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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]
Proof
editThe proof is a consequence of the more widely known inequalities
- ,
- and
- .
Proof of the first inequality
editUsing 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 inequality
editNow 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
See also
editNotes and references
edit- ^ Toomer, G. J. (1998), Ptolemy's Almagest, Princeton University Press, p. 54, ISBN 0-691-00260-6
External links
edit- Leibowitz, Gerald M. "Hellenistic Astronomers and the Origins of Trigonometry" (PDF). Archived from the original (PDF) on 2011-09-27. Retrieved 2019-06-24.
- Proof of the First Inequality
- Proof of the Second Inequality