How many angles have integer measure in degrees and are interior angles of an integer-sided triangle?

Question

Determine how many possible angles $ \alpha $ such that

i) The measure of $ \alpha $, in degrees, is rational

ii) $ \alpha $ is an inner angle of some triangle with integer sides

Could someone give me a hint for this problem? (I don't know if he's easy or not)

I think the answer is 3. It's just not because he doesn't say that sine is rational, but what he said is equivalent because it shows that cosine is rational so sine is also

Answer 1

For all the side to be integer, the cosine of the angle need to be rational. The only Three values where the angle is an integer and the cosine is rational are

• $60^°$ equilateral triangles, but there are other examples;

• $90^°$ pytagorean triangles, such as the $3-4-5$ triangle;

• $120^°$ such as the $3-5-7$ triangle.

I don't have a formal proof of it and I'll be happy to see one. I used Excel to search for such triangles.

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Edit : Following the comment of @URL

Thank's @URL, I had never heard of Niven's theorem. For those who are interested, a proof can be found here.