词条 | Aristarchus's inequality |
释义 |
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] ProofThe proof is a consequence of the more known inequalities , and . Proof of the first inequalityUsing 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 inequalityNow 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 references1. ^{{Citation|title=Ptolemy's Almagest|last1=Toomer|first1=G. J.|authorlink=Gerald J. Toomer|publisher=Princeton University Press|page= 54|year= 1998|ISBN =0-691-00260-6}} External links
2 : Trigonometry|Inequalities |
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