trigonometric identity


Equality and Equivalence




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,,x n)(x 1,,x n)p(x_1, \ldots, x_n) \in \mathbb{R}(x_1, \ldots, x_n) is interpreted as a partial function nDf n\mathbb{R}^n \hookleftarrow D \stackrel{f}{\to} \mathbb{R}^n where DD is the “natural domain” of pp (see rational function for more discussion); these are partial analytic functions. The basic trigonometric functions sine (sin\sin) and cosine (cos\cos) are (total) analytic functions \mathbb{R} \to \mathbb{R}. All of these may be interpreted as partial functions n n\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.


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

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

  • exp(z+w)=exp(z)exp(w)\exp(z + w) = \exp(z)\exp(w),

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

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

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

which in turn implies

(1)sin(α)cos(β)=12(sin(α+β)+sin(αβ)) \sin(\alpha) \cos(\beta) \;=\; \tfrac{1}{2} \left( \sin(\alpha + \beta) + \sin(\alpha - \beta) \right)



See also

Revised on November 26, 2017 10:32:48 by Urs Schreiber (