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. BEST
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