Need a bit of help: Sin[theta] is rational iif theta=30 deg.

[daon]

This one has got me scratching my head. Showing that if \(\displaystyle \theta\) = 30, then \(\displaystyle Sin\theta\) is rational is obvious, but how to show that If \(\displaystyle Sin \theta\) is rational then \(\displaystyle \theta\) =30 is not (to me, anyway).

Ideas:

Should I start with \(\displaystyle Sin \theta \,\, = \frac{a}{b}\) for a,b in Z, and attempt to get a=1, b=2, so that I can say theta=30 deg?

Or would contradiction be the right route here?

-Daon

 

[pka]

Please reread your posting.
Is that really what you mean to be saying?
If so, what are the conditions on theta?
I am not sure that you really mean what you wrote.

 

[daon]

DOH! I forgot to mention that theta is rational and between 0 and pi/2, exclusive. Sorry, I've been reading math all day, and I'm really worn down!

 

[pka]

Can you give us some clues from what you might have been studying?
What topics? What context does the from?

 

[daon]

We have just finnished going over the existance of irrational numbers and proving things such as \(\displaystyle \sqrt{2} \,\, \notin \,\, \L \mathbb{Q}\), the Archimedean Property, 1/n principle. We assume to know all "high-school algebra" properties of the real numbers.

I hope thats enough information. Thanks.

 

[pka]

Sorry that I could not answer before this.
The only way that I have ever seen is with continued fractions:
http://mathworld.wolfram.com/Sine.html (scroll well down the page).

I spent some time to see if I could do it with power series, a topic I though you might be able to use. But I had no luck. I would be interested to see what you were expected to do on this problem.