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Approximating Sine and Cosine functions using Taylor Series

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In this example, we calculate an approximation of the cosine functions using

the `Taylor series `_:

.. math::

    \cos (x) = \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n)!} x^{2n}

The following example uses the Taylor Series approximation to generate the Sine

and Cosine functions. Successive terms of the taylor series are calculated

until successive approximations agree to within a small degree. A Sine

function is also synthesised using the identity :math:`sin(x) \equiv cos(x-\pi/2)`

.. code-block:: c

    /* taylor.c */

    /* Jonathan P Dawson */

    /* 2013-12-23 */

    /* globals */

    float pi=3.14159265359;

    /* approximate the cosine function using Taylor series */

    float taylor(float angle){

        float old, approximation, sign, power, fact;

        unsigned count, i;

        approximation = 1.0;

        old = 0.0;

        sign = -1.0;

        count = 1;

        power = 1.0;

        fact = 1.0;

        for(i=2; approximation!=old; i+=2){

            old = approximation;

            while(count<=i){

                power*=angle;

                fact*=count;

                count++;

            }

            approximation += sign*(power/fact);

            sign = -sign;

        }

        return approximation;

    }

    /* return the cosine of angle in radians */

    float cos(float angle){

        return taylor(angle);

    }

    /* return the sine of angle in radians */

    float sin(float angle){

        return cos(angle-(pi/2));

    }

    /* test routine */

    void main(){

        float x;

        float step=pi/25;

        for(x=-2*pi; x <= 2*pi; x += step){

           file_write(x, "x");

           file_write(cos(x), "cos_x");

           file_write(sin(x), "sin_x");

        }

    }

A simple test calculates Sine and Cosine for the range :math:`-2\pi <= x <= 2\pi`.

.. image:: images/example_2.png