Reformulating Exact Values without Trigonometric Components or Imaginary Numbers

[Zetison]

Homework Statement

Find exact values for z without any trigonometrical components or imaginary numbers
$$x = \frac{79}{60} + \frac{1}{30} \sqrt{6121} cos\left(\frac{1}{3}arccos(z)\right)$$
where
$$z = \frac{473419}{6121\sqrt{6121}}$$

Homework Equations

$$cos\left(\frac{1}{3}arccos(z)\right) = \frac{\left(z + \sqrt{z^2-1}\right)^{1/3}}{2} + \frac 1 {2\left(z+\sqrt{z^2-1}\right)^{1/3}}$$

The Attempt at a Solution

I pretty fast get imaginary numbers from the equation above. But they shall, as I know, cancel one way or the other. So this is what I get:

$$x = \frac{79}{60} + \frac{1}{30} \sqrt{6121} cos\left(\frac{1}{3}arccos(z)\right) = \frac{79}{60} + \frac{1}{30} \sqrt{6121} \frac{\left(z + \sqrt{z^2-1}\right)^{1/3}}{2} + \frac 1 {2\left(z+\sqrt{z^2-1}\right)^{1/3}}$$

$$x = \frac{79}{60} + \frac{1}{30} \sqrt{6121} \left(\frac{\left({ \frac{473419}{6121\sqrt{6121}} + \sqrt{ -\frac{5207760000}{229333309561}}}^{1/3}}{2} + \frac {1}{2\left({ \frac{473419}{6121\sqrt{6121}} + \sqrt{ -\frac{5207760000}{229333309561}}}\right)^{1/3}}\right)$$$$x = \frac{79}{60} + \frac{1}{60} \sqrt{6121} \left(\left({ \frac{473419}{6121\sqrt{6121}} + i \sqrt{\frac{5207760000}{229333309561}}}\right)^{1/3} + \frac 1 {\left({ \frac{473419}{6121\sqrt{6121}} + i \sqrt{\frac{5207760000}{229333309561}}}\right)^{1/3}}\right)$$$$x = \frac{79}{60} + \frac{1}{60} \left({473419 + 600 i \sqrt{14466}}\right)^{1/3} + \frac {6121}{60\left({473419 + 600 i \sqrt{14466}}\right)^{1/3}}$$

Let
$$u = 473419 + 600 i \sqrt{14466}$$

Then I have

$$x = \frac{79}{60} + \frac{1}{60} {u}^{1/3} + \frac {6121}{60{u}^{1/3}}$$

$$x = \frac{79}{60} + \frac{{u}^{2/3} + 6121}{ {60{u}^{1/3}}}$$

$$x = \frac{79}{60} + \frac{{u}^{4/3} + 6121{u}^{2/3}}{ {60u}}$$But from here, I can not see how I can get rid of the i-s...

 

[tiny-tim]

… But from here, I can not see how I can get rid of the i-s...

Hi Zetison!

(btw, to make LaTeX brackets that fit, type "\left(" and "\right)" )

Don't you se where that formula comes from?

If cosy = x, then (x + √(x² - 1))^1/3 = (cosy + isiny)^1/3 = cosy/3 + isiny/3, and 1/that = cosy/3 - isiny/3.

So you can either add and divide by 2, or you can just take the real part of one of them.

 

[Zetison]

But that doesn't give me an answer without trigonometrical components and imaginary numbers...

And btw:
[PLAIN]http://folk.ntnu.no/jonvegar/images/imaginary9.jpg

Thank you for LaTeX tip

 

[tiny-tim]

And btw:
[PLAIN]http://folk.ntnu.no/jonvegar/images/imaginary9.jpg[/QUOTE]

No, I meant cos(y/3) + isin(y/3)

 

[Mentallic]

I don't wish to hijack this thread but I have a query about something in here so if you can help solve my problem in my thread https://www.physicsforums.com/showthread.php?t=407917" I'd appreciate it.

 

[Zetison]

I don't think it is possible to solve this problem, all tho my grandfather say he has done it. (he can't remember how, but I guess he cheated...)

 

[Mentallic]

I don't think it is possible to solve this problem

Why do you say that?

 

[Zetison]

Because I have tried for several years now :) But if you think it can be solved, be my guest ;)

 

[Mentallic]

Because I have tried for several years now :) But if you think it can be solved, be my guest ;)

Are you referring to the exact same question? That is, to answer the given question without complex or trigonometric manipulations?

 

[Zetison]

Are you referring to the exact same question? That is, to answer the given question without complex or trigonometric manipulations?

Yes ;)

 

[Mentallic]

Yes ;)

Ahh, well then I'd be wondering why you are so curious in answering this problem without complex or trig. Why spend years on such an endeavour?

 

[Zetison]

Well, it's just for fun actually. My grandfather is claim that he has solved it, but he can't recall how. So I only got he's word for it. But I really want to solve the problem. He claimed that he solved it when he was at my age (21), so I got a pressure to so too ;)

 

[Mentallic]

Fair enough, but if you started trying to solve it years ago, then you must've been solving a slightly different or more general question?

Such as $$x=a\cdot \cos\left(\frac{\cos^{-1}\left(b\cdot a^{-3}\right)}{3}\right)$$

where b has some kind of special relationship to a which I can't quite see - if you take a look at the OP in this thread, $b=473419$ and $a=\sqrt{6121}$

 

[Zetison]

Well, the original problem is to solve for x here:

[PLAIN]http://folk.ntnu.no/jonvegar/images/flaggstangoppgaven.jpg

Which lead to:

[PLAIN]http://folk.ntnu.no/jonvegar/images/math.gif

This will lead to a 3.degree-equation. And the result is a answer with either trigonometrical components or imaginary numbers, which is a solution, but not the solution my grandfather got...

 

[Mentallic]

Oh I see, but what do you mean by:

And the result is a answer with either trigonometrical components or imaginary numbers, which is a solution, but not the solution my grandfather got...

If your grandfather got a different solution then he can't be right, because there is only one distinct solution to this problem, whichever way it is solved. If he claims he was able to find the exact solution by another method which didn't involve any use of complex or trig then I'd be quite sceptical about it, considering the cubic formula was derived using these methods.

 

[Zetison]

Oh I see, but what do you mean by:


If your grandfather got a different solution then he can't be right, because there is only one distinct solution to this problem, whichever way it is solved. If he claims he was able to find the exact solution by another method which didn't involve any use of complex or trig then I'd be quite sceptical about it, considering the cubic formula was derived using these methods.

Yes, I know. I'm a bit sceptical my self. But he is a really smart guy and has also study math like me. The problem is that he is no so old that he can't remember anything :P

But if there is no solution without trig or i-s, then we shuld be able to prove it?

 

[Mentallic]

Yes, I know. I'm a bit sceptical my self. But he is a really smart guy and has also study math like me. The problem is that he is no so old that he can't remember anything :P

Fermat also claimed he had a proof for Fermat's Last Theorem, but that claim has been disputed and regarded as false amongst the masses :tongue:

That's not to say that he didn't solve it, because I don't know of any existent side-steps you could possibly take in solving such a problem, but it's unlikely in my books.

But if there is no solution without trig or i-s, then we shuld be able to prove it?

Sorry I didn't quite catch that, do you mean we should be able to prove/disprove that one can't find a solution without the use of complex numbers or trig?

 

[Zetison]

Yes. My english sucks, I know :P