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Curious Composition a Matter of Degrees

Date: 02/18/2011 at 22:12:31
From: Mike
Subject: Nomenclature dealing with trigonometry

Dr. Math,

I have found that the cosine of sine of tangent of theta in degrees is
equal to approximately one; that is,

   cos(sin(tan([?][?]))) ~= 1

Actually, I wrote a simple program that inputs multiple random values for
theta, and it keeps coming out with an answer of about 1.

I don't understand why these trigonometric actions seem to yield 1. I'm
wondering not only the reasoning behind this, but also if there is a
specific name for this pattern, kind of like a trigonometric version of
the Fibonacci sequence or Golden ratio.

From looking at this site a couple of times, I realize this may be an
unorthodox question, but I can tell you are knowledgeable in answering, as
well as clear in your explanations -- and neither my teacher nor any
"Google-ing" was sufficient to learn if this phenomenon had any name to 
it -- so I figured I'd give this a shot.


Date: 02/18/2011 at 23:04:25
From: Doctor Peterson
Subject: Re: Nomenclature dealing with trigonometry

Hi, Mike.

Actually, it's very simple: you're using the trig functions with degree
inputs. You'd get very different results using radians (which is what most
programs will do by default -- so I'm curious what language you were using).

Take it step by step.

First, tan(theta) rises from 0 to infinity as theta goes from 0 to 90
degrees. But tan(89) is only 57, so numerically it isn't rising very fast.

Second, the sine of any angle is less than 1. As its input rises from 0 to
57 degrees (that is, theta going from 0 to 89 degrees), the sine rises
from 0 to about 0.8. But as the input continues on to infinity, the sine
will just oscillate infinitely many times between -1 and 1. So if you
tried exclusively whole degrees for theta, less than 90 degrees, the sine
would only rise to 0.8.

Now you are taking the cosine of that! What is the cosine of an angle that
is less than 1 degree? It's very close to 1. That's all you're seeing: the
cosine of very small angles. If you let theta get up to about 80 degrees,
you'll see the output of your function start to drop as the input to the
cosine passes 5 or 10 degrees.

So there's really nothing very interesting about this -- but it was very
interesting discovering why it is not as interesting as it looked!

- Doctor Peterson, The Math Forum

Date: 03/16/2011 at 23:31:58
From: Mike
Subject: Thank you (Nomenclature dealing with trigonometry)

Thank you very much for your response.

I actually made the program using the TI-83 calculator; and after seeing
your explanation, I now can see why it is so simple.

Upon closer examination of the graph of the equation, I notice a "normal"
but "highly compressed" or "shrunken" sinusoidal curve with a max of 1,
and min of ~0.9998476592.

Again, I appreciate your help.
Associated Topics:
High School Trigonometry

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