..  _cosc1336-lab10:

Lab 10 - Building a module
##########################

..  vim:filetype=rst spell:

In the lecture this week we explored building a module named ``mymath.py``. In
this lab we will add another function to the module and learn a bit about how
the computer figures out those funny trigonometric functions along the way.
(Have not taken trig yet? Don't worry, you will before you get out of this
school!) 

Computing the sine of an angle
******************************

Long ago, mathematicians wanted to calculate the trig functions that were
needed to figure out how long the edges of triangle would be. Triangles were
very useful, but they could come in many different shapes. For a
``right-triangle``, one with two of the three edges coming into one corner 90
degrees from each other, they came up with a few formulas like this:

..  image:: Triangles.png
    :align: center

Now, in this class, we do not really care about all those funny functions like
``sin``, ``cos``, or even ``tan``, but the relationships fascinated the math
folks. Enough so that they started wondering how to calculate something like
``sin(x)`` (where ``x`` is the angle between the base and the edge leading up
to the far corner). What they came up with is interesting as well. They
discovered that if you computer the terms in a ``series`` of simple expressions
and summed them up, you would get the value you were interested in. For the
``sin`` function, what they came up with was this:

..  image:: SinSeries.png
    :align: center

The only problem with all this is that the angle between those two edges needs
to be expressed in terms of something called ``radians``. The formula for
``radians`` is this:

    * radians = degrees * PI / 180.0

That solves that problem!

Look closely at this ``series``. There is a pattern in what we see. Let's see
if we can see the pattern:

  * The series is a sequence of simple terms combined together

  * After the initial ``x`` term, each part has ``x`` raised to an odd power
    starting with 3, and the power goes up by 2 each term.

  * The denominator is the ``factorial`` of the power!

  * The sign of the term alternates between plus and minus

The magnitude of each term gets smaller and smaller as we go, eventually
heading toward zero. When it gets there, we will not really know

Could we come up with some code that generates this sequence, and use that code
to build our own ``sin`` function? Of course we can!

Building the ``sin`` function for our ``mymath`` module
*******************************************************

What you need to do is set up a loop that will compute each term (after the
first one) adding in the new term. The problem we face is figuring out when to
stop the calculations. For our experiments we will stop when the new term has a
value less than 0.00000001. In order figure this out, we will need to generate
four variables in our loop:

    * power

    * denominator

    * sign

    * term

The first one is easy, we just start off with a value of 2 and add two to the
power each time we run through the loop. For the sign, we can use a statement
like ``sign = - sign`` and start off with sign set to -1 to start. That will
make the value flip between plus one and minus one. Finally the term is the
angle (``x`` - in radians) raised to the power and divided by the factorial
part. We multiply that term by the sign and add it in to the result we are
after. See how it works?  

That leaves the factorial. Shoot, that looks like another function.

Factorial function
==================

The factorial of 5 is 5 * 4 * 3 * 2 * 1 or 1 * 2 * 3 * 4 * 5

A simple function to calculate the factorial looks like this:

..  code-block:: python

    def factorial(n):
        fact = n
        for val in range(1,n):
            fact = fact * val
        return fact

Does that look right?

Testing your code
*****************

Since we are building a module, you are to build a test program that uses your
module and the real ``math`` module to see how well your code is doing. You
should test the ``sin`` function over the range of angles from 0 to 360, in one
degree increments. Look at the example code in the ``mymath`` module in the
lecture to see how to get the real ``sin`` function in the ``math`` module to
give you the real value and see if you are close. (You should be).

Your test program will need to import both your new module and the real
``math`` module:

..  code-block:: python

    import math
    import mymath

We will discuss how Python finds these files in class.

What to turn in
***************

You will end up building two files:

    * ``mymath.py``
    * ``test.py``