Geometry vs. Trigonometry

Date: 07/14/97 at 04:49:10
From: Anonymous
Subject: Description

Is there any sort of easy to comprehend explanation of the difference
between trigonometry and geometry?  Are they the same thing but at 
different levels?

Date: 07/16/97 at 02:21:40
From: Doctor Sydney
Subject: Re: Description

Hello!

Geometry is a field of study that has its roots in Euclid's original 
work which attempted to look at things on a plane. He was interested 
in studying points, lines, planes, lengths, areas, angles and so on.  
This led to the study of more complex things like congruency, parallel 
lines, polygons, angles inside of polygons, circles, and other such 
stuff.  

Recently, mathematicians have begun to study geometry that is in other 
spaces besides the plane. For instance, some mathematicians specialize 
in studying geometry on a sphere. As you might imagine, notions of 
length, angles, and area have slightly different meanings on the 
sphere than they do on the plane. Geometry such as this geometry on 
the sphere is called non-Euclidean geometry. If you are interested in 
learning more about this, search our archives for old questions and 
answers on this subject. To search our archives, go to:

      http://mathforum.org/dr.math/   

From there, you can browse through the archives or conduct a search on 
specific topics, like non-Euclidean geometry. 

So, that is a brief description of geometry. How is it similar to or 
different from trigonometry? Well, trig and geometry are actually 
intricately related: trig helps us to solve geometrical problems. In 
its most basic form, trig helps us to determine the relationships 
between angles in a triangle and the lengths of the sides of the 
triangle. It does this by defining six functions: sine, cosine, 
tangent, cotangent, secant, and cosecant. 

For instance, suppose we have a right triangle ABC as follows:

        C
        |\
        | \
        |  \
        |   \a
       b|    \
        |     \
        |      \
        |       \
        A--------B
           c

In the diagram above, a, b, and c represent the lengths of sides CB, 
AC, and AB respecively. The angle CAB is a right angle. 

Call the angle CBA theta. Then the trig functions sine and cosine can 
help us to determine relationships between the angles of the triangle 
and the ratios of the sides of the triangle:

   sin(theta) = b/a
   cos(theta) = c/a
   tan(theta) = b/c

As you can see, these trig functions can be very helpful for solving 
problems of geometry.

More advanced trigonometry leaves classical geometry behind. For 
instance, we can use trig functions to help us understand complex 
numbers better (numbers that have the square root of -1 in them). 
In addition, because trig functions are functions, we can use them in 
varied settings - anywhere mathematics deals with functions, trig 
will be involved.  

This is just a brief overview of how geometry and trigonometry are 
related. I would recommend that you search the Dr. Math archives for 
more information on these topics. There I found some interesting 
information about trig.

You will find information on the history and uses of trig at:

  http://mathforum.org/dr.math/problems/pietruschka10.html      

You'll find an answer to the question "What is trig?" at:
 
  http://mathforum.org/dr.math/problems/aziebell.8.20.96.html    

I hope this helps to answer your question.

-Doctors Bernard and Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 07/18/97 at 06:07:08
From: Anonymous
Subject: Re: Description

An interesting phrase, "anywhere mathmetics deals with functions." 
Where doesn't mathematics deal with functions? Up to this point, I 
have been a math dunce. Function of what? I have currently enrolled in 
algebra 2. However, it will take much effort.

Date: 07/18/97 at 10:11:31
From: Doctor Sydney
Subject: Re: Description

Hello again!

You made an excellent point about functions. Many mathematicians say 
that functions are one of the uniting features of all the different 
fields of mathematics; the function is one of the most fundamental 
concepts in mathematics.  

So, in this way trig is used in a much wider variety of settings than 
is geometry. However, somehow saying that trig is used in a much wider 
variety of settings than geometry seems to be inaccurate - trig 
FUNCTIONS can be seen in many settings, but the "field" of trig itself 
really is not present in these varied settings except as a tool for 
the analysis of the trig functions being used.  

The intersections and domains of the various fields of math is a 
fascinating topic to study and your question was a very good and 
interesting one. You're not a math "dunce" if you took the time to 
write back and ask about such a mathematically-rich topic! 
Furthermore, the insight that functions are everywhere in math 
demonstrates that you are far from being a math dunce. I'm sure that 
if you work hard and have some confidence in yourself, algebra 2 will 
go just fine.

Thanks for writing back!  If you need help next year in algebra 2, be 
sure to check our archives of past questions and answers. We have a 
big algebra section.  Or, if you can't find help there, please do 
write back to us and we will try to help.  Good luck!

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

Date: 07/19/97 at 01:45:46
From: Anonymous
Subject: Re: description

Thanks so much for the encouragement. Although I have a doctorate from 
the mid-seventies, I have found it necessary to redeem my math skills 
for certification in secondary teaching (social studies). I find that 
I am facing a situation wherein I know I can do it, I've done it
before, but I have forgotten a lot of math. This brings the axiom 
"use it or lose it " to mind. I will keep in touch.

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