# Thread: Decide whether α must be a rational number. HOW!?

1. ## Decide whether α must be a rational number. HOW!?

Let α be the real number that tan(α · π) = √2. Decide whether α must be a rational number.

2. ## Re: Decide whether α must be a rational number. HOW!?

For what values of x is tan(x) equal to $\sqrt{2}$?

3. ## Re: Decide whether α must be a rational number. HOW!?

I do not know

4. ## Re: Decide whether α must be a rational number. HOW!?

something about 55 degrees, it seems

5. ## Re: Decide whether α must be a rational number. HOW!?

Since this is not a "triangle" problem, you at not taking the tangent of any angle- you are dealing with the function f(x)= tan(x). And in that function x is a number, not an angle at all, so is not measured in degrees. Technically, it is not measured in radians either (it isn't "measured" at all) but the definitions of the sine, cosine, tangent functions (have you seen the "circle" definition of the trig functions?) you can get the values by using the "radian" setting on you calculator.

6. ## Re: Decide whether α must be a rational number. HOW!?

I've seen the "circle" definition of the trig functions, but how I can do that

7. ## Re: Decide whether α must be a rational number. HOW!?

How did you get "about 55 degrees"? Using a calculator? Just make sure you calculator is set to "radian" mode rather than "degree" mode.

8. ## Re: Decide whether α must be a rational number. HOW!?

from the trigonometric tables. √2 there is between 54 and 55 degrees. α · π=(54;55), so α belongs (54/π ; 55/π). And we know, beetwen 2 irrationals numbers there is one irrational number.

9. ## Re: Decide whether α must be a rational number. HOW!?

Originally Posted by TobiWan
And we know, beetwen 2 irrationals numbers there is one irrational number.
There are a lot more than one! And there's a bunch of rational numbers as well.

-Dan

10. ## Re: Decide whether α must be a rational number. HOW!?

do you know how to do it?

11. ## Re: Decide whether α must be a rational number. HOW!?

If I did I'd have posted a solution. Or at least some kind of hint.

-Dan

12. ## Re: Decide whether α must be a rational number. HOW!?

I have no clue on this one as well. If it was tan(α · π) = √3 it would be a piece of cake.

To the OP: you posted this in trigonometry - are you a high school student? If so, it makes me think the problem is misstated. Please verify that it's correctly written. Thanks.

13. ## Re: Decide whether α must be a rational number. HOW!?

of course correctly, Let α be a real number that tan(α · π) = √2. Decide whether α has to be a rational.

14. ## Re: Decide whether α must be a rational number. HOW!?

Theorem. the degree of $\tan \left(\frac{m}{n}\pi \right)$ over $\mathbb{Q}$ is either $\phi (n)$ or $\phi (n)/2$
(0 < m < n and gcd(m,n) = 1 )

This theorem can be used to show that $\tan \left(\frac{m}{n}\pi \right) =\sqrt{2}$ has no solution (m and n natural numbers)

is there an elementary trig identity proof of this statement?
I don't know

15. ## Re: Decide whether α must be a rational number. HOW!?

can I ask a link for this theorem? I don't understand it

Page 1 of 2 12 Last