Trigonometric rational function
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In mathematics, a trigonometric rational function is a rational function in the functions sin θ and cos θ. Equivalently, it is a ratio of trigonometric polynomials. The simplest examples are the tangent and cotangent functions.
This article is concerned with the discussion of the limiting behaviour of such functions, which is an issue when the denominator becomes 0 at a given value of θ.
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A statement of Beck's rule
In the case θ = 0, there is Beck's rule: simply ignore the functions themselves. Stating this more accurately, in a case of a non-trivial limit (indeterminate form) of such a rational function and a simple (not repeated) zero of the denominator, it is permissible to replace sin nθ by n, and cos nθ directly by its value 1. What remains is the limit. This law was developed in texts by Richard Sharpless Beck. A proof can be based on L'Hôpital's rule.
Example
For example:
- <math>\lim_{x \to 0}\frac{\sin{7x}}{\tan{11x}} = \frac{7}{11}. <math>
General theory
To develop some general theory, assume given trigonometric polynomials
- P(sinθ, cosθ) and Q(sinθ, cosθ)
with real coefficients, and write
- R(θ) = P/Q.
In considering limits of R, it is no real loss of generality to consider limits as
- θ → 0.
That's because the addition theorem for sin and cos may be applied: if we require instead a limit as
- θ → α,
a change of variable and new choice of R is covered by that case.
Then it is a help to the theory to rely on a substitution of rational functions of another variable t for sinθ and cosθ. This is a classical operation, traditionally known as the use of tan half-angle formulae. Geometrically, we can say this: for any point on the unit circle draw the line passing through it and the point (−1,0) on the circle. If its gradient is t, then the equation for the intersection of the line and circle is a quadratic equation involving t. Since we know one solution is at (−1,0), the other one on solving will be (cosθ, sinθ) written as rational functions of t.
The relation to the limit problem near (1,0) (that is, θ = 0) is therefore that we have now to evaluate a limit of R*(t) (which is R having the substitution made, and written as a rational function P*/Q* in t after clearing denominators). The limit for θ has turned into the limit as
- t → 0
since the gradient t will be 0 at (1,0). All trigonometry has now been removed from the question, and t is a local parameter for the circle near the point in question. In these terms it is easy to justify Beck's rule since it leads to ratios like
- [at + o(t)]/[bt + o(t)] → a/b.
But when there are equal powers of t of order higher than 1, it isn't hard to see that it may break down.
Using local algebra
For some general theoretical support for the idea of substitutions in limits: consider that inside the field F of all such rational functions, there is a valuation ring V of those that take a finite value at the given point (1,0) at which we set t = 0. Here V is a local ring with maximal ideal I of functions taking the value 0 at that point. The evaluation of the limit here corresponds to calculating modulo I. That is, V/I is the underlying field (which we took to be the real numbers) up to a field isomorphism.
What that does for us is to assure a certain consistency. If we calculate modulo I, and find a way of evaluating a given limit, the element of I we used is available also to be applied to some other limit; and we shall never be misled, in that a substitution procedure in terms of t-expansions will either remain undefined, or give a correct answer.
In short, Beck's rule and similar devices enjoy a kind of partial correctness.