Chapter 17 - The Sine and
Cosine Function
### Chapter 17 - The Sine and
Cosine Function

* I see crowds of People,
walking around in a Ring. T.S. Eliot Wasteland *

*
A current under sea Picked his bones in
Whisper. As he rose and fell
*

He passed the stages of his
age and youth

Entering the whirlpool. Wasteland

Your
memory of trigonometry is probably filled with meaningless identities and
strange acronyms to help remember what all the Trig. Functions refer to.
Some
of you may recall that infamous Indian Tribal Chief, SohCahToa, whose name
helped you pass your equally notorious quizzes and tests. ( **S**ine **o**pposite over **h**ypotenuse, **C**osine **a**djacent over **h**ypotenuse,
**T**angent, **o**pposite over **a**djacent; hence
SohCahToa) Others may remember being presented with a circle, called the
Unit Circle
for reasons soon to be seen. In this circle any point on it had the
co-ordinates ( Cos Ø, Sin Ø) where x= Cos Ø and y= Sin Ø. Often times the
circle added more confusion to your understanding of Sines and Cosines. At
such
a point your mind must have justifiably asked, . What do Indian Tribal
Chiefs
and Circles have to do with the study of the Sine function, a function
whose
application arises everywhere from the ten second swaying period of a supertall skyscraper to the mechanisms that translate the back and forth
motions
of the watch spring into time. In almost any study of oscillating or
vibrating
systems, the Sine function arises, for reasons that are **not** attributed to mere coincidence and
chance.

In
all simplicity the Sine function is the complex mathematical function to
describe circles. From the lines from T.S Eliot's Wasteland and from your
own
experience circles represent a never ending, repetitive cycle. Motion
along a
circle passes through the same point infinite times. It is this complex
nature
of the circle, that causes its study to be so intense and intricate.
Before we
can talk about circle, we need to first understand
triangles.

Let
us begin our study of the Sine function with a look at right triangles. In
all
simplicity the Sine of an angle (
The
issue of what is an angle and how to describe it will be dealt with later)
is
the ratio of the opposite side to the hypotenuse:

Or more directly:

From the calculator or a Trig. Table we can find the
angle q approximately. The Cosine functions similar to the Sine function
except that it measures the
adjacent
side, not the opposite side, ratio to the hypotenuse. For
example:

In right triangle ABC,

The
last trig. Function, the Tangent, is the ratio of the opposite side to the
adjacent side. Thus in right-triangle ABC,

The
remaining Trigonometric functions; Secant, Cosecant, and Cotangent are
simply
the reciprocal values of the Cosine, Sine, and Tangent respectively.

It is not important to remember the reciprocal
functions as they unnecessarily add confusion to one.s understanding of
Sine.s
and Cosine.s and Tangent.s. It is enough to remember that they are just 1
over
the value for the Sine, Cosine or Tangent.

To further
make life easier one can express Tan (q) as just Sin (q) over Cos (q). This is easy to see
as:

This reduces the study of Trigonometry to only two
functions, the Sin (q) and Cos (q). Any other Trig function can be expressed as a variation of these
two.
To state them again they are:

At
this stage your mind should only be focused on knowing what the Sine and Cosine of an angle mean.
Sine
is opposite over hypotenuse, and Cosine is adjacent over
hypotenuse.

Now
there are two very important concepts to understand about the Sine and
Cosine
of a given angle. First, Sines and Cosines apply only to **right triangles**, when there is one right angle of 90 degrees,
whose
opposite side is called the hypotenuse. If one were asked to find the Sine
of
the following angle.

Then the answer is not 2.8/5.5 or 2.8/7 as triangle
ABC is clearly not a right triangle and side AC is not a hypotenuse. To
find
Sin (q) we
would have to somehow construct a right triangle, probably the easiest way
by
drawing an altitude from C perpendicular to AB, like
this:

The values for h and x can be found by writing two
simultaneous equations based on Pythagoreans's
theorem:

The
second important point to understand about Sines and Cosines of angles is
that
their values are independent of the dimensions of the triangle. What this
means
is that the Sine of a 62 degree angle will always be .883, regardless of
the
size of triangle it is measured in.

Size
is a meaningless quantity unless it is used with reference to something
else.
If you examine a right triangle individually then you will find it
impossible to
describe its size. It is only when you look at it with respect or relative
to
another triangle that you can correctly say the following concerning these
two
right triangle.

Triangle A is large. (Relative to B only)

Now if we were to compare Triangle A with let us say
another Triangle C we would say:

Triangle A is small. ( relative to C)

Finally we can compare Triangle A with two other
triangles and say the following:

Triangle A is of average size. ( relative to b and
C).

This
brings us to a logical question; how can Triangle A be small, large, and
even
average? Since Triangle A is unique and does not change then it is our
adjectives which are at fault in describing the attribute of size of the
triangle. We are now in a position to state an important philosophical
idea:
Adjectives are only as accurate as the number of objects an object is
described
relative to. In other words to describe something one must have at least a
standard to compare it to.

As we
mentioned before descriptions are only as accurate as the number of
objects it
is compared relative to. For this reason calling Triangle A average was
more
accurate than calling it large or small. This is because, in describing
Triangle A as average we were comparing it relative to two triangles and
not
just one as in the other two cases.

The
importance of understanding relativity cannot be understated in describing
any
situation, phenomena, object or attribute. To name but a few examples from
the
natural world, weight, velocity, size, distance and objects and light.
Relativity can also be taken a step further to ask what in this world is
absolute, or whose existence is independent of anything else. In the SI
units
of measurement, every quantity except length, time and mass are considered
derived. For example volume, density , force etc.. can all be expressed in
terms of the fundamental quantities of length, time and mass. But are
length,
mass and time, independent of each other?

Returning
back to our discussion of the Sine function we saw that the Sine of an
angle
was simply the opposite side over the hypotenuse. We also now see how all
other
trigonometric functions can be expressed as variations of the Sine
function
alone, with the help of the
Pythagorean
theorem. Therefore before continuing let us go through a simple proof of
the Pythagorean
theorem, a theorem which as you already know, arises almost everywhere in
Mathematics.

We
begin this proof by drawing an altitude in a right triangle
ABC,

The altitude with height, h, divides the triangle
into two other right triangles, both similar to triangle
ABC.

Viewing these three triangles together:

As these triangles are all similar we can state two
identities:

Adding a^{2 } with b^{2} gives us:

Taking the theorem a step further we can prove the
following important identity relating sine with cosine. In any right
triangle
ABC

Cos (q) would then be equal to a/c; however by the Pythagorean theorem;
, we can then re-write a as

Hence:

Squaring both sides gives us:

Remember that

This identity holds true for
__any __ right triangle. For
remembrance
sake it can also be written as;
This
useful identity tells us that even the Cosine of an angle can be expressed
in
terms of the Sine of the angle, therefore the Sine function is the
fundamental
Trigonometric function, as all other functions can be derived from it,
including the Cosine function.

Until now we have restricted
our study to triangles, where the Sine of an angle is the ratio of the
opposite side to the hypotenuse. We can therefore define the Sine function
to
be the function that outputs a unique ratio for an inputted angle. But we
must
ask ourselves what is an angle, and how can we express the sine of an
angle
mathematically without having to look up Trig. Tables? The solution is
left for
the next chapter that will go into detail to define the Sine function
mathematically by using a circle and arclength.