rigorous definition of trigonometric functions
Likewise define a sequence by the conditions and
Next, define a sequence of matrices as follows:
Using the recursion relations which define and , it may be shown that , More grenerally, using induction, this can be generalised to .
It is easy to check that the product of any two matrices of the form
is of the same form. Hence, for any integers and , the matrix will be of this form. We can therefore define functions and from rational numbers whose denominator is a power of two to real numbers by the following equation:
From the recursion relations, we may prove the following identities:
From the fact that and as , it follows that, if and are two sequences of rational numbers whose denominators are powers of two such that , then and . Therefore, we may define functions by the conditions that, for any convergent series of rational numbers whose denominators are powers of two,
By continuity, we see that these functions satisfy the angle addition identities.
Application. Let us use the definitions above to find and . Let for every positive integer . Then we need to find and . We use the matrix above defining and , and set :
As a result, . Similarly, .
|Title||rigorous definition of trigonometric functions|
|Date of creation||2013-03-22 16:22:11|
|Last modified on||2013-03-22 16:22:11|
|Last modified by||CWoo (3771)|