Proposition 6

Tag 0DOD, permanent. Cite it as [Bourbaki, Tag 0DOD].

Algebra, chapter IX, § 10. Angles

Let d be the isomorphism of $S^+/H$ onto $O^+$ defined in Prop. 3. For every element $\varphi$ of $S^+/H$ such that $t = t(\varphi)$ is finite, one has

(6)

s(d(\varphi)) = 2t/(1 - \delta t^2), \qquad c(d(\varphi)) = (1 + \delta t^2)/(1 - \delta t^2).

In fact $\varphi$ is the class mod. $H$ of the similitude $1 + t\omega$ , and the rotation $d(\varphi)$ is therefore $(1 + t\omega)^2/N(1 + t\omega) = (1 + \delta t^2 + 2t\omega)/(1 - \delta t^2)$ (Prop. 3), which proves (6).

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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