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Re: Trigonometry challenge
Posted:
Feb 22, 2006 9:49 PM
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Dear Jimmy,
Thank you for your efforts to clarify whether rational trig is indeed
simpler than classical trig. Let me respond to your points below:
I am proposing that students eventually only learn rational trigonometry
when they study triangles. However in the near term, it will require a
(probably lengthy) transitional period while people start to appreciate
the power of the new way of thinking. In that period, angles and the
basic connections between the two theories will be taught. However, if
you look at a standard trig textbook, you will see a large number of
trig identities that currently students grapple with. I would dispense
with the majority of these. I would dispense with all that time spent on
graphing the trig functions, and on learning the special values
(tan(pi/3), arccos(-1/2) etc).
To understand basic circular motion, only a fraction of these standard
trig facts are needed. That's what I mean when I say that the two
subjects of triangles and circles are better studied separately.
Furthermore, it is quite appropriate that the circular functions are
introduced after complex numbers, and related to the complex exponential
function. This is the mathematically elegant way, and is conceptually
much simpler. However it is not part of what I consider rational
trigonometry.
You are quite right to point out that rational trig still relies on
square root functions. That's because quadratic equations are key to the
subject. However it is worth pointing out that if you start with a
geometric configuration in which all the points are defined over the
rational number field (ie fractions) then ALL of the calculations end up
with rational answers. So the example you asked me, spread of 1/3 and
1/5, do not come from a rational triangle, since the third spread had a
sqrt in it.
One of the papers I posted at the Authors Corner at http://wildegg.com
is called `The Wrong Trigonometry' and it clarifies just how complicated
the usual trig functions and their inverses are when viewed
mathematically. These are quite sophisticated special functions, and
there is no way a student without a calculator using classical trig
could compete with a student using rational trig for the majority of
trig problems. In comparison, the sqrt function is rather mild. If you
have found out ways to do the Survivor questions without a calculator,
my bet is that you have avoided calculating angles. Many experienced
high school maths teachers will know how to do this to an extent--which
is to say that they already know some rational trigonometry!
In the second problem the classical guy was at that point starting with
the lengths and angles. I would be happy to see your calculations re the
Survivor paper. Could you send them to me? If they illuminate the issue,
I would be happy to post them.
Cheers,
Norman Wildberger
Jimmy wrote:
Thank you for your candor.
It appears you are proposing students learn the rational trigonometry
theory as well as classical trigonometry or some parts thereof. That
is, students use the rational trig for solving only triangle-related
problems, and use the classical trig for solving circular or rotational
problems, or for converting from rational trig?s nonlinear units to
classical trig?s linear units (such as when working with vectors within
a linear system of measure).
1. You state one of the principle advantages of rational trigonometry
is that it easier for students to calculate rational answers, they being
more accurately expressed numerically. But one notices that rational
trig generally must express answers in terms of the square root function
which only gives the appearance of a rational result. When the function
is evaluated it generally produces an irrational number.
Classical trigonometry, on the other hand, can express its answers in
terms of the sine function and square root function which too gives the
appearance of a rational result. Only after the functions are evaluated
do they produce an irrational number.
Why are we making a distinction and claiming that rational trigonometry
produces rational results and is more accurate, when the claim can also
be made for classical trigonometry?
2. Unlike classical trigonometry, when one examines the calculations
required by rational trigonometry, it often requires the student to
solve simultaneous quadratic or linear equations and/or use the
quadratic formula. In contrast, classical trigonometry can achieve the
same rational result (as described above), but without the having to
solve the simultaneous quadratic or linear equations or deal with the
quadratic formula. How is it that rational trigonometry is easier?
I reworked the same four problems you present in your paper, ?Survivor:
the Trigonometry Challenge?, using more appropriate classical formulas,
and in all four problems the classical trigonometry calculations were
decidedly less and easier when expressed rationally. If calculators
were allowed on the island, the classical trigonometry would still
clearly win even after evaluating the functions. BTW, the second
problem requires the classical guy to calculate the angle to determine
its sine, when it was unnecessary.
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