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Re: Spherical Law of Cosines
From: Trevor Kenchington
Date: 2002 Oct 27, 21:11 -0400

Herbert Prinz wrote:

> While removing Dan Allen's confusion about the cosine theorem, Trevor
> Kenchington adds some of his own:
>
>
>>If it were true that:
>>
>>cos(c) = sin(a)*sin(b) + cos(a)*cos(b)*cos(ab)
>>
>>and:
>>
>>cos(c) = cos(a)*cos(b) + sin(a)*sin(b)*cos(ab)
>>
>>then it would necessarily be true that:
>>
>>sin(a)*sin(b) + cos(a)*cos(b)*cos(ab) = cos(a)*cos(b) +
>>sin(a)*sin(b)*cos(ab)
>>
>>since both are equal to cos(c). And so we would have to suppose that
>>sin(a)=cos(a), which is obviously absurd.
>>
>>
>
> Trevor is meaning to say
>
>      sin(a) = cos(b)

Just for the record, I meant nothing of the kind and did mean exactly
what I wrote.

Herbert has, however, pointed out that the two equations for cos(c)
could be simultaneously true if sin(a) was equal to cos(b), as well as
if sin(a) was equal to cos(a). I had missed the one while noting the
other. He has also pointed out that sin(a) _is_ equal to cos(b) for a
limited set of values of a and b. But then again, sin(a) is also equal
to cos(a), provided that a is 45 degrees.

Clearly, what I meant to say (but failed to do explicitly) was that the
two equations for cos(c) cannot both be universally true over all
possible values of a and b. On that, I think Herbert and I are in full
agreement.

Trevor Kenchington

--
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