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Long Post on Trig, Limits
Posted:
Apr 2, 1997 11:07 PM


This is a Long Post about proving the basic limits related to trig functions. I try to keep within the spirit of the historical development, but bypass some of the more intricate geometrical results by using the later techniques of coordinate geometry. As indicated in a previous post, the main point was the *marriage* of geometry and algebra via trigonometry and coordinate geometry. The basic limits *lead* to calculus, using calculus techniques to prove the basic limits is a *circular reasoning*. If you are not interested in any of these, please delete at this point.
Han Sah, sah@math.sunysb.edu *********************************************************************
Comments on Trigonometry
0. Introduction.
It is a difficult task to provide a *coherent* treatment of the trigonometric functions. A large part of the problem rests with the definition and properties of the real numbers. In particular, it is essential to discuss the question of an *angle*. The classical Euclidean approach is qualitative. We may speak of degree measurement and bisection of an angle. With just the primitive tools of an unmarked ruler and a compass, it was not even possible to construct a 20 degree angle (this was only shown to be impossible close to 2,000 years later). Thus, even when we try to define sine and cosine in terms of ratios of a right triangle, we are still faced with the problem of assigning a numerical value to the *angle*. Usually, textbooks give the concept of *arclength* by invoking the use of some *physical devices* such as a *tape measure*. In basic calculus courses, this is typically carried out after integration has been introduced. Along the way, the definition of arc length for a *parametrized* curve where a point P on the curve has coordinates defined by the use of two functions so that P = (x,y) = (f(t),g(t)). It is then tacitly assumed that f and g are differentiable functions of the parameter t so that as t varies over the closed interval [a,b], the arc length is then defined to be:
int_a ^b of sqrt([dx/dt]^2 + [dy/dt]^2)^1/2 *dt
This integral can not be evaluated exactly except in a small number of special cases.
It is also tacitly assumed that for distinct values of t in [a,b], we get distinct points P on the curve. Implicit in this approach is a definition of arc length and the need to show that the arc length so introduced does not depend on the choice of parametrization. [This uses the chain rule as well as the assumption that for distinct values of t in [a,b], we get distinct points on the curve.]
By this time, both the teacher and the students have had to use a large amount of machinery and everyone is exhausted.
Aside from everything else, there is at least one step of the definition that requires justification. Namely, the arc length is defined to be the least upper bound (provided that it exists) of the lengths of *inscribed* polygonal path determined by selecting a *finite* but arbitrary set of points on the curve. The concept of *circumscribed* polygonal paths is not welldefined unless we make the assumption that the curve has a tangent line at each point. This is actually stronger than the assumption of the existence of a parametrization by differentiaable functions.
Example: Consider the following two functions with t in [1,1].
x = f(t) = t^3
y = g(t) = t^3
These two functions are differentiable in t (but not three times differentiable).
The curve it traces out is the graph of y = x and does not have a tangent line at the origin. However, the arc length is well defined and can be computed by the integral formula or even by the use of the distance formula.
Thus, the textbook approach to arc length is cumbersome to the extreme.
In contrast, the concept of *area* is somewhat simpler. Namely, we can carry out an outside as well as inside approximation by using graph paper with finer and finer mesh. As long as we can cover the boundary of a bounded region by a large number but finite number of small rectangles so that the total sum of the areas of the rectangles can be made as small as we like, then the bounded region will have a well defined area. We still can not escape the need to know some basic properties of the real numbers. Nevertheless, this approach is historically justifiable. It is the method used by Greeks, Chinese, etc to get a good approximation of the value of \pi. This is defined to be the ratio A(r)/r^2, where r is the radius of a circle and A(r) is its area. The area A(r) can be shown to exist by way of the basic *sandwich* principle using approximations by way of inscribed and circumscribed regular polygons with 3*2^n sides, n = 1, 2, .... A second example may help to explain the difficulties.
Example: Consider the following curve:
On the xyplane mark off points P(n) = (1/2^n,0), n = 0, 1, 2, etc. Using P(n+1), P(n) as base, erect an isoscles triangle with height 1/n. The top edges of these isoscles triangles define a continuous function f on [0,1] where we define f(0) to be 0. Our isosoles triangles have bases of length 1/2^n, n = 1, .... and area 1/n*2^{n+1} < 2^{n+1}. Thus the total area underneath the graph of f is at most 1 and finite by using the geometric series with ratio 1/2. The length of the top two edges, by the theorem of Pythagoras total at least 2/n for n = 1, 2, ..., Thus, using the divergence of the harmonic series, the arc length of the graph of f is in fact infinite.
