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Aristarchus's inequality (after the Greek
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
The proof is a consequence of the more widely known inequalities
- ,
- and
- .
Proof of the first inequality
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 inequality
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
See also
Notes and references
External links