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Anonymous Sun 02 Jan 20:58:57 2022 No.14048602 View Reply Original Report

Quoted By:

Is sin(1) algebraic? (in radians, of course)

Anonymous

Anonymous Sun 02 Jan 2022 21:05:36 No.14048628 Report

Quoted By: >>14048638 >>14048922

i is algebraic, so e^i = cos1 + isin1 is transcendental by Lindemann-Weierstrass. If sin1 were algebraic, then so would cos1 = sqrt(1 - (sin 1)^2), and hence so would e^i. A contradiction. Therefore sin(1) is transcendental.
https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem

Anonymous

Anonymous Sun 02 Jan 2022 21:07:20 No.14048638 Report

Quoted By:

>>14048628
Thank you based anon.

Anonymous

Anonymous Sun 02 Jan 2022 22:29:00 No.14048922 Report

Quoted By: >>14048993

>>14048628
You are claiming that because e^i is transcendental that the coefficients of a+bi must also be transcendental, which I don't think follows from Lindemann-Weierstrass

Anonymous

Anonymous Sun 02 Jan 2022 22:51:09 No.14048993 Report

Quoted By: >>14049081

>>14048922
i is algebraic and nonzero, so e^i is transcendental.
If e^i = a+bi is transcendental, then at least one of a and b must be transcendental.
You agree so far?

Anonymous

Anonymous Sun 02 Jan 2022 23:16:42 No.14049081 Report

Quoted By: >>14049122

>>14048993
Why must one of a and b be transcendental?

If that is required then yes the rest of your proof follows naturally

Anonymous

Anonymous Sun 02 Jan 2022 23:30:32 No.14049122 Report

Quoted By:

>>14049081
Algebraic + algebraic = algebraic.

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