/sci/ - Science & Math » Thread #12403022

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Anonymous Mon 30 Nov 12:52:56 2020 No.12403022 View Reply Original Report

Quoted By: >>12403037 >>12403102 >>12403179 >>12403180

Is it possible to represent $\sin \left( \sqrt{2} \right)$ in closed form using only elementary functions, without using the imaginary constant or other trig functions?

Anonymous

Anonymous Mon 30 Nov 2020 13:01:21 No.12403037 Report

Quoted By: >>12403043 >>12403047

>>12403022
No, because it is irrational.

/thread

Anonymous

Anonymous Mon 30 Nov 2020 13:02:20 No.12403040 Report

Quoted By: >>12403048

$\frac{ \left(e^{ \sqrt{2}i}-e^{- \sqrt{2}i} \right)i}{2}$

Anonymous

Anonymous Mon 30 Nov 2020 13:03:29 No.12403043 Report

Quoted By:

>>12403037
Nigga, expressing in closed form doesn't require it to be rational or even algebraic for that matter.

Anonymous

Anonymous Mon 30 Nov 2020 13:04:26 No.12403047 Report

Quoted By:

>>12403037
But $\sqrt{2}$ and it is elementary

Anonymous

Anonymous Mon 30 Nov 2020 13:04:43 No.12403048 Report

Quoted By:

>>12403040
>without using the imaginary constant

Anonymous

Anonymous Mon 30 Nov 2020 13:07:41 No.12403055 Report

Quoted By: >>12403058

$sin(x)$ is periodic so it's probably impossible to represent in closed form with only real numbers. Though I don't know of a formal proof for this.

Anonymous

Anonymous Mon 30 Nov 2020 13:10:55 No.12403058 Report

Quoted By: >>12403090

>>12403055
I only asked for the case sqrt(2), not a general representation

Anonymous

Anonymous Mon 30 Nov 2020 13:22:19 No.12403090 Report

Quoted By:

>>12403058
Also probably impossible

Anonymous

Anonymous Mon 30 Nov 2020 13:29:50 No.12403102 Report

Quoted By: >>12403120

>>12403022
>is it possible to represent an elementary function in closed form using only elementary functions

Anonymous

Anonymous Mon 30 Nov 2020 13:37:07 No.12403120 Report

Quoted By: >>12403136

>>12403102
>without using the imaginary constant or other trig functions
>or other trig functions

Anonymous

Anonymous Mon 30 Nov 2020 13:40:35 No.12403136 Report

Quoted By: >>12403141

>>12403120
>other

Anonymous

Anonymous Mon 30 Nov 2020 13:42:45 No.12403141 Report

Quoted By: >>12403178

>>12403136
Maybe I assumed the reader would be reasonable enough to know sin(sqrt(2)) = sin(sqrt(2)) was not the answer I was looking for.

Anonymous

Anonymous Mon 30 Nov 2020 13:52:17 No.12403169 Report

Quoted By:

If wolframalpha doesn't have such a representation no one else does

https://www.wolframalpha.com/input/?i=sin%28sqrt%282%29%29

Anonymous

Anonymous Mon 30 Nov 2020 13:56:20 No.12403178 Report

Quoted By:

>>12403141
That's a you problem.

Anonymous

Anonymous Mon 30 Nov 2020 13:56:32 No.12403179 Report

Quoted By: >>12403189

>>12403022
Not possible. See https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem

Anonymous

Anonymous Mon 30 Nov 2020 13:56:36 No.12403180 Report

Quoted By: >>12403185

>>12403022
It's already expressed in a form that you want

Anonymous

Anonymous Mon 30 Nov 2020 13:58:19 No.12403185 Report

Quoted By:

>>12403180
This.

Anonymous

Anonymous Mon 30 Nov 2020 13:59:54 No.12403189 Report

Quoted By:

>>12403179
This theorem is about algebraic representations. OP asked for closed form which is a weaker requirement.

https://en.wikipedia.org/wiki/Closed-form_expression#Comparison_of_different_classes_of_expressions

Anonymous

Anonymous Mon 30 Nov 2020 14:26:10 No.12403251 Report

Quoted By:

0.9877659459927355270691340720789426559067931295160371960387636855

If you need more digits than this you're a faggot

Anonymous

Anonymous Mon 30 Nov 2020 16:07:41 No.12403486 Report

Quoted By:

Why do you want this?

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