Archimedes' Limits to Value of Pi/Lemma 2

Lemma for Archimedes' Limits to Value of Pi

Let:

$\cot \phi = \dfrac p q$

Then:

$\csc \phi = \dfrac 1 q \cdot \sqrt {p^2 + q^2}$

Proof

We have:

\(\ds \csc^2 \phi - \cot^2 \phi\)

\(=\)

\(\ds 1\)

Difference of Squares of Cosecant and Cotangent

\(\ds \leadsto \ \ \)

\(\ds \csc \phi\)

\(=\)

\(\ds \sqrt {1 + \cot^2 \phi}\)

rearranging

Let:

$\cot \phi = \dfrac p q$

Then:

\(\ds \csc \phi\)

\(=\)

\(\ds \sqrt {1 + \paren {\dfrac p q}^2}\)

\(\ds \)

\(=\)

\(\ds \sqrt {\dfrac {q^2 + p^2} {q^2} }\)

$1 = \dfrac {q^2} {q^2}$

\(\ds \csc \phi\)

\(=\)

\(\ds \dfrac 1 q \cdot \sqrt {p^2 + q^2}\)

rearranging

$\Box$