why is cos(2/7) an irrational transcendental number

Question

for any integers p and q, p≠0, q≠0, cos(p/q) is an irrational transcendental number.

explain how to prove.

Answer 1

cos(2/7) if measured in radians

arccos(1/7) = about 1.281044625 radians by calculator. digits don't terminate or repeat. that makes it irrational

cos(2/7) measured in degrees

= arccos(2/7)=73.3984504 degrees

= .4077691689pi radians

the 1st two forms of the arccos never terminate and are irrational

even if the 3rd terminated it would still be irrational given the pi factor

Answer 2

and e^p/q and e^ip/q , p and q integers, p≠0, q≠0: are transcendental

Answer 3