[Daily puzzle] is this number rational \ stacker news ~science

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1053 sats \ 10 comments \ @south_korea_ln 25 Oct 2024 science

After this week's hiatus, hope to pick up again more regularly.

Today's question:

Is the tangent of

1\degree

rational, i.e.

 $\tan(1\degree) = \frac{p}{q} ?$

with

p

and

q

integers.

Provide a proof for it.

Previous iteration: #732372

Thanks for posting this puzzle too: #734034

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542 sats \ 9 replies \ @Scroogey 25 Oct 2024

We can use the trigonometric addition formula

 $tan(a+b) = \frac{tan(a)+tan(b)}{1-tan(a)*tan(b)}$

in an inductive proof by contradiction:

Assumption: $tan(1°)$ is rational.
Step: if $tan(n)$ is rational then $tan(n+1)$ must also be rational, because the above formula is the ratio of two rational numbers.
But $tan(30°) = \frac{1}{\sqrt{3}}$ which is irrational, hence the assumption must be wrong.

11 sats \ 2 replies \ @south_korea_ln OP 25 Oct 2024

Nice job. This was supposedly an entrance exam question for the University of Kyoto in 2006.

160 sats \ 1 reply \ @Rothbardian_fanatic 25 Oct 2024

I understand the Kyodai and Todai exams are literal nightmares! They have people going to special exam schools for several years on end to get the score high enough to be accepted.

0 sats \ 0 replies \ @south_korea_ln OP 26 Oct 2024

Yeah looks pretty extreme. You really need to be trained on this to be able to solve this kind of problem with tight time constraints.

1 sat \ 5 replies \ @SimpleStacker 25 Oct 2024

Nice. Are you a mathematician, @Scroogey?

0 sats \ 4 replies \ @Scroogey 25 Oct 2024

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1 sat \ 3 replies \ @SimpleStacker 25 Oct 2024

Pulling out an inductive proof by contradiction, along with an obscure (to me) formula, is impressive for someone who doesn't do proofs on a regular basis :)

0 sats \ 2 replies \ @Scroogey 25 Oct 2024

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1 sat \ 1 reply \ @SimpleStacker 25 Oct 2024

Proof by contradiction was my favorite method in college. I think it's easier because you know where to start and you just move forward. So even to prove that things are true, I'd just assume the converse and show a contradiction.

Because a non-contradiction based proof seems harder. You know where you want to go, but you don't know where to start.

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