[[!redirects Sandbox > history]] [[!redirects Sandbox]] < [[nlab:Sandbox]] Forget about real analysis. If we are already using Archimedean ordered Artinian local $R$-algebras, we could continue and use the complexification $C = R[i]/(i^2 + 1)$ of the Archimedean ordered field $R$ instead. Synthetic complex analysis to define the complex exponential function on $C$, and the complex sine and cosine function straight into trigonometry. There is a ring homomorphism $h:R \to C$, and an element $z:C$ is **purely real** if it is in the image of $h$, and an element $z:C$ is **purely imaginary** if $-i z$ is in the image of $h$. There exist functions $\Re:C \to R$ and $\Im:C \to R$ such that for all real numbers $x \in R$, $\Re(h(x)) = x$ and $\Im(i h(x)) = x$, and for all complex numbers $z \in C$, $z = h(\Re(z)) + i h(\Im(z))$. Given a Archimedean ordered field $R$, whose elements we shall call **real numbers**, let $C = R[i]/(i^2 + 1)$ be the ring of **complex numbers**. An **exponential function** over $C$ is a function $\exp:C \to C$ such that for all Artinian local $C$-algebras $A$ with unique $C$-algebra homomorphisms $h:C \to A$, there is a function $\exp_A:A \to A$ such that $h(\exp(z)) = \exp_A(h(z))$ for all $z \in C$ and for all natural numbers $n \in \mathbb{N}$ and elements $\epsilon \in A$, $\epsilon^{n + 1} = 0$ implies that $$\exp_A(h(z) + \epsilon) = h(\exp(z)) \sum_{i = 0}^{n} \frac{1}{i!} \epsilon^i$$ The sine function on the complex numbers is defined as $$\sin(z) \coloneqq -i \frac{1}{2} (\exp(i z) - \exp(- i z))$$ and the cosine function on the complex numbers is defined as $$\cos(z) \coloneqq \frac{1}{2} (\exp(i z) + \exp(- i z))$$ The exponential, sine, cosine functions on the real numbers is defined as $$\exp(x) \coloneqq \Re(\exp(h(x)))$$ $$\sin(x) \coloneqq \Re(\sin(h(x)))$$ $$\cos(x) \coloneqq \Re(\cos(h(x)))$$ category: redirected to nlab