Re: Trig Fomula

On Aug 9, 3:45 pm, quasi <qu...@xxxxxxxx> wrote:
On Thu, 09 Aug 2007 12:29:47 -0500, Robert Israel



<isr...@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote:
Deep <deepk...@xxxxxxxxx> writes:

Any comment and reference about the correctness of the following
assertion will be appreciated.

Assertion: If (1) is valid then (2) is also valid under the given
conditions.

Tan kD = (u/v)^(1/2) (1)

Tan D = (m/n)^(1/2) (2)

Conditions: 0 < D < pi/2; u, v, m, n are integers each > 1 and none is
a pefect square; (u, v) = 1, (m, n) = 1.
Prime k > 3.

Why don't you try a few small examples before making such assertions?

I second this complaint.

However, based on many similar prior posts, I think that Deep probably
lacks the skills and technology to test such conjectures.

In any case, here's a simple counterexample ...

Let u=3,v=2, and let D be the smallest positive real number such that
tan(5D)=sqrt((3/2)). Thus, (1) is satisfied.

It's straightforward to show that tan^2(D) is a root of the equation

2*x^5-115*x^4+520*x^3-530*x^2+110*x-3 = 0

but by the rational root test, the above equation has no rational
roots. Thus (2) fails.

quasi

Yes, you are right in your assessment. Having said so I request you to
kindly identify the error in the following analysis. I have used [1]
as a reference.
[1] Hobson,E.W, A Treatise on Plane Trigonometry(1928)pp111.

Tan kD = A/B (1)
A = N1tan D - N3(tanD)^3 + N5(tan D)^5 - ... (2)

B = 1 - N2(tanD)^2 + N4(tanD)^4 - ... (3)

where N1 = k, N2, N3, ... all are divisible by k
(1) can be regarded as an equation in tan D having k roots which are
tan D, tan(D+pi/k), ... , tan(D+(k-1)pi/k)
From (1) it is seen that if tan D = (u/v)^(1/2) then tan kD = (U/
V)^(1/2)
From this I assert that if tan kD = (U/V)^(1/2) then tan D = (u/
v)^(1/2)
According to you, as I understand,this assertion is wrong.
I would greatly appreciate if you would kindly tell why this assertion
is not correct.

With thanks and gratitude.
Deep


.

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