Now since
x
2
+
y
2
= 1 we know that (sin (
θ
))
2
+ (cos (
θ
))
2
= 1. This fact is certainly worth
memorizing, but if one remembers where sin and cos come from there is no need to memorize
anything.
The two periodic functions can be combined in various ways and indeed you will see this on
your assignments. However for now we will define
tan(
θ
) =
sin (
θ
)
cos (
θ
)
.
This function will also be periodic, but it is also undefined whenever cos (
θ
) = 0 (at
π
2
and

π
2
for example). The function gets very large in absolute value near these points.
Incidentally the proper definition of a periodic function goes something like this:
DEFINITION:
A function
f
(
x
) is periodic with period
L
if for any integer
n
and any valid
input
x
,
f
(
x
+
nL
) =
f
(
x
).
Thus we have further enriched our mathematical language with the ability to describe at
least some periodic phenomena. Of course the nicely behaved sin and cosine functions may
describe a pendulum pretty well, but can’t possibly describe something as complex as a
heartbeat.
Note however that there is nothing stopping us from linking our functions together in a
string of inputs and outputs, something like
f
(
x
) =
p
(sin (
x
))
for a polynomial
p
, where we take
x
, feed it into the sine machine then take the output and
feed that into a polynomial function,
p
(
.
), to get another output. While it will take a couple
of years before you see why, these polynomials made up of sines and cosines let you describe
almost any measurement in a compact and succinct way. Here’s a little sneak preview.
Consider
s
1
(
x
) = cos (
x
),
s
2
=
s
1
+
1
4
cos (2
x
),
s
3
(
x
) =
s
2
+
1
9
cos (3
x
) and so on.
Pretty
simple pattern, right?
You might want to check that the general formula can be written
recursively as
s
n
(
x
) =
s
n

1
+
1
n
2
cos
nx.
(3)
Of course we need to also know that
s
1
(
x
) = cos (
x
), but once we know this, we can build
s
n
(
x
) for any integer
n
. You can also use sigma notation to get
s
n
(
x
) =
k
=
n
X
k
=1
1
k
2
cos (
kx
)
.
(4)
18
This would be a big waste of time if it wasn’t for the result shown in the following picture.
x
3
2
1.5
0.5
1
1
0.5
1
1
0
0
2
3
1 term
10 terms
200 terms
Here I have plotted
s
1
(
x
),
s
10
(
x
) and
s
200
(
x
) for

π < x < π
. You can see that
s
1
(
x
) is just
cos
x
, but that
s
10
and
s
200
are essentially the same. This means that if I had a function
that was kind of pointy near
x
= 0 but round near
x
=
±
π
then instead of finding some new
form for this function I could just use
s
200
(
x
) as an
approximation
(actually most of the
time I could get away with
s
10
(
x
)). All it takes is a recipe for choosing the numbers in front
of cos (
x
), cos (2
x
), cos (3
x
) and so on.
You might be thinking this whole business of
p
(cos (
x
)) is too complicated, why not just use
polynomials
p
(
x
) to approximate? Toward this end consider the following example that you
will be able to understand in full by the end of first year. I claim that for
x
pretty close to
zero (can you make this more precise?) that I can approximate cos(
x
) by the polynomial
p
n
(
x
) =
k
=
n
X
k
=0
(

1)
k
x
2
k
(2
k
)!