Real sine and cosine as real restrictions of complex trigonometric functions

For $z\in \mathbb{C}$, let $E(z)={\sum }_{n=0}^{\mathrm{\infty }}{z}^n\mathrm{/}n!$ be the complex exponential and set $s_{\mathbb{C}}(z)=(E(iz)-E(-iz))\mathrm{/}(2i)$ and $c_{\mathbb{C}}(z)=(E(iz)+E(-iz))\mathrm{/}2$, where $i^2=-1$. For $t\in \mathbb{R}$, define the real-valued functions $S(t)=\mathrm{Re}(s_{\mathbb{C}}(t))$ and $C(t)=\mathrm{Re}(c_{\mathbb{C}}(t))$, with $\mathrm{Re}(x+iy)=x$. These are the real restrictions of the complex sine and cosine; on real inputs they agree with the usual real sine and cosine.

The complex exponential by its power series

The complex exponential series converges absolutely for every complex argument

Complex sine and cosine: Complex sine, cosine, hyperbolic sine, and hyperbolic cosine from the complex exponential.

Real part of a complex number: Real and imaginary parts, complex conjugation, and modulus.

Agreement with real sine and cosine on real arguments: The exponential formulas, real restrictions, and trigonometric-hyperbolic dictionary over $\mathbb{C}$.

Complex sine and cosine power series and their real restrictions: Stein and Shakarchi, Complex Analysis, Chapter 1, §2.3, printed pp. 16–17 (PDF pp. 34–35).

Complex exponential and its convergence on all of $\mathbb{C}$: The complex exponential by its power series and The complex exponential series converges absolutely for every complex argument.