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ntaraxq

ntaraxq

Answered question

2022-07-06

cos cos ( A π ) where A is an irrational algebraic number

Answer & Explanation

Pranav Greer

Pranav Greer

Beginner2022-07-07Added 13 answers

Let A be algebraic irrational. We claim that cos ( A π ) transcendental. Assume not: cos ( A π ) is algebraic. Then sin ( A π ) = ± 1 cos 2 ( A π ) is algebraic. Then e i A π = cos ( A π ) + i sin ( A π ) is algebraic. Also e i A π 0 and e i A π 1. Now 2 / A is algebraic irrational. Note
( e i A π ) ( 2 / A ) = e 2 π i = 1
is algebraic. This contradicts Gelfond-Schneider.

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