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Aristarchus's inequality (after the Greek
astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either o ...
and
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
Aristarchus of Samos; c. 310 – c. 230 BCE) is a law of
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
which states that if ''α'' and ''β'' are
acute angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles a ...
s (i.e. between 0 and a right angle) and ''β'' < ''α'' then : \frac < \frac < \frac.
Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of import ...
used the first of these inequalities while constructing his table of chords.


Proof

The proof is a consequence of the more widely known inequalities : 0<\sin(\alpha)<\alpha<\tan(\alpha) , : 0<\sin(\beta)<\sin(\alpha)<1 and : 1>\cos(\beta)>\cos(\alpha)>0.


Proof of the first inequality

Using these inequalities we can first prove that : \frac < \frac. We first note that the inequality is equivalent to :\frac < \frac which itself can be rewritten as :\frac < \frac. We now want show that :\frac<\cos(\beta) < \frac. The second inequality is simply \beta<\tan\beta. The first one is true because : \frac = \frac < \frac = \cos(\beta).


Proof of the second inequality

Now we want to show the second inequality, i.e. that: : \frac <\frac. We first note that due to the initial inequalities we have that: : \beta<\tan(\beta)=\frac<\frac Consequently, using that 0<\alpha-\beta<\alpha in the previous equation (replacing \beta by \alpha-\beta<\alpha ) we obtain: : <=\tan(\alpha)\cos(\beta)-\sin(\beta). We conclude that : \frac=\frac+1< \frac+1 = \frac.


See also

* Aristarchus of Samos *
Eratosthenes Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ;  – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexand ...
*
Posidonius Posidonius (; grc-gre, wikt:Ποσειδώνιος, Ποσειδώνιος , "of Poseidon") "of Apamea (Syria), Apameia" (ὁ Ἀπαμεύς) or "of Rhodes" (ὁ Ῥόδιος) (), was a Greeks, Greek politician, astronomer, astrologer, geog ...


Notes and references


External links

*
Proof of the First Inequality

Proof of the Second Inequality
Trigonometry Inequalities {{elementary-geometry-stub