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How does cos(theta) produce irrational values for 60 deg <= theta <= 90 deg?

That is what I knew. For example- cos 75 = (√2(√3 – 1))/4

cosine is a continuous function. Rationals are dense in reals etc. eg, arccos(1/3)

Yes, arccos(1/3) = 70.5287794 degrees

The important point is that

cos(arcos(1/3)) = cos(70.5287794... degrees) = 1/3

that is a rational value.