Exercise 19, § 1

One knows (A, IX, §10, exerc. 2) that in the complex plane $\mathbf{C}^2$ the group $A$ of angles of directed lines is isomorphic to the orthogonal group $\mathbf{O}_2(\mathbf{C})$, the canonical isomorphism associating the angle $\theta$ with rotation through the angle $\theta$; one transports the topology of $\mathbf{O}_2(\mathbf{C})$ (considered as a subspace of the space $\mathbf{M}_2(\mathbf{C})$ of matrices of order 2 over $\mathbf{C}$) to the group $A$ by this isomorphism, so making $A$ a locally compact topological group;

further, the map $\theta \mapsto \cos \theta + i \sin \theta$ is an isomorphism of the topological group A onto the multiplicative group $\mathbf{C}^*$ of complex numbers $\neq 0$, the inverse isomorphism being defined by the formulae $\cos \theta = \frac{1}{2}(z + 1/z)$, $\sin \theta = \frac{1}{2i}(z - 1/z)$. Deduce from these relations that every continuous homomorphism $z \mapsto \varphi(z)$ of the additive group $\mathbf{C}$ onto A such that the complex functions $\cos \varphi(z)$, $\sin \varphi(z)$) are differentiable on $\mathbf{C}$, is defined by the relations $\cos \varphi(z) = \cos az$, $\sin \varphi(z) = \sin az$ ($a$ a complex number).