With this example at hand, we actually have to provide some argument to be convinced that the circumference of a circle has a finite total length. However, there is no difficulty giving an argument showing that the area of a circle exists and is finite.
The present exposition will try to follow a historical approach with a few modern conveniences tossed into the pot: Hindsight is better than foresight. One of the purposes in teaching is to take advantage of the hindsight so as to provide a reasonably convincing argument that can be made rigorous without a huge expenditure of effort.
Section 1. Definition of an angular measure.
We draw a circle of radius r and two radii to shade off a sector of the circle. The shaded part may be convex or nonconvex. Using the same argument, this shaded part will have an area. We may therefore, with hindsight, define the angular measure by:
\theta = 2* [area/r^2]
The multiplication factor of 2 is a matter of convention. It is used to make things come out right. As such, it disregards the orientation of the angle. We note immediately that this definition is
invariant under change of scale in the measure of length.
As things stand, a change of scale in distance measurement has *no effect* on the angular measure (this is the radian measure).
Section 2. Signed angle in the Cartesian coordinate system.
Here, we use the usual convention for the Cartesian coordinate system so that angles are measured positively in the *counterclockwise direction* and negatively in the *clockwise* direction. The reference ray is the positive xaxis. Because of the invariance under change of scale, we can limit ourselves to a circle of radius 1 defined by the equation:
x^2 + y^2 = 1.
[Note: This uses the Theorem of Pythagoras and the resulting distance formula.]
Thus, given a point (x,y) on this unit circle, we have an angle determined by the two rays emanating from (0,0):
the starting ray determined by (1,0). the end ray determined by (x,y).
The angle \theta is positive if we rotate in the counterclockwise direction, and negative in the clockwise direction. A full circle yields an angle of 2\pi or 2\pi, where \pi is the area of the full circle of radius 1. For our purposes, it is NOT essential to know the precise numerical value of \pi, just that it exists and can be approximated to any degree of accuracy as we like.
Section 3. Definition of sine and cosine and fundamental properties.
Once we have an angle with value \theta, we denote the values x and y as functions of \theta and give them names:
x = cos (\theta) y = sin (\theta)
Using reflections and the definition, it is immediate that:
cos( \theta) = cos(\theta) (3.1) sin( \theta) =  sin(\theta) cos 0 = 1 and sin 0 = 0. (cos \theta)^2 + (sin \theta)^2 = 1.
As is known historically, the angle is completely determined by the chordal length joining the two points on the circle. In particular, congruent chords determine the same angular measure (for the convex angle). This leads to the fundamental observation that the angle is
invariant under rotation.
It is also the underlying principle that led to the *basic* ADDITION FORMULAS:
(3.2) cos(A + B) = cos(A)*cos(B)  sin(A)*sin(B) sin(A + B) = cos(A)*sin(B) + sin(A)*cos(B)
We can give a rapid proof of these by way of the reflection principle and the Theorem of Pythagoras:
Namely, recall A + B means addition of *signed area* up to a factor of 2. If we replace A, B independently by their negatives, then (3.1) shows that it is enough to prove (3.2) for A, B > 0.
We now consider the two angles defined by the two pairs of points on the circle:
(1,0), (cos (AB),sin (AB)), the angle is AB.
(cos B, sin B), (cos A, sin A), the angle is AB.
(A+B)  B = A.
Remember that angular measure is twice the area of the circular sector. It follows that the chordal lengths must be the same. In other words, the two distances using the corresponding pairs must be the same. We may use the distance formula to compute the square of the distances. The first pair yields:
(1cos(AB))^2 + (sin(AB))^2 = 2  2*cos(AB)
The second pair yields:
(cos A  cos B)^2 + (sin A  sin B)^2 = 2  2*[cos A * cos B + sin A * sin B]
Equating these two equations, cancelling out 2 from both sides and divide the result by 2, we get cos(AB) = cos A * cos B + sin A * sin B
Replacing B by B and using (3.2), we get the addition formula for cosine. If we replace A by \pi/2  A and do the exercise:
cos(\pi/2  A) = sin A sin(\pi/2  A) = cos A
we quickly obtain the addition formula for sine.
