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LECTURE 10: TRIGONOMETRIC FUNCTIONS
1. Introduction
In this lecture we will use the calculus we have developed to introduce a rigorous
definition of the familiar trigonometric functions and derive some of their basic
properties.
We begin by a slightly less formal discussion. Let S 1 denote the unit circle in
R2 centred at 0; that is,
S 1 :“ tpx, yq P R2 : x2 ` y 2 “ 1u.
Given a point P P S 1 , if the length of the arc S 1 from p1, 0q to P has length θ, then
pcos θ, sin θq are defined to be the co-ordinates of P .
To make this simple discussion more precise we would have to develop the theory
of lengths of curves. This is done in several exercises throughout the course textbook, but it is easier for us to observe that cos and sin can equivalently be defined
in terms of areas of circlar sectors.
Again let P P §1 and θ be the length of the arc from p1, 0q to P . The total length
of the circumference of the circle is (by definition) 2π and so the length of the
sector is θ{2π of the total length. Accordingly, the area Apθq of the sector should
be θ{2π ˆ Areaofthecircle which is
θ
θ
ˆπ “ .
2π
2
Thus, one may define pcos θ, sin θq to be the co-ordinates of the point P such that
the resulting sector has area θ{2.
With this in mind, we now begin our rigorous treatment of trigonometry.
2. A rigorous approach to trigonometry via calculus
The portion of the circumference lying in the upper semi-circle may be parametrised
as the graph of the function ϕ : r´1, 1s Ñ r0, 1s given by
a
ϕpxq :“ 1 ´ x2 .
Thus, by definition, the area of the upper semi-circle should correspond to the
integral of this function, prompting:
Definition.
π :“ 2
ż1 a
1 ´ t2 dt
´1
In order to define the trigonometric functions we will
? have to compute the areas
of sectors. To do this, first suppose the point P “ px, 1 ´ x2 q defining the sector
lies in the first quadrant. Then the area of the sector from p1, 0q to P is the area
under the graph of ϕ over the interval rx, 1s together with the area of the triangle
with vertices p0, 0q, P and px, 0q. Thus, it is given by the formula
?
ż1a
x 1 ´ x2
`
1 ´ t2 dt.
2
x
If P lies in the second quadrant (so ´1 ď x ď 0), then the term in the above formula
corresponding to the area of the triangular region becomes negative. A quick sketch
1
2
LECTURE 10: TRIGONOMETRIC FUNCTIONS
will convince the reader that in this case the area of the triangle should indeed be
subtracted and that the formula is valid for P in either sector.
Definition. Define A : r´1, 1s Ñ r0, π{2s by
?
ż1a
x 1 ´ x2
Apxq :“
`
1 ´ t2 dt.
2
x
Lemma. The function A : r´1, 1s Ñ r0, π{2s is differentiable on p´1, 1q and
´1
A1 pxq “ ?
for x P p´1, 1q.
2 1 ´ x2
Recall, from our informal discussion we know pcos θ, sin θq should correspond to
the co-ordinates of the point P P S 1 which defines a sector of area θ{2. This is
made precise by the following.
Definition. If 0 ď x ď π, then cos θ is the unique number in r´1, 1s such that
Apcos θq “
θ
2
and
sin θ :“
a
1 ´ cos2 θ.
Theorem. If 0 ă θ ă π, then cos and sin are differentiable at θ and
cos1 pθq “ ´ sin θ;
sin1 pθq “ cos θ
Using this theorem one can sketch the graphs of cos and sin. In particular, note
that since cos1 pθq “ ´ sin θ ă 0 for 0 ă θ ă π, the function should be decreasing
from cos 0 “ 1 to cos π “ ´1. By the intermediate value theorem, there must be
some point y such that cos y “ 0; that is
ż1a
y
1 ´ t2 dt.
“ Ap0q “
2
0
Since
ż1a
ż0 a
π
1 ´ t2 dt “
1 ´ t2 dt “ ,
4
0
´1
it follows that y “ π{2. A similar analysis can be applied to sin.
Definition. The functions cos, sin are extended to functions cos, sin : R Ñ r´1, 1s
as follows:
‚ If π ď x ď 2π, then
sin θ :“ ´ sinp2π ´ θq
cos θ :“ cosp2π ´ θq.
‚ If θ “ 2πk ` θ for some k P Z and θ P r0, 2πs, then
1
sin θ :“ sinpθ1 q
cos θ :“ cospθ1 q.
The differential identities then also extend to the whole of R.
Theorem. The functions cos, sin : R Ñ r´1, 1s are differentiable and
cos1 pθq “ ´ sin θ;
sin1 pθq “ cos θ
for all θ P R.
LECTURE 10: TRIGONOMETRIC FUNCTIONS
Proof. Exercise.
3
Jonathan Hickman, Department of mathematics, University of Chicago, 5734 S. University Avenue, Eckhart hall Room 414, Chicago, Illinois, 60637.
E-mail address: jehickman@uchicago.edu