Rational and irrational series consisting of special denominators
Rational approximations to π and some other numbers
Rational values of the arccosine function
We give a short proof to characterize the cases when arccos(√r), the arccosine of the squareroot of a rational number r ∈ [0, 1], is a rational multiple of π: This happens exactly if r is an integer multiple of 1/4. The proof relies on the well-known recurrence relation for the Chebyshev polynomials of the first kind.
Remarks on the Irrationality and Transcendence of Certain Series.