91Ó°ÊÓ

Verify. $$\sqrt{5+2 \sqrt{5}}-\sqrt{5-2 \sqrt{5}}=\sqrt{10-2 \sqrt{5}}$$ Also, show that none of these three nested radicals is in \(Q(\sqrt{5}) .\) This is another example of nonuniqueness of the standard form.

Express sin \(72^{\circ}\) as nested radicals in standard form. Check by computing decimal equivalents with a calculator.

\text { Same problem for } \cos 36^{\circ}, \sin 36^{\circ}.

(Theorem of three reflections). (a) Given three lines \(a, b, c\) through a point \(\mathrm{O},\) show that there exists a unique fourth line \(d\) such that $$ \sigma_{c} \sigma_{b} \sigma_{a}=\sigma_{d} $$ where \(\sigma\) denotes the reflection in a given line. Hint: Let \(A\) be a point of \(a\), and take \(d\) to be the perpendicular bisector of \(A C,\) where \(C=\sigma_{c} \sigma_{b}(A) .\) (see Proposition 41.2 for an analogous result in hyperbolic geometry.) (b) Given three lines \(a, b, c\) perpendicular to a line \(l,\) show that there exists a unique fourth line \(d\) such that \(\sigma_{c} \sigma_{b} \sigma_{a}=\sigma_{d}\)

Find \(\cos 24^{\circ}, \sin 24^{\circ}, \cos 12^{\circ}, \sin 12^{\circ},\) and the side of the regular 15 -sided polygon inscribed in the unit circle. Express in standard form, and check decimal equivalents with the calculator.

Find the side of a regular pentagon circumscribed around a unit circle.

Given a regular pentagon of side \(1,\) find the distance from the center to a vertex, in standard form.