Section 4. Two fundamental limits.
(4.1) lim_{t>0} (sin t)/t = 1 (4.2) lim_{t>0} (1cos t)/t = 0
Note: Since sin (t) = t, it is enough to consider t > 0+ in (4.1). Since the asserted limit in (4.2) is 0, and cos(t) = cos t, we may also assume t > 0+ in (4.2).
It is NOT appropriate to invoke l'Hopital's rule at this point. We have not yet introduced the definition of a derivative. Even if we had, there remains the question about the calculation of the derivative of the sine and of the cosine function. It should be noted that (4.1) *is* the derivative calculation of sin x at x = 0. Namely:
sin t/t = (sin t  sin 0)/(t  0)
is the Newton quotient used in the derivative calculation. Similarly, (4.2) *is* the derivative calculation of cos x at x = 0.
For these calculation, the *hindsight definition* now comes in handy. Namely, (1/2)*sin t* cos t is just the area of the triangle with vertices: (0,0) (cos t, 0) and (cos t, sin t). This is evidently less than the area of the angular sector namely t/2. Note: t was defined to be the twice the area of the angular sector.
By extending the ray througgh (cos t, sin t) until it meets the vertical line through (1,0) at (1, (sin t)/(cos t)), we see that angular sector has area less than the area of the triangle with vertices (0,0), (1,0) and (1, (sin t)/(cos t)). Thus, we have the critical *sandwiching inequality for t > 0:
0 < (1/2)*sin t * cos t < t/2 < (1/2)*(sin t)/(cos t).
Multiplying by 2 and dividing by sin t ( it is positive for t > 0 and close to 0), we get
0 < cos t < (sin t)/t < 1/cos t < 1
Now, as t > 0+, cos t increase to 1. This gives (4.1).
(4.2) follows from (4.1) by multiplying (1cos t)/t by (1+cos t)/(1+cos t) and use (3.1) so that it becomes:
[(sin t)/t]* [(sin t)/(1+cos t)]
By using (4.1) and the fact that sin t goes to 0 as t goes to 0 and cos t goes to 1 as t goes to 0, we get (4.2). In the same manner, we also get the more precise:
(4.3) lim_{t>0} (1cos t)/t^2 = 1/2.
Section 5. Derivative of sin x and cos x.
d (sin x)/ dx =: lim_{h>0) [sin (x+h)  sin x]/h
Use the addition formula for sin x, the Newton quotient is
[sin x * cos h + cos x * sin h  sin x]/h
= sin x * (cos h  1)/h + cos x * (sin h)/h
Using (4.1) and (4.2), the derivative of sin(x) is cos(x). Similarly, using the addition formula for cosine, we get d (cos x)/ dx =  sin x.
Comments:
(1) This does not address the issue about arc length.
(2) It also does not address the issue of the definition and the basic properties of the real numbers. The latter has to be dealt with in order to give meaning to the concept of limit. However, for an initial introductory course in calculus, we can gloss over it lightly.
(3) The issue of arc length, as indicated, is more subtle. We can jazz up the example of infinitely many isoscles triangles. Consider the function k*f defined over [0,k] with k > 0+. No matter how small k is, the length of the graph of k*f over [0,k] remains infinite. Namely, multiplying the terms of a divergente infinite series by any positive number k, the resulting infinite series still diverges. Leaving out a finite number of terms in a divergent series still yields a divergent series. Now as k shrinks, our curve is crunched into a right triangle with base length k and height k. We don't have to make it an isoscles triangle. By lowering the height faster than shrinking the base, we can make it very *thin*. It is then apparent that we have a situation that a thin sliver of a triangle can have a curve of infinite length crunched up inside. Thus, the assertion that a circle must have a finite circumference does involve some special feature of the circle.
One might object that the function is not differentiable. This can be repaired, we just have to replace f by the function:
g(x) = x^2 * cos(1/x), x not 0 = 0 , x = 0
g is now differentiable, but its derivative oscillates wildly near the origin. It has the same 'shape' as the previous example.
On a graphing calculator, one can zoom in as much as one wants, the derivate is roughly like sin (1/x).



