Aristarchus's inequality

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

: $\frac < \frac < \frac.$

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

: $\frac < \frac.$

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
* Posidonius

Notes and references

External links

*

Proof of the First Inequality

Proof of the Second Inequality

Trigonometry
Inequalities

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