Aristarchus's inequality explained

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

(\sin\alpha)/(\sin\beta) < (\alpha)/(\beta) < (\tan\alpha)/(\tan\beta) .

Ptolemy 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

(\sin(\alpha))/(\sin(\beta)) < (\alpha)/(\beta) .

We first note that the inequality is equivalent to

(\sin(\alpha))/(\alpha) < (\sin(\beta))/(\beta) which itself can be rewritten as(\sin(\alpha) - \sin(\beta))/(\alpha - \beta) < (\sin(\beta))/(\beta) .

We now want show that

(\sin(\alpha) - \sin(\beta))/(\alpha - \beta) <\cos(\beta) < (\sin(\beta))/(\beta) .

The second inequality is simply

\beta<\tan\beta . The first one is true because(\sin(\alpha) - \sin(\beta))/(\alpha - \beta) = (2 ⋅ \sin\left( (\alpha - \beta)/(2 \right)\cos\left( (\alpha + \beta)/(2 \right))))/(\alpha - \beta) < (2 ⋅ \left( (\alpha - \beta)/(2 \right)) ⋅ \cos(\beta))/(\alpha - \beta) = \cos(\beta).

Proof of the second inequality

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

(\alpha)/(< \beta) (\tan(\alpha))/(\tan(\beta)) .

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

\beta<\tan(\beta) = (\sin(\beta))/(< \cos(\beta)) (\sin(\beta))/(\cos(\alpha))

Consequently, using that

0 <\alpha - \beta<\alpha in the previous equation (replacing \beta by \alpha - \beta<\alpha ) we obtain:{\alpha - \beta}<{ (\sin(\alpha - \beta))/(\cos(\alpha)) }=\tan(\alpha)\cos(\beta)-\sin(\beta).We conclude that(\alpha)/(= \beta) (\alpha - \beta)/(\beta) +1 < (\tan(\alpha)\cos(\beta) - \sin(\beta))/(\sin(\beta)) +1 = (\tan(\alpha))/(\tan(\beta)) .

See also

• Aristarchus of Samos
• Eratosthenes
• Posidonius

External links

• Web site: Hellenistic Astronomers and the Origins of Trigonometry . Gerald M. . Leibowitz . 2019-06-24 . 2011-09-27 . https://web.archive.org/web/20110927130936/https://www2.math.uconn.edu/~leibowitz/math2720s11/Greek_Trig.pdf . dead .
• Proof of the First Inequality
• Proof of the Second Inequality