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

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SUMMARY

The discussion centers on proving that sin(10 degrees) is irrational. Participants suggest using proof by contradiction, starting with the assumption that sin(10) is rational, represented as a ratio of two integers. A key hint involves utilizing the identity sin(30) = 1/2 to derive a polynomial that sin(10) satisfies, ultimately demonstrating that it has no rational roots. This approach is essential for establishing the irrationality of sin(10).

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  • Understanding of trigonometric functions, specifically sine.
  • Familiarity with proof by contradiction techniques.
  • Knowledge of polynomial equations and rational roots.
  • Basic understanding of angle addition formulas in trigonometry.
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  • Study proof by contradiction in mathematical contexts.
  • Learn about polynomial equations and the Rational Root Theorem.
  • Explore trigonometric identities, particularly angle addition formulas.
  • Investigate the properties of irrational numbers in trigonometry.
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Mathematics students, educators, and anyone interested in advanced trigonometric proofs and the properties of irrational numbers.

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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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