Re: how to get the angle from the cosine, etc.
From: Stan Brown (the_stan_brown_at_fastmail.fm)
Date: 10/06/04
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Date: Wed, 6 Oct 2004 07:36:50 -0400
"Sean Hunt" <seanstewarthunt@hotmail.com> wrote in sci.math:
> I'm actually looking for a way to get the angle (in radians), given
>that you already have the sine and cosine of the angle. I'm trying to
>derive an equation to go back and forth rather than a look-up process,
What do you mean by "an equation"? I'm guessing that you mean some
sort of polynomial, or at worst something involving only powers,
multiplication/division, and addition/subtraction. No such equation
can exist, because cos( ) is a transcendental function and so is its
inverse.
The only finite equation that is an answer to your question is
angle = arccos(x)
where x is the cosine. (arccos is sometimes written as cos with a
superscript -- not exponent -- of -1.)
What we are forced to do is to _define_ the inverse cosine as a new
function. But since cosine is a many-to-one function, inverse cosine
would be a one-to-many function Therefore we pick an interval and
say that arccos(x) is always between 0 and pi. This has the good
effect that arccos( ) is now a function, but it does mean that we
can't write arccos(cos(theta)) = theta.
There are some issues with that. For instance, if the cosine is 1/2,
you might be tempted to say that the angle is pi/3. And indeed
arccos(1/2) = pi/3. But many other angles also have a cosine equal
to 1/2. The original angle could also be 5pi/3, or 2pi + pi/3, etc.
If you also know the sine, then you have a unique solution within
the interval [0, 2pi), but overall there are still an infinite
number of solutions.
> On the surface it looks like a fairly easy correlation, but I
>haven't been able to calculate the angle given sine and cosine. Now
>I'm beginning to think that this might be a calculus problem rather
>than a trigenometry problem.
Well, if you're willing to accept an infinite series, the calculus
can help. But no closed-form solution exists.
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com
Fortunately, I live in the United States of America, where we are
gradually coming to understand that nothing we do is ever our
fault, especially if it is really stupid. --Dave Barry
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