Is sin(10) Irrational? Proving the Irrationality of sin(10) in Degrees

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Homework Help Overview

The discussion revolves around proving the irrationality of \(\sin{10}\) degrees. Participants are exploring the implications of this statement and the methods to approach the proof.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Some participants express confusion about how to start the proof, while others suggest using proof by contradiction. There are discussions about assuming \(\sin(10)\) is rational to derive a contradiction. Additionally, one participant mentions finding a polynomial that \(\sin(10)\) satisfies and showing it has no rational roots.

Discussion Status

Participants are actively engaging with the problem, sharing their thoughts on how to approach the proof. Some guidance has been offered regarding the use of contradiction and polynomial roots, but there is no consensus on a specific method yet.

Contextual Notes

Participants note the lack of provided equations or specific methods in the original problem statement, which may affect their ability to formulate a clear approach.

SrEstroncio
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Homework Statement


Prove \sin{10}, in degrees, is irrational.


Homework Equations


None, got the problem as is.


The Attempt at a Solution


Im kinda lost.
 
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SrEstroncio said:

Homework Statement


Prove \sin{10}, in degrees, is irrational.


Homework Equations


None, got the problem as is.


The Attempt at a Solution


Im kinda lost.

Okay, are you familiar with proof by contradiction?

Start by contradicting the statement :

Suppose that sin(10) is irrational.
 
SrEstroncio said:

Homework Statement


Prove \sin{10}, in degrees, is irrational.


Homework Equations


None, got the problem as is.


The Attempt at a Solution


Im kinda lost.

Let x=sin(10). You'll want to find a polynomial x satisfies and show it has no rational root. Hint: sin(30)=1/2=sin(10+10+10).
 
I should suppose sin(10) is rational, if i am to contradict the statement, shouldn't i?
 
yes and that means that sin(10) can be represented by the ratio of two integers a and b sin(10) = a/b

next step...
 
SrEstroncio said:
I should suppose sin(10) is rational, if i am to contradict the statement, shouldn't i?

Zondrina's hint contains nothing of any real substance to start solving the problem. Mine does. I suggest you start trying it.
 

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