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Long Post on Trig, Limits
Posted:
Apr 2, 1997 11:07 PM
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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 *arc-length*
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 well-defined 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 xy-plane 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 non-convex. 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
*counter-clockwise direction* and negatively in the *clockwise*
direction. The reference ray is the positive x-axis. 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 (A-B),sin (A-B)), the angle is A-B.
(cos B, sin B), (cos A, sin A), the angle is A-B.
(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:
(1-cos(A-B))^2 + (sin(A-B))^2 = 2 - 2*cos(A-B)
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(A-B) = 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} (1-cos 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 *hind-sight 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 (1-cos 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} (1-cos 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).
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