### Chapter 18 - The Sine Function- Definition

Until now we have restricted
our study of the Sine function to triangles where:

Remember a function is by definition only a function when
the output is directly related to the input. For example the following function
gives the Volume of water in a bowl as function of height:

Input any height
and one can easily calculate the volume. However the function,

is not a
function as none of the variables; opposite, hypotenuse, or angle (theta) are
related in any way. Give me an angle and I can not give you the Sine of it,
that is without looking it up in some table.

To begin our study of the Sine function,
we need to first find a way of defining it as a mathematical function, such
that the Sine of an angle is a function of the angle inputted. It therefore
behooves us to define what exactly is an angle. Till now we have assumed an
understanding of angles and we looked at Triangles to show what they mean.
Before defining what an angle is, let us now take an important look at circles
to see how angles can be defined by them.

From the Pythagorean theorem we
showed that in any right triangle,
. The equation :

Is therefore the
set of all points a distance r from the origin, from the distance formula which
is based on the Pythagorean theorem. The important point to realize here is
that the co-ordinates of each point on the circle gives us the dimension of a unique
right-triangle with base, x , height, y and hypotenuse r. Each point satisfies
the Pythagorean theorem for right triangles,
; though the radius or hypotenuse remains fixed, the x and
y.s can take on infinite values.

To visualize these concepts, the
Graph of the circle

Let us now look at how to define an
angle in terms of the circle. An angle in all simplicity is the measurement of
rotation between two intersecting lines. This rotation can be expressed in
terms of the arc length of the circle. If you follow the path of a point at
(3,0) to (0,3) along the arc of the circle then you will notice that the radius
drawn to each point on the path will increase at a steady rate.

You can notice two points here.
First the circumference of this circle is given by the formula:

This tells us
the total arc length of this circle is just
, where r is the radius. Now if we define there to be 360
degrees in a circle, what this means is that a line that rotates a full circle
to return to its original position will have covered 360 degrees ( a completely
arbitrary number). Since degrees are a measurement of an angle, or the amount a
line or r in the graph has rotated then we must figure out some method of
relating the arc length, a measurement of rotation to degrees.

If we define there to be 360 degrees
in one full rotation the ratio of some number of degrees
q
to 360 degrees must equal
the ration of the arc length covered to the
total arc length of the circle.

Multiplying
terms out to solve for degrees gives us:

There is still
one thing left, this definition of degrees is entirely dependent on the radius
of the circle. If you recall from our discussion of the relativity of size in
mathematics as being meaningless without reference to something else, then
theoretically 30 degrees should be independent of the size of the circle it is
measured in. To account for this we introduce an important new concept, the
radian. The radian is a relativistic measurement for arc length that gives us
the arc length measurement in terms of the radius of the circle.

For example in
a circle of radius 2, the circumference is 4
p.
An arc length of 2 radians
is then just 4 units long or twice the radius. Since the circumference of the
circle is directly related to its radius then an arc length in radians will
always have the same relative size to its circle, regardless of the size of the
circle. This probably sounds more confusing than it really is, but this is all
because we are looking for a way to define degrees in terms of arc length. If
we replace arc length with radians*r, where r represents the constant radius
and radians is any number which could be a fraction we get:

This important
equation now tells us how to define an angle in terms of the radian measurement
of an arc length, measured in radians. We can re-write it as:

The important
point to realize throughout this discussion of what an angle is, is that angles
can only be first defined in terms of arc length rotations between two lines
measured in radians. Having established radians, only then can we also use
degrees, which is just another unit of measurement of angles entirely based on
radians. Instead of calling an angle .86 radians we can say 60 degrees:

Degrees are
directly related to radians, and are more often used when there is no reference
circle to define an angle easily.

