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### 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
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

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

-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!

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
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.

BEST

Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry
High School Non-Euclidean Geometry
High School Trigonometry

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