Newton-Raphson Method, Simultaneous Equations and Euclidean Geometry

      I recently came across an interesting, or at least I thought it was, problem aimed at those students familiar with calculus and the Newton-Raphson method. (the textbook is at the bottom as a reference) I personally didn't find the problem too challenging, but for those unfamiliar with the method, I imagine it will be quite difficult.

      Where:

1. O is the centre (origin of the circle); 
2. OA and OB are both radii: from the first statement;
3. BA is the chord.

      The arc length from B to A is 5cm and the chord, B to A, is 4cm.

Find the angle BOA in radians to 4dp.

Well, we know the arc length is given by the product of the radius and theta:

Since the arc length is 5 then:

And from the cosine rule:

A is theta, b and c are OA dnd OB and a is the chord BA

We now have two equations, with two unknown variables: theta and the radius. These are simultaneous equations. The values for theta and the radius must satisfy both equations.
This allows us to substitute in the arc length equation into the bottom equation, to get a single solvable function of theta. Which yields:

We need to solve this to find our angle AOB:

The graph of the function looks like this:

We can see the solution to 3 dp, shame not 4!

Equations like the one above are extremely difficult to solve by hand, however, mathematicians use methods to estimate these equations, one such method is the Newton-Raphson method. This is an iterative formula that allows for a better and better approximation of x. Although, it involves taking the derivative of the function:

We have to choose an initial value of x: based on the graph we can say x is roughly 2. This will be our first x. Now to differentiate:

Here I used the quotient rule to differentiate, then I simplified the expression. Since we now have the function and its derivative, we can carry out the iteration:

We can see the iteration converges to 2.2622 radians, only after 3 iterations, with an initial approximation of 2. 2.2622 radians or 129.6 degrees is our answer for angle AOB.

If you have got this far and understood it well done, but don't be afraid to check out a few of the linked videos if you didn't. This was a very difficult problem: I found it in 'Stewart: 3rd Edition Calculus' (what I believe to be degree level). Please comment with any questions and queries,  and of course, stay to see the next blog!-naBla5040