tangent 
function is, what is the tangent of 90 degrees?  Well, it is *
undefined*.  The reason for this is if you look at the first triangle 
ABC, and we wanted to find the tangent of BAC, we can never make BC 
long enough to get angle BAC to become exactly 90 degrees.  Now, if 
you want to look at it through another view, what is the sine and 
cosine of 90 degrees?  Here we look at the triangle AOB in the circle.  
When AOB becomes 90 degrees, triangle AOB degenerates into a single 
line segment, OA, since B coincides with O.  Then sin(AOB) = 1, since 
AB = OA.  Also, cos(AOB) = 0, because OB has length 0.  Thus

     tan(AOB) = 1/0 = undefined!

since division by 0 is not allowed.

So there was my introduction to sine and cosine (and tangent).  
There's a lot more to be covered with these amazing and useful 
functions, but perhaps we'll get into that when you become 
sufficiently familiar with the concepts.

-Doctor Pete,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/