Let α be the real number that tan(α · π) = √2. Decide whether α must be a rational number.
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.
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.
advanced algebra answer
Theorem. the degree of $\displaystyle \tan \left(\frac{m}{n}\pi \right) $ over $\displaystyle \mathbb{Q}$ is either $\displaystyle \phi (n)$ or $\displaystyle \phi (n)/2$
(0 < m < n and gcd(m,n) = 1 )
This theorem can be used to show that $\displaystyle \tan \left(\frac{m}{n}\pi \right) =\sqrt{2}$ has no solution (m and n natural numbers)
is there an elementary trig identity proof of this statement?
I don't know