Let us now return to our discussion
of circles. Having established just what an angle refers to conceptually we now
need to define an angle mathematically. We shall select for our study a circle
of radius one, which is often called the unit-circle, for reasons soon to be
seen. If we draw a small triangle in the unit-circle we can see that:

In such a
triangle

What this means
is that for any point
on the unit circle,
the Sine of the **arc length **measured
from (0,) to (x, y) will be the y value of that particular arc length. For
example:

In terms of
triangles and degrees, this says that the ratio of the opposite side to the hypotenuse
of a 45 degree angle is just
. It is extremely important to forget about triangles and
degrees for the moment but just concentrate on circles and arc lengths which
are both mathematically defined shapes. In terms of the unit circle whose the
angle measurement 45 degrees is expressed in terms of the arc length of a
circle whose radius is 1. pi/4 then refers to an arc
length of 0.78 radii.

** Error in picture below, it should say 0.78, not 1.56**

We are now ready to define the Sine
function as that function that outputs the y-value for an inputted arc length of a unit circle. The cosine functions
gives us the x-value for that same arc
length. It is here where the difficulty arises. We need to find a way of
expressing the arc length of a circle a unit circle in terms of x and y. By
being able to calculate the arc-length in terms of x and y values, the Sines
and Cosines will just be the x and y values of the point until which we are
calculating the arc length. Therefore the Sine function is really an arc length
function as we shall soon see.

To begin let us recall our formula
for calculating the length of curves of graphs of equations. It was defined as
the following differential that gives the length of a hypotenuse used to
approximate the graph over a small interval of x.

Integrating it
to find the total arc length.

For example the
length from 0 to 1 of the following graph of y= 5x is:

The length of
the graph from 0 to 1 therefore has length 5.22.

Relating this to our study of the
circle, the arc length can be expressed as follows:

The derivative
of this function for a circle is:

The arc length from
zero to a where, a is less than or equal to 1 is:

In the circle
this translates to:

There are two important points to
understand here. First for any
inputted the output
will be an arc length measured in radians since we are dealing with the unit
circle. For example if I were to evaluate the integral from x=0 to x=1 I should
get
or 1.57 radii. The
second point to keep in mind is that
is the same as
and
is therefore

Returning to our integral we have:

Simplifying the
integral:

The expression in the radical,
can be further
reduced to
by adding the two
fractions:

This now gives
us:

The question now
arises, . how do we interpret this or relate it to the Sine function. To answer
this let us study the graph of the circle once more. Since we commonly measure arc
lengths in radians or degrees beginning at (1,0) it will require a moments
thought to realize that now we are measuring angles from (0,1).

First take a
look at section 1. It represents the way we initially defined the Sine
Function, where Sine( arc length) = y. Now examine section 2. It is exactly the
same as section 1 excepts instead of y we have x. The sections are identical
because the graph of a circle is symmetric about the line y= x. We can
therefore define the Sine function as follows.

Since x is our
output and not our input here, we need to find a better way of defining the
Sine function. If you recall from our study of inverse functions, we saw that
the inverse function always yields x when inputted in the original function
since a function and its inverse were essentially the same function except that
the x and y were switched around, and the two functions were then graphed in
the same x-y plane. For example if we had
then its inverse was
found by replacing y with x to get
or
.
Since any point (x, y) on the graph of
has a corresponding
point on (y, x) on the graph of its inverse;
then inputing y
values from
in
would output x values. For example:

This may sound
more complicated then it really is but we can now conclude that since the Sine
of the arc length integral outputs x, then that integral is the inverse of the
Sine function:

The inverse sine
function is commonly called the arcsine function which is analogous to asking
what angle or arc length has sin x?

At this point is important to stop
thinking about angles, degrees, and triangles. What is essential now are
circles and arc lengths measured in radians. The Sine function, mathematically
speaking is then just the function gives the corresponding x value of an arc
length measured from to 0 to x on the circle. The inverse sine of a number x is
then just the arc length from 0 to x on the graph of a circle. This is all it
is, nothing more, nothing less. The inverse sine function is therefore our most
important function since it serves as the base for calculating arc lengths for
any

This concludes our definition of the
Sine function as mathematical function to express the arc length of a unit
circle, or any circle. You can already begin to see the problems we will
encounter in integrating it since the arc length approach infinity towards x=1.