Trignometry With Pure Python

Bhaskaras’s Appoximation Formula

Discovered by 7th century mathematician Bhaskara I, this formula is elegant and simpler than the next method that I am gonna discuss. While it does not give the perfect value, it is still a very good approximation

$\sin (x) = \dfrac { 4 \times x ( 180 - x ) } { 40500 - x (180 - x) }$

The graphs of sin x and the approximation formula are visually indistinguishable and are nearly identical.

Python Implementation

angle = 30
def sin(x):
    return ( 4*x*(180-x) ) / ( 40500 - x *(180-x) )
def cos(x):
    return sin(90 - x)
def tan(x):
    return sin(x) / cos(x)

Taylor Series

So lets say you have a robot that can draw any function you want but its kinda like AI and gets better with more functions we feed it. The “sine” is a function that goes infinitely up and down. But the robot doesn’t know how to draw the sine curve directly.

So to teach it how to draw a complex curve, we give it a set of simpler instructions called the Taylor Series. These instructions tell robot how to draw a bunch of smaller, simpler curves called “polynomials.”

Each polynomial is like a tiny part of the sine curve. The more polynomials we use, the more accurate our drawing becomes. It’s like putting together puzzle pieces to make a bigger picture.

Eventually there will be a point where there would be enough polynomials to make the robots drawing almost identical to the actual sine curve

$\sin(x) = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + \dfrac{x^9}{9!} - \ldots$

pi = 3.14592653589793238462643383279502
def f(n):  # Factorial Function
    if n == 1 or n == 0:
        return 1
    else:
        return n * f(n - 1)

def deg(x): # convert deg to radians
    rad = x * pi/180
    return rad

def sin(x):  # Taylor Expansion of sinx
    x = deg(x)
    k = 0
    sinx = 0
    while x >= pi:
        x -= pi
    if pi > x > pi / 2:
        x = pi - x
    while k < 15:
        sinx += (-1)**k * x**(2*k + 1) / f(2*k + 1)
        k += 1
    return sinx

Credits

Namish

29 Jul 2023