construction of tangent function from addition formula
It is possible to define trigonometric functions rigorously using a procedure based upon the addition formula for the tangent function. The idea is to first note a few purely algebraic facts and then use these to show that a certain limiting process converges to a function which satisfies the properties of the tangent function, from which the remaining trigonometric functions may be defined by purely algebraic operations.
Define the algebraic functions and and as follows:
Calculemus! On the one hand,
On the other hand,
These quantities are equal. ∎
For all , we have .
Since , we have
By the binomial identity, the right-hand side equals . Taking square roots of both sides,
Subtracting from both sides,
Dividing by on both sides,
or . ∎
Let be a positive real number. Then the sequence
converges to .
Having made these preliminary observations, we may now begin making the construction of the trigonometric function. We begin by defining the tangent function for successive bisections of a right angle.
Define the sequence as follows:
By the forgoing theorem, this is a decreasing sequence which tends to zero. These will be the values of the tangent function at successive bisections of the right angle. We now use our function to construct other values of the tangent function.
Define the sequence by the following recursions:
There is a subtlety involved in this definition (which is why we did not specify the range of and ). Since is only well-defined when , we do not know that is well defined for all and . In particular, if it should happen that is well defined for some and but that , then will be undefined for all .
Suppose that , , and are all well-defined. Then .
We proceed by induction on . If , then is defined to be , and it is easy to see that .
Suppose, then, that we know that . By definition, and, by theorem 2, we have
If , then is well-defined, , and .
We shall proceed by induction on . To begin, we note that because . Also note that, if , then is the only value for which the condition happens to be satisfied. The condition is not relevant when .
Suppose that we know that, for a certain , when , then is well-defined and . We will now make an induction on to show that if , then is well-defined, and . When , we have, by definition, so the quantity is defined and it is obvious that and .
Suppose we know that, for some number , we find that is well-defined, strictly less than and equals . By theorem 4, since and , we may conclude that and , which implies that , so is well-defined. By definition, , so . Recall that . By theorem 1, we have
By theorem 2, equals which, in turn, by our induction hypothesis, equals . Combining the results of this paragraph, we may conclude that:
which means that is defined and equals .
Moreover, by definition,
Since, as we have just shown, and, as we already know, , we have , so is well-defined. Furthermore, we may evaluate this quantity using theorem 1:
Hence, we have .
|Title||construction of tangent function from addition formula|
|Date of creation||2013-03-22 16:58:39|
|Last modified on||2013-03-22 16:58:39|
|Last modified by||rspuzio (6075)|