[Bongle]

You can approximate sin very well between -PI/2 and PI/2 with the function f(theta) = theta - theta³/6 + theta⁵/120.

Once you've got that approximation, you can basically compute any sin value by using the periodic properties of sin.

sin(x) = sin(PI - x) between PI/2 and 3PI/2 (this recursively uses the bit we can accurately approximate)
sin(x) = sin(x - 2PI) between 3PI/2 and 2PI.

Here's the code for sin:

#define PI 3.14159

float ApproxSin(float theta)
{
    while(theta > (3*PI)/2)
    {
        theta -= 2*PI;
    }
    while(theta < -PI/2)
    {
        theta += 2*PI;
    }
    // theta is now between -PI/2 and 3*PI/2

    if(theta > -PI/2 && theta <= PI/2) // basic approximation
    {
        return theta - theta*theta*theta/6 + theta*theta*theta*theta*theta/120;
    }
    else if(theta > PI/2 && theta < 3*PI/2)
    {
        return ApproxSin(PI - theta);
    }
}

An approximation for cos is f(x) = 1 - x²/2 + x⁴/24, you can do the same thing with it, more or less.