TITLE: A "Matrix Trigonometry"
QUESTION [6 upvotes]: $e^X$ for matrix $X$ is defined as an always-converging taylor series (provided that $X$ is a $n \times n $ complex matrix):
$$e^X:=\sum_{k=0}^{\infty}\frac{X^k}{k!} $$
A thought occurred to me that we might as well define $\cos(X):=\frac12(e^{iX}+e^{-iX})$ and $\sin(X):=\frac1{2i}(e^{iX}-e^{-iX})$. Now some obvious questions arise:

Is there a generalization for $2\pi$, the period of sine and cosine? Perhaps the best way to do so is to generalize the Euler's Identity $e^{2i\pi}=1$; Is there matrix $T$ such that $e^T=1$? This implies that $\cos (X+T)=\cos (X), \sin(X+T)=\sin (X)$.
A simple calculation shows that $\cos^2(X)+\sin^2(X)=I$. Can we generalize other trigonometric identities any further?
Can this concept be used further to derive some useful results? My senses tell me this should find its place in applied mathematics.

If there's any previous reference (which I think is likely) please inform me.

REPLY [3 votes]: The holomorphic functional calculus provides an algebra homomorphism from functions analytic in a neighbourhood of $\sigma(X)$ (the spectrum of $X$) to the closed subalgebra of the $n \times n$ matrices generated by $X$ (more generally, this is true in any complex Banach algebra). 
One way to do this is $$f(X) = \frac{1}{2\pi i} \oint_\Gamma f(z) (zI - X)^{-1}\ dz$$ where $\Gamma$ is a contour that surrounds $\sigma(X)$ in an open set where $f$ is analytic.
 All identities true for ordinary analytic functions carry over to functions of $X$.  
In particular, if we choose a locally constant $f$ whose values in a neighbourhood of $\sigma(X)$ are integer multiples of 
$2 \pi i$, then $f(X)$ satisfies $e^{f(X)} = I$, $\cos(X + f(X)) = \cos(X)$, etc.