Showing that tan(1) is irrational

[Mr Davis 97]

Homework Statement

Prove that $\tan (1^\circ)$ is irrational.

Homework Equations

The Attempt at a Solution

Suppose for contradiction that $\tan (1^\circ)$ is rational. We claim that this implies that $\tan (n^\circ)$ is rational. Here is the proof by induction: We know by supposition that the base case holds. So, suppose that $\tan (n^\circ)$ is rational. Then $\displaystyle \tan (n^\circ + 1^\circ) = \frac{\tan(n^\circ) + \tan(1^\circ)}{1-\tan(n^\circ)\tan(1^\circ)}$, and this is the ratio of two rational numbers, and so is rational. So by mathematical induction $\tan (n^\circ)$ is rational.

However, this implies that $\tan(30^\circ) = \sqrt{3}$ is rational, which is a contradiction.

 

[Orodruin]

Do you have a question?

 

[Mr Davis 97]

Do you have a question?

I guess I was just attempting a solution. Seems like it's correct though

 

[haruspex]

tan(30^∘)=√3

Umm, no.

 

[RPinPA]

Summarizing, if $\tan(1)$ is rational then it follows by induction that $\tan(n)$ is rational for all $n \in N$. An inductive proof by contradiction is a structure I haven't seen a lot, in fact I can't offhand recall such a proof. But it certainly seems valid to me.

The only issue is, @haruspex pointed out, is that $30^\circ$ is not the angle you want to use in your final line of argument.