A trig problem

Originally posted by blacknight1985
so it seems my manual calculation is right...😀

Any of you guys ever heard of a TI-83?....🙂

I've seen people knowing how to work with calculators, producing approximations. But noone yet producing an answer. Isn't there any?

Originally posted by FabianFnas
I've seen people knowing how to work with calculators, producing approximations. But noone yet producing an answer. Isn't there any?

Let me try to "show" there probably isn't:

arctan(x) = 1/(2x)
tan(arctan(x) = tan(1/(2x))
x = tan(1/(2x))

From Lindemann-Weierstrass we know that the tangent of any non-zero algebraic number is transcendental, so if x was algebraic then 1/2x is algebraic and therefore tan(1/(2x))=x should be transcendental, so we get a contradiction.

x must then be transcendental and we can easily check that it's not a multiple of Pi.

Originally posted by Palynka
Let me try to "show" there probably isn't:

arctan(x) = 1/(2x)
tan(arctan(x) = tan(1/(2x))
x = tan(1/(2x))

From Lindemann-Weierstrass we know that the tangent of any non-zero algebraic number is transcendental, so if x was algebraic then 1/2x is algebraic and therefore tan(1/(2x))=x should be transcendental, so we get a contradiction.

x must then be transcendental and we can easily check that it's not a multiple of Pi.

Good explanation!

So to find the solution of the problem, numerical methods can be our only tool? An approximation can be the only answer?

Originally posted by FabianFnas
Good explanation!

So to find the solution of the problem, numerical methods can be our only tool? An approximation can be the only answer?

I can't say. 😀

I only proved x is transcendental, but the answer could be a multiple of another transcendental number or a log or...

Although the nature of the tangent I would expect that unless it is a multiple of Pi, then probably numerical methods are the only tool.