Contents

# Contents

## Idea

A trigonometric identity is (formally) a commutative diagram in the category of cartesian spaces and partial functions whose edges are labelled by rational functions (or sometimes algebraic functions) and trigonometric functions.

Slightly more precisely: each rational function $p(x_1, \ldots, x_n) \in \mathbb{R}(x_1, \ldots, x_n)$ is interpreted as a partial function $\mathbb{R}^n \hookleftarrow D \stackrel{f}{\to} \mathbb{R}^n$ where $D$ is the “natural domain” of $p$ (see rational function for more discussion); these are partial analytic functions. The basic trigonometric functions sine ($\sin$) and cosine ($\cos$) are (total) analytic functions $\mathbb{R} \to \mathbb{R}$. All of these may be interpreted as partial functions $\mathbb{R}^n \rightharpoonup \mathbb{R}^n$, and generate a class of functions by applying the monoidal category structure on the category of partial functions that is induced by the cartesian product on cartesian spaces. A trigonometric identity is then (formally) an equality of morphisms in the monoidal category thus generated.

Of course, this is complete overkill; category theorists are not oblivious to the fact that this is exactly the kind of description lampooned in Linderholm’s Mathematics Made Difficult. It’s just a formal way of covering bases. So let us add that in practice, a trigonometric identity usually involves functions obtained by substituting trigonometric functions into rational functions, or substituting rational linear (affine) functions into trigonometric functions: the class of functions considered is usually fairly limited in scope.

## Examples

Virtually all trigonometric identities can be seen as arising from suitable exponential function identities on complex numbers such as

• $\exp(i t) = \cos(t) + i\sin(t)$ (Euler's formula);

• $\exp(z + w) = \exp(z)\exp(w)$,

• $\widebar{\exp(z)} = \exp(\widebar{z})$.

These imply relations between the sine function and the cosine function as follows:

$\sin\left( \alpha + \beta \right) \;=\; \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta)$
$\cos(\alpha + \beta) \;=\; \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)$

which in turn implies

(1)$\sin(\alpha) \cos(\beta) \;=\; \tfrac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right)$

etc.