Aristarchus's inequality

Aristarchus's inequality (after the Greek astronomer and mathematician Aristarchus of Samos; c. 310 – c. 230 BCE) is a law of trigonometry which states that if α and β are acute angles (i.e. between 0 and a right angle) and β < α then

Ptolemy used the first of these inequalities while constructing his table of chords.[1]

Contents

ProofEdit

The proof is a consequence of the more known inequalities  ,   and  .

Proof of the first inequalityEdit

Using these inequalities we can first prove that

 

We first note that the inequality is equivalent to   which itself can be rewritten as  

We now want show that

 

The second inequality is simply  . The first one is true because

 

Proof of the second inequalityEdit

Now we want to show the second inequality, i.e that:

 

We first note that due to the initial inequalities we have that:

 

Consequently, using that   in the previous equation (replacing   by  ) we obtain:

 

We conclude that

 

Notes and referencesEdit

  1. ^ Toomer, G. J. (1998), Ptolemy's Almagest, Princeton University Press, p. 54, ISBN 0-691-00260-6

External linksEdit