No I don't believe so. Here is the best solution I can give you:

Use Excel to create a spreadsheet.

First column numbers 1-90. I put "1' in block A1 then in A2 put "=1+1" and then copied and pasted down to 90

Second column convert from degrees to radians. I put "=pi()*A1/180" in B1, then copied like above.

Third column find sin of second column. I put "=sin(B1) in C1 then copied like above.

I then formatted column C to be a number to 15 decimal places.

In reviewing the outputs, the only angles that appeared to have rational values (ratio of integers) was 0, 30, 90.

0 and 90 don't make a triangle. Sin of 30 is rational, but the third side would then equal the hypotenuse times sin(60) which is not rational.

After 15 decimal places, they all appeared to just have zeros after that, suggesting they might be rational. But I believe this is just a limitation of Excel.

I would suggest doing the same on a calculator that you can switch between rational (exact) and decimal approximation answers. If for the sine of any integer angle X, you get an exact rational answer, you should then check the sine of its Complimentary Angle (90-X). If both answers are rational, you then have the 2 angles of a right triangle that would meet your requirement.

In writing this answer, I just realized that there was no stipulation that it had to be a right triangle. However, the above proof still works for the most part. If, as I suspect, the only rational value of sin is sin(30) = 1/2 (0,90 don't help as mentioned earlier), then 30-60-90 doesn't work and 30-30-120 doesn't work.

Hope that helps and makes sense.

Joe P.

03/